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Inverse Kinematics

Inverse Kinematics. The Problem. Forward Kinematics: Joint angles  End effector. Inverse Kinematics: End effector  Joint angles. Robotic applications: cutting/welding. Animation Applications ( more ). IK Solutions.

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Inverse Kinematics

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  1. Inverse Kinematics

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

  3. Animation Applications (more)

  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)

  5. L2 q2 y L1 q1 x Case Study: Two-Link Arm Analytic (closed form) solution Case:3-link arm

  6. L2 q2 L1 q4 q3 q1 2-link Arm (analytic solution2) y x

  7. Unreachable Targets • No joint angles can satisfy the target If this value is > 1 or < -1, no solution exists

  8. * i i g = Statement of the IK Problem Want Error at ith iteration q: joint angles Seek correction Dq to fix the error 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

  9. Error at End Effector

  10. L2 q2 y L1 q1 x Jacobian Inverse Simple case: Cramer’s rule suffices General case: pseudo inverse

  11. L3 q3 L2 q2 y L1 q1 x 3-link 2D arm Jacobian inverse is underdetermined

  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

  13. About Pseudo Inverse A+ A: row space → column space A+: column space → row space C(AT) C(A) A+ A N(A) N(AT) R3 R2

  14. Using GSL (Gnu Scientific Library) Full rank Full row rank Full column rank

  15. L2 q2 y L1 q1 x Forward Kinematics Using Coordinate Transformation End effector at the local origin Rotate(z,q1) Translate(L1,0) Rotate(z,q2) Translate(L2,0)

  16. Computing Jacobian reference wi: unit vector of rotation axis

  17. L2 L2 q2 q2 y y L1 L1 q1 q1 x x Example r1 r2

  18. About Joint Space Redundancy Joint Space End Effector m n If m > n Redundant manipulator

  19. Example q3 L3 L2 q2 L1 q1 qmax

  20. 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)

  21. CCD-1 (Cyclic Coordinate Descent)

  22. CCD-2

  23. CCD-3 Implementing joint limits in CCD is straight forward: simply clamp the joint angle

  24. Pcurrent L2 q2 Pdest f P1 L1 q1 Pc L2 q2 Pd L1 f P0 q1 CCD (2-link arm)

  25. 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)

  26. 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

  27. Summary [Details] A b x

  28. DLS (cont) (continued) Shown next page Therefore Instead of computing inverse, solve For 3-link 2D arm: m = 2, n = 3 2x2 How to choose l? Affect convergence rate

  29. Damping Effects

  30. Properties of Pseudo Inverse A+ (From Wikipedia)

  31. Challenging IK Cases:Multiple Targets

  32. Types of IK

  33. Types of IK (cont)

  34. 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.

  35. Kinematic chain

  36. Support Materials

  37. L3 q3 L2 q2 y L1 q1 x 3-link 2D arm Jacobian inverse is underdetermined

  38. 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) Summary Projection onto a Space Recall least square problem b e=b–p a2 p a1

  39. Projection (cont) Known as the “normal equation” P: projection matrix

  40. Extra material Jacobian transpose Constraint dynamics

  41. pc pd L2 q2 y L1 q1 x IK General (Jacobian Transpose)

  42. L2 q2 y L1 q1 x Jacobian Transpose

  43. Differences

  44. P2 L2 q2 P1 L1 q1 [Constraint Dynamics]

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