FORWARD KINEMATICS INTRODUCTION TO ROBOTICS: DISCUSSION 2 ROBERT PETER MATTHEW 20150908
Aug 19, 2020
FORWARD KINEMATICS
INTRODUCTION TO ROBOTICS DISCUSSION 2ROBERT PETER MATTHEW
20150908
rpmatthew ltatgt berkeleyedu
ADMINISTRIVADISCUSSION 2 FORWARD KINEMATICS
bull Course text is available on bCoursesbull Course grading policybull Piazza questions
bull Discussion sections Tues 1000-1100 Wed 1100-1200bull Office hours Mon Thurs 1100-1200bull Homeworks Released Due and Graded by Thursdaybull DSP Students Letters of Accommodation required asap
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EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872
119877119877119883119883 1205791205791
119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
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EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
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AXIS ANGLEDISCUSSION 1 ROTATIONS
119877119877 = 1201281201283 +120596120596120596120596 2
119904119904119904119904119904119904 120579120579 +1205961205962
120596120596 22 1 minus 119888119888119888119888119904119904 120579120579
Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula
(a derivation of this equation will be given next week)
120596120596 =120596120596112059612059621205961205963
=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
120596120596 2 = 119894119894=1
119899119899119909119909119894119894 2
1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992
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EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
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Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
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Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
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TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
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HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
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EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
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EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
ADMINISTRIVADISCUSSION 2 FORWARD KINEMATICS
bull Course text is available on bCoursesbull Course grading policybull Piazza questions
bull Discussion sections Tues 1000-1100 Wed 1100-1200bull Office hours Mon Thurs 1100-1200bull Homeworks Released Due and Graded by Thursdaybull DSP Students Letters of Accommodation required asap
2104
rpmatthew ltatgt berkeleyedu
EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872
119877119877119883119883 1205791205791
119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
3104
rpmatthew ltatgt berkeleyedu
EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
4104
rpmatthew ltatgt berkeleyedu
AXIS ANGLEDISCUSSION 1 ROTATIONS
119877119877 = 1201281201283 +120596120596120596120596 2
119904119904119904119904119904119904 120579120579 +1205961205962
120596120596 22 1 minus 119888119888119888119888119904119904 120579120579
Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula
(a derivation of this equation will be given next week)
120596120596 =120596120596112059612059621205961205963
=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
120596120596 2 = 119894119894=1
119899119899119909119909119894119894 2
1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992
5104
rpmatthew ltatgt berkeleyedu
EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
6104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872
119877119877119883119883 1205791205791
119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
3104
rpmatthew ltatgt berkeleyedu
EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
4104
rpmatthew ltatgt berkeleyedu
AXIS ANGLEDISCUSSION 1 ROTATIONS
119877119877 = 1201281201283 +120596120596120596120596 2
119904119904119904119904119904119904 120579120579 +1205961205962
120596120596 22 1 minus 119888119888119888119888119904119904 120579120579
Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula
(a derivation of this equation will be given next week)
120596120596 =120596120596112059612059621205961205963
=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
120596120596 2 = 119894119894=1
119899119899119909119909119894119894 2
1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992
5104
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EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
6104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS
Find the net rotation described by (21205871205873
minus1205871205872
minus1205871205873
) in Z-Y-X form
4104
rpmatthew ltatgt berkeleyedu
AXIS ANGLEDISCUSSION 1 ROTATIONS
119877119877 = 1201281201283 +120596120596120596120596 2
119904119904119904119904119904119904 120579120579 +1205961205962
120596120596 22 1 minus 119888119888119888119888119904119904 120579120579
Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula
(a derivation of this equation will be given next week)
120596120596 =120596120596112059612059621205961205963
=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
120596120596 2 = 119894119894=1
119899119899119909119909119894119894 2
1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992
5104
rpmatthew ltatgt berkeleyedu
EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
6104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
AXIS ANGLEDISCUSSION 1 ROTATIONS
119877119877 = 1201281201283 +120596120596120596120596 2
119904119904119904119904119904119904 120579120579 +1205961205962
120596120596 22 1 minus 119888119888119888119888119904119904 120579120579
Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula
(a derivation of this equation will be given next week)
120596120596 =120596120596112059612059621205961205963
=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
120596120596 2 = 119894119894=1
119899119899119909119909119894119894 2
1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992
5104
rpmatthew ltatgt berkeleyedu
EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
6104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
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EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =100
=0 0 00 0 minus10 1 0
Consider a rotation about the x axis
6104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =110
=0 0 10 0 minus1minus1 1 0
7104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS
120596120596 =111
=0 minus1 11 0 minus1minus1 1 0
8104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS
JOINT POSITION TORQUE
END EFFECTOR POSITION TORQUE
Forward Kinematicsndash Given joint positions find end
effector coordinatesInverse Kinematics
ndash Given end effector coordinates find required joint positions
Forward Dynamicsndash Given joint torques find end
effector forcestorquesInverse Dynamics
ndash Given a desired end effector forcetorque find required joint torques
9104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
10104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
11104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Translation of Origin Relative Rotation
12104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
13104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Homogeneous Coordinates
Note all configurations are RELATIVE
14104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Points Vectors
15104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies
that make up the manipulator
Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints
and the joint angles
16104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
17104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
18104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
19104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
20104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120788120788120782120782120782120782
21104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
22104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
23104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
24104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
25104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119912119912 =120785120785120782120782120782120782
26104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
27104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
28104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
29104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
30104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
31104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
32104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
33104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
34104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
35104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119936119936
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119912119912 =120785120785120782120782120782120782
36104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 11199541199541198601198601
1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862
120782120782 11199541199541198621198621
The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator
Note all configurations are RELATIVE
37104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
38104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
39104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
40104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
41104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
42104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
43104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120782120782
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119937119937
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
44104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
45104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
46104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
47104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
48104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
49104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
50104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119929119929119912119912119912119912 = 119929119929119937119937
119953119953119912119912119912119912 =120782120782120782120782120785120785
119954119954119914119914 =120785120785120782120782120782120782
119929119929119912119912119914119914 = 119929119929119936119936
119953119953119912119912119914119914 =120785120785120782120782120782120782
1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860
120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1
1199541199541198621198621
51104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
coordinate transforms
52104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Rigid body motion as
solutions to differential equationscoordinate transforms
53104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS
Twists and screws describe the same concept
Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis
-Chasles Theorem
These rotations and translations can be found using twists andor screw theory
54104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS
Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo
Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo
55104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
56104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
57104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS
Given the differential equation 119902 = 120596120596 times 119902119902 119905119905
We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0
Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579
where
120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0
58104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Using similar analysis we can derive
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
where
120585120585 = 120596120596 119907119907120782120782 0
bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE
coordinates
59104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
bull lsquoWedgersquo 120585120585^ = 119907119907120596120596
^= 120596120596 119907119907
120782120782 0 = 120585120585
bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0
or= 119907119907
120596120596 = 120585120585
bull Note the ^ symbol refers to two different operations
60104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0
Compute the matrix exponent 119890119890120585120585120579120579
bull Analytically
119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1
bull Computationally expm
61104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907
120596120596
Special cases
bull Pure Rotation120585120585 = minus120596120596 times 119902119902
120596120596
bull Pure Translation120585120585 = 119907119907
0
62104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
63104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
64104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Rotational velocity
120596120596 =001
65104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Linear velocity119907119907 = minus120596120596 times 119902119902
119902119902 is ANY point on the rotational axis
119907119907 = minus001
times011989711989710
=119897119897100
66104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Twist
119907119907 =119897119897100
120596120596 =001
120585120585 = 119907119907120596120596 =
119897119897100001
67104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Exponential
119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
68104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579
120782120782 1
bull Initial Configuration119892119892 0 = 1198771198770 1198791198790
120782120782 1
119892119892 0 = 120128120128011989711989710
120782120782 1
69104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
70104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS
Example Single rotation about the 119885119885 axis
bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0
119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0
0 0 1
1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579
0120782120782 1
120128120128011989711989710
120782120782 1
119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0
0 0 1
0l10
120782120782 1
71104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
72104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
73104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
74104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
75104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283600
120782120782 1
76104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
77104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
78104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
79104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
80104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283300
120782120782 1
81104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
82104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
83104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
84104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
85104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
000001
119892119892(0) = 1201281201283303
120782120782 1
86104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
87104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
88104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
89104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
90104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0
1205851205851 = minus120596120596 times 119902119902120596120596 =
minus300010
119892119892(0) = 1201281201283303
120782120782 1
91104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
The effect of multiple RBMs can be found via the composition of multiple matrix exponents
For any reference frame at a zero configuration we can write
Note all configurations are in ABSOLUTE coordinates
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
92104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
93104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
94104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
95104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
96104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
97104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283600
120782120782 1
1205851205851 =
000001
1205851205852 =
0minus30001
98104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
99104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
100104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
101104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
102104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
103104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
rpmatthew ltatgt berkeleyedu
FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS
119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0
119892119892(0) = 1201281201283603
120782120782 1
1205851205851 =
000001
1205851205852 =
minus303010
104104
Kinematics of Robot Manipulator