역운동학의 구현과 응용 Implementation of Inverse Kinematics and Application

1. 역운동학의 구현과 응용Implementation of Inverse Kinematics and Application 서울대학교 전기공학부 휴먼애니메이션연구단 최광진 kjchoi@graphics.snu.ac.kr

2. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

3. ? End Effector Base What is Inverse Kinematics? • Forward Kinematics

4. End Effector Base What is Inverse Kinematics? • Inverse Kinematics

5. ? End Effector Base What does looks like?

6. Infinite number of solutions ! Solution to • Our example Number of equation : 2 Unknown variables : 3

7. Our example • System DOF = 3 • End Effector DOF = 2 Redundancy • System DOF > End Effector DOF

8. Redundancy • A redundant system has infinite number of solutions • Human skeleton has 70 DOF • Ultra-super redundant • How to solve highly redundant system?

9. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

10. What is NLP? • Non Linear Programming • Method to optimize a nonlinear function • Example • Objective function • Constraint • Iterative algorithm

11. NLP-based Method • Inverse Kinematics problem as non-linear optimization problem • Minimization of Goal Potential Function • Zhao and Badler, 1994, ACMTOG

12. Goal Potential Function • “Distance” from the end effector to the goal • Function of joint angles : G(q)

13. Goal distance End Effector Base Our Example

14. Goal Potential Function Position Goal Orientation Goal Position/Orientation Goal

15. Our Example • Goal Potential Function

16. Nonlinear Optimization • Recasted Constrained Optimization Problem

17. Nonlinear Optimization • Available NLP Packages • LANCELOT • DONLP2 • MATLAB • Etc...

18. Quiz • Will G(q) be always zero? • No : Unreachable Workspace • Will the solution be always found? • No : Local Minima/Singular Configuration • Will the solution be always unique? • No : Redundancy

19. Handling Singularity

20. Singular Configuration • Causes infinite joint velocity • Occurs when any cannot achieve given • Example • Fully stretched limbs

21. Remedy • For Jacobian-based method • Damped pseudo inverse • Dexterity measure • Clamping • For nonlinear optimization method • Try again with another initial value

22. Parametric Singularity • Gimbal Lock in Euler angle representation • When a degree of freedom is lost, the gimbals is said to “lock” • Consider a y-roll of 90 degrees which aligns the x and z axis • Remedy : Quaternion

23. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

24. Differential Kinematics • : Jacobian Matrix • Linearly relates end-effector change to joint angle change

25. Differential Kinematics • Our Example

26. Differential Kinematics • Is J always invertible? No! • Remedy : Pseudo Inverse

27. Null space • The null space of J is the set of vectors which have no influence on the constraints • The pseudoinverse provides an operator which projects any vector to the null space of J

28. Utility of Null Space • The null space can be used to reach secondary goals • Or to find comfortable positions

29. Null Space

30. Calculating Pseudo Inverse • Gaussian Elimination • Singular Value Decomposition

31. How Can We Get From ?

32. Integrating • Problems • Initial tracking error • Numerical drift • Remedy • Error feedback

33. Discretization

34. Open-loop vs. Closed-loop scheme

35. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

36. Redundancy Is Evil • Multiple choices for one goal • What happens if we pick any of them?

37. Redundancy Is Good • We can exploit redundancy • Additional objective • Minimal Change • Similarity to Given Example • Naturalness

38. Minimal Change • Pseudo Inverse Solution • Minimal velocity norm solution • Minimal acceleration norm solution • Penalty Term in Goal Potential Function

39. Similarity to Given Example • Adding homogeneous solution term to the pseudo inverse solution • New Goal Potential Function

40. Naturalness • Based on observation of natural human posture • Neurophysiological experiments • Example • Pointing task with pen stylus • Linear mapping between shoulder joint and pen stylus in spherical coordinate system • Analytic solution for human arm • endgame.mov

41. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

42. Multiple Goals

43. Conflict Between Goals ee 2 ee 1 base

44. Conflict Between Goals Goal 1 ee 2 ee 1 base

45. Conflict Between Goals Goal 2 ee 2 ee 1 base

46. Conflict Between Goals Goal 2 Goal 1 ee 2 ee 1 base

47. Conflict Between Goals Goal 2 Goal 1 ee 2 ee 1 base

48. Handling Multiple Goals • Weighted sum of each goal potential function • Jacobian-based method • Same formulation • No weighting

49. Content • What is Inverse Kinematics? • Redundancy • Basic Method • NLP-based method • Jacobian-based method • Issues • Resolving Redundancy • Multiple Goals • Application : Motion Retargetting

50. Summary • Inverse Kinematics • Solver • NLP-based Solver • Jacobian-based Solver • Issues • Resolving Redundancy • Multiple Goals • Motion Retargetting