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©2011 Stevens Institute of TechnologyP. 2/3 | 01/01/11
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Date: By:
ME 598: Introduction to Robotics
Lecture 3: Velocity Kinematics‐ The Jacobian
Stevens Institute of Technology Dr. Mishah U. Salman
Fall 2013
Slides adapted from Dr. David J. CappelleriSome slides courtesy of Jonathan Fiene, University of Pennsylvania
©2011 Stevens Institute of TechnologyP. 2/3 | 01/01/11
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Review: Forward Kinematics
Fall 2013 ME 598, Lecture 32
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Review: Denavit‐Hartenberg Convention
Fall 2013 ME 598, Lecture 33
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Review: Denavit‐Hartenberg Convention
Fall 2013 ME 598, Lecture 34
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Review: Denavit‐Hartenberg Transform
Fall 2013 ME 598, Lecture 35
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Review: Inverse Kinematics
Fall 2013 ME 598, Lecture 36
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Review: Algebraic Decomposition
Fall 2013 ME 598, Lecture 37
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Review: Geometric Analysis
Fall 2013 ME 598, Lecture 38
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Review: Kinematic DecouplingInverse kinematics =
inverse position kinematics + inverse orientation kinematics
Two sub problems:
• Find position of the intersection of the wrist axes (oc)
• Find orientation of the wrist
Fall 2013 ME 598, Lecture 39
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z1
z2 x2
z5, z6
z4 z3
x1
θ1
d2
θ2
x3
θ3
x4,
θ4
x5θ5
θ6
d6
x6
Forward Kinematics
Link ai αi di θi1 0 90 13” θ1
2 8” 0 d2 03 8” 90 0 04 0 -90 0 θ4
5 0 90 0 θ5
6 0 0 d6 θ6
Review: Example P3.10‐ PUMA 360 Manipulator
Fall 2013 ME 598, Lecture 310
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z0
x0
y0
dx
dy
dz
θ1
Given: d = [dx, dy, dz]T
Find: θ1, d2, and d3 as functions of dx, dy, dz
Inverse Kinematics
Review: Example P3.13‐ Cylindrical Manipulator
Fall 2013 ME 598, Lecture 311
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z0
x0
y0
ocz1
x1
y1
y2x2 z2
z4
z3, z5, z6x4 ,x5θ4
θ5
θ6
d6
y3x3
Link ai αi di θi1 0 0 1 θ1*2 0 -90 d2* 03 0 0 1+d3* 04 0 -90 0 θ4*5 0 90 0 θ5*6 0 0 d6 θ6*
Review: Example P3.15‐Cylindrical Manipulator + Spherical Wrist
Fall 2013 ME 598, Lecture 312
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Velocity Kinematics: The Jacobian
Fall 2013 ME 598, Lecture 313
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The Jacobian: Differential Motion
Fall 2013 ME 598, Lecture 314
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Example 1: Planar RRThe Jacobian: Position Jacobian
Fall 2013 ME 598, Lecture 315
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Example 1: Planar RRThe Jacobian: Position Jacobian
Fall 2013 ME 598, Lecture 316
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100010000
31
122111212
122111212
03 dd
sasacscacasc
Tj1
j2
j3
xo
yo
x1y1
z2
x2y2
a1
a2
x3y3
d3d1
Example 2: SCARA
3
2
1
dq
31
12211
12211
ddzsasaycacax
321
321
321
dzzzdyyydxxx
J p
10000
12212211
12212211
cacacasasasa
The Jacobian: Position Jacobian
Fall 2013 ME 598, Lecture 317
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[i.e. all columns of J are not linearly independent]
The Jacobian: Singularities
Fall 2013 ME 598, Lecture 318
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[http://en.wikipedia.org/wiki/Determinant]
The Jacobian: Identifying Singularities
Fall 2013 ME 598, Lecture 319
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Example 1: Planar RRThe Jacobian: Singularities
Fall 2013 ME 598, Lecture 320
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10000
12212211
12212211
cacacasasasa
J
Example 2: SCARAThe Jacobian: Singularities
Fall 2013 ME 598, Lecture 321
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Manipulator Singularities = Wrist Singularities + Arm Singularities
Compute Jacobian using oc instead of on
θ5 = 0 or π(z3 and z5 are collinear)
The Jacobian: Decoupling of Singularities
Fall 2013 ME 598, Lecture 322
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The Jacobian: Jacobian Transpose
Fall 2013 ME 598, Lecture 323
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Example 1: Planar RR
The Jacobian: Jacobian Transpose
Fall 2013 ME 598, Lecture 324
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Specifies the end‐effector velocity that will result when the joints move with velocity
The Jacobian relationship:
Inverse problem: Find the joint velocities that produce the desired end‐effector velocity
pJq 1)( qp
q
q
[Hard if have non‐square J pseudo‐inverse (pinv)]
The Jacobian: Inverse Jacobian
Fall 2013 ME 598, Lecture 325
©2011 Stevens Institute of TechnologyP. 2/3 | 01/01/11
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Announcements
Fall 2013 ME 598, Lecture 326
©2011 Stevens Institute of TechnologyP. 2/3 | 01/01/11
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• Various artistic robotic assignments throughout the course term– Labs– Midterm Project– Multiple specific events towards end of the semester
• Mixed media, sculpture, dance, etc.– Kinematics, path planning– Localization, image processing– Coordination
Announcements:Term Project Theme‐ Robotic Art Installation
Fall 2013 ME 598, Lecture 327
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• Homework # 3• Reading
– Spong Ch. 4 (today’s lecture)– Spong Ch. 5 (next lecture)
• Lab 1– Art Installation Logo– Kinematic robot arm, – 2 setups in EAS 001
• Teams must take turns• Many preliminary steps of lab may be done concurrently by different teams (do not require operating computer/robot)
Announcements
Fall 2013 ME 598, Lecture 328