Animation CS 551 / 651

1. AnimationCS 551 / 651 Kinematics Lecture 09 Sarcos Humanoid

2. Kinematics • The study of object movements irrespective of their speed or style of movement

3. Degrees of Freedom(DOFs) • The variables that affect an object’s orientation • How many degrees of freedom when flying? • So the kinematics of this airplane permit movement anywhere in three dimensions • Six • x, y, and z positions • roll, pitch, and yaw

4. Degrees of Freedom • How about this robot arm? • Six again • 2-base, 1-shoulder, 1-elbow, 2-wrist

5. Configuration Space • The set of all possible positions (defined by kinematics) an object can attain

6. Work Space vs. Configuration Space • Work space • The space in which the object exists • Dimensionality • R3 for most things, R2 for planar arms • Configuration space • The space that defines the possible object configurations • Degrees of Freedom • The number of parameters that necessary and sufficient to define position in configuration

7. More examples • A point on a plane • A point in space • A point moving on a line in space

8. A matter of control • If your animation adds energy at a particular DOF, that is a controlled DOF Low DOF, high control High DOF, no control

9. Hierarchical Kinematic Modeling • A family of parent-child spatial relationships are functionally defined • Moon/Earth/Sun movements • Articulations of a humanoid • Limb connectivity is built into model (joints) and animation is easier

10. Robot Parts/Terms • Links • End effector • Frame • Revolute Joint • Prismatic Joint

11. More Complex Joints • 3 DOF joints • Gimbal • Spherical (doesn’t possess singularity) • 2 DOF joints • Universal

12. Hierarchy Representation • Model bodies (links) as nodes of a tree • All body frames are local (relative to parent) • Transformations affecting root affect all children • Transformations affecting any node affect all its children ROOT

13. Forward vs. Inverse Kinematics • Forward Kinematics • Compute configuration (pose) given individual DOF values • Good for simulation • Inverse Kinematics • Compute individual DOF values that result in specified end effector position • Good for control

14. Forward Kinematics • Traverse kinematic tree and propagate transformations downward • Use stack • Compose parent transformation with child’s • Pop stack when leaf is reached

15. Denavit-Hartenberg (DH) Notation • A kinematic representation (convention) inherited from robotics

16. Z-axis aligned with joint

17. X-axis aligned with outgoing limb

18. Y-axis is orthogonal

19. Joints are numbered to represent hierarchy Ui-1 is parent of Ui

20. Parameter ai-1 is outgoinglimb length of joint Ui-1

21. Joint angle, qi, is rotation of xi-1 about zi-1 relative to xi

22. Link twist, ai-1, is the rotation of ith z-axis about xi-1-axis relative to z-axis of i-1th frame

23. Link offset, di-1, specifies the distance along the zi-1-axis (rotated by ai-1) of the ith frame from the i-1th x-axis to the ith x-axis

24. Inverse Kinematics (IK) • Given end effector position, compute required joint angles • In simple case, analytic solution exists • Use trig, geometry, and algebra to solve

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

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

27. ? End Effector Base What does look like?

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

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

30. Analytic solution of 2-link inverse kinematics x2 y2 (x,y) O2 2 y0 y1 a2 2 x1  a1 O1  1 x0 O0

31. Failures of simple IK • Multiple Solutions

32. Failures of simple IK • Infinite solutions

33. Failures of simple IK • Solutions may not exist

34. Iterative IK Solutions • Frequently analytic solution is infeasible • Use Jacobian • Derivative of function output relative to each of its inputs • If y is function of three inputs and one output • Represent Jacobian, J(X), as a 1x3 matrix of partial derivatives

35. Jacobian • In another situation, end effector has 6 DOFs and robotic arm has 6 DOFs • f(x1, …, x6) = (x, y, z, r, p, y) • Therefore J(X) = 6x6 matrix

36. Jacobian • Relates velocities in parameter space to velocities of outputs • If we know Ycurrent and Ydesired, then we subtract to compute Ydot • Invert Jacobian and solve for Xdot

37. Turn to PDF slides • Slides from O’Brien and Forsyth • CS 294-3: Computer GraphicsStanfordFall 2001

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

39. 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

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

41. Calculating Pseudo Inverse • Singular Value Decomposition

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

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

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

45. Naturalness • Based on observation of natural human posture • Neurophysiological experiments

46. Conflict Between Goals ee 2 ee 1 base

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

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

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

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