1. Inverse Kinematics

2. The Problem Forward Kinematics: Inverse Kinematics:
Joint angles  End effector
Inverse Kinematics:
End effector  Joint angles
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Robotic applications: cutting/welding

3. Animation Applications (more)
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4. IK Solutions
Analytical solutions are desirable because of their speed and exactness of solution.
For complex kinematics problems, analytical solutions may not be possible
Use iterative methods
Optimization methods (e.g., minimize the distance between end effector and goal point)
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5. Case Study: Two-Link Armq1 q2 L1 L2 x y
Analytic (closed form) solution
Case:3-link arm
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6. 2-link Arm (analytic solution2)q1 q4 q3 q2 L2 L1 y
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7. Unreachable Targets No joint angles can satisfy the target
If this value is
> 1 or < -1,
no solution exists
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8. Statement of the IK Problem*
Want
Error at ith iteration i
q: joint angles
Seek correction Dq to fix the error
g =
Omit high order terms…
Jacobian matrix
xRn, qRm, Jnm
Note: here Dq are in radians!
m: joint space dimension
n: space where end effector is in
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9. Error at End Effector
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10. Jacobian Inverse L2 q2 y L1 q1 x Simple case: Cramer’s rule suffices
General case: pseudo inverse
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11. Jacobian inverse is underdetermined
3-link 2D arm
Jacobian inverse is underdetermined q1 q2 L1 L2 x y L3 q3
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12. Pseudo Inverse and Ax=b
A panacea for Ax=b
Full rank: A-1 exist; A+ is the same as A-1
Underdetermined case: many solutions; will find the one with the smallest magnitude |x|
Overdetermined case: find the solution that minimize the error r=||Ax–b||, the least square solution
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13. About Pseudo Inverse A+
A: row space → column space
A+: column space → row space
C(AT)
N(A) R3
C(A)
N(AT) R2 A A+
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14. Using GSL (Gnu Scientific Library)
Full rank
Full row rank
Full column rank
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15. Forward Kinematics L2 q2 y Using Coordinate Transformation L1 q1 x
End effector at the local origin
Rotate(z,q1)
Translate(L1,0)
Rotate(z,q2)
Translate(L2,0)
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16. q1 q2 L1 L2 x y
[Computing Jacobian]
This is wrong!!
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17. Computing Jacobian reference wi: unit vector of rotation axis
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18. Example q1 q2 L1 L2 x y r1 q1 q2 L1 L2 x y r2
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19. About Joint Space Redundancy
If m > n
Redundant
manipulator
Joint Space
End Effector m n
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20. Example q3 L3 L2 q2 L1 q1
qmax
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21. IK Solution (CCD)
CCD: cyclic coordinate descent; initially from C. Welman (1993)
From the most distal joint, solve a series of one-dimensional minimization analytically to satisfy the goal (one joint at a time)
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22. CCD-1 (Cyclic Coordinate Descent)
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23. CCD-2
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24. CCD-3
Implementing joint limits in CCD is straight forward: simply clamp the joint angle
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25. CCD (2-link arm) Pcurrent L2 q2 Pdest P1 f L1 q1 Pc L2 q2 Pd L1 f P0
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26. IK Solutions (DLS) For 3-link 2D arm: m = 2, n = 3
DLS (damped least square)
Position of k end effectors
target of end effectors
error of end effectors
Joint angles
Jacobian matrix (mn)
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27. DLS (Damped Least Square)
Joint angle to correct error
Minimize damped least square
Rewrite error as
Normal equation for
least square problem
Simplifying
and get
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28. [Details] A x b
Summary
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29. Affect convergence rate
DLS (cont)
(continued)
Shown next page
Therefore
Instead of computing inverse, solve
For 3-link 2D arm: m = 2, n = 3
How to choose l?
2x2
Affect convergence rate
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30. Damping Effects
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31. Properties of Pseudo Inverse A+ (From Wikipedia)
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32. Challenging IK Cases: Multiple Targets
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33. Types of IK
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34. Types of IK (cont)
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35. Inverse Kinetics (Boulic96)
The constraint on the position of the center of mass is treated as any other task, and solved at the differential level with a special-purpose Jacobian matrix that relates differential changes of the joint coordinates to differential changes of the Cartesian coordinates of the center of mass.
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36. Kinematic chain
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39. Support Materials

40. Jacobian inverse is underdetermined
3-link 2D arm q1 q2 L1 L2 x y L3 q3
Jacobian inverse is underdetermined
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41. Projection onto a Space
Recall least square problem
Ax=b may not have solution (if b is not in C(A))
Solve instead where p is the projection of b onto C(A) b
e=b–p a2 p a1
Summary
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42. Projection (cont) P: projection matrix Known as the “normal equation”
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43. Jacobian transpose Constraint dynamics
Extra material
Jacobian transpose
Constraint dynamics

44. IK General (Jacobian Transpose)q1 q2 L1 L2 x y pc pd
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45. Jacobian Transpose q1 q2 L1 L2 x y
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50. Differences
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51. [Constraint Dynamics]q1 q2 L1 L2 P1 P2
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53. All joints rotate w.r.t. local zq2 g L1 F2 q1 F1 FB
All joints rotate w.r.t. local z
CCD step 1: Find 2g; project og to XY plane (as og’); rotate OE to og’
CCD step 2: Find 1g; project og to XY plane (as og’); rotate OE to og’
Fall 2012