Mechatronics 1 Week 3 & 4
Dec 20, 2015
Learning Outcomes
• By the end of week 3-4 session, students will understand kinematics of industrial robots.
Course Outline
• Forward Kinematics
• Homogeneous Transformation Matrix.
• Denavit Hartenberg (D-H) parameters.
• D-H Transformation Matrix
• Interpretation and application of homogeneous tranformation matrix
• Inverse kinematics.
Homogenous Transformation Matrix
• It provides information on position and orientation in a 4x4 matrix.
1000333231
232221
131211
RRRp
RRRp
RRRp
Tz
y
x
Link i-1 Link i
Joint i-1 axisJoint i axis Joint i+1 axis
a i-1
ai
i
Zi'
Xi'
Yi'
iZi
Yi
Xi
Yi-1Zi-1
Xi-1
di
Ref. Lee, Fu, Gonzalez, 1987
Coordinate Frame Placement(Denavit Hartenberg Concept)
ai : length of link i di : offset distance at joint i
i
i
: joint angle of link i
: twist angle of link i
Rotation Matrix with D-H Parameters
ii
iiiii
iiiii
ii
cossin0
cossincoscossin
sinsinsincoscos1R
ii
iiiii
iiiii
ii
cossin0
cossincoscossin
sinsinsincoscos
1R
Homogenous Transformation Matrix with D-H Parameters
1000
cossin0
sincossincoscossin
cossinsinsincoscos
1
iii
iiiiiii
iiiiiii
ii
d
a
a
A
1000
cossin0
sincossincoscossin
cossinsinsincoscos
1iii
iiiiiii
iiiiiii
ii
d
a
a
A
Kinematics & Inverse Kinematics
Direct Kinematics
Inverse Kinematics
Joint Coordinates
Joint Coordinates
Position & Orientation of End Effector
Link Parameters
Link Parameters
Inverse Kinematics
• Given a desired position (P) & orientation (R) of the end-effector.
• Find the joint variables which can bring the robot to the desired posture.
),,( 21 nqqqq
x
y
z
Inverse Kinematics
• More difficult
• Solution not unique• Redundant robot• Elbow-up/elbow-down
configuration A
B
1
2
A shorter path is generally desirable
Kinematics(Mitsubishi RV-M1)
L2 L3
Assumed position of endeffector
Base
J2 J3
J1
z0
y0 z1
x0 x1
y1
z2
y2
x2
z3
x3
y3
Kinematics (RV-M1)
Limitation on
Movement of Robot RV-M1
Total Span Min Max
Waist 300o -150o 150o
Shoulder 130o -100o 30o
Elbow 110o 0o 110o
i ai αi di θi
1 0 -90º 0 θ1
2 a2 = L2 0 0 θ2
3 a3 = L3 0 0 θ3