My Adventure with Inverse Kinematics

1. My Adventure with Inverse Kinematics Chris Hecker definition six, inc. checker@d6.com

2. What is this lecture about? • the path I took while solving a problem with inverse kinematics • how I use math to model and [sometimes] solve problems • decisions I made, both good and bad • learn from my successes and failures • work in progress!

3. What isn’t this lecture about? • not an IK tutorial or introduction • not going to give you the One True Way to do IK • has nothing to do with graphics • not about my rock climbing game • Experimental Gameplay WorkshopFriday, 4pm - 7pm

4. Takeaway • You will get insight into how IK on a whole human body is different from IK on a single arm. • You will see how IK is a building block, but not a complete solution to character animation.

5. Prerequisites • must be very comfortable with math • doesn’t mean you know tons of math • does mean you follow quickly and aren’t afraid of it • probably best if you’ve implemented a simple IK system • foot placement, arm grabbing door, etc.

6. Talk Structure(linear with occasional branches) • ... • attempted solution • problem statement and discussion • related theory • attempted solution • problem statement and discussion • related theory • ...

7. Problem: How to move a guy with the mouse? • move like a [rock climbing] human • interactive • direct body/limb control • did not want to use physics

8. g Theory: How to move a guy with the mouse? • inverse kinematics vs. forward kinematics • FK: p=f(q) • IK: q = f-1(p) • p is Euclidean, angles • q is configuration space p p = (x, y) q = (q1, q2) q2 q1

9. p q2 q1 Theory: How to move a guy with the mouse?(cont.) • degrees of freedom (DOF) • incredibly important concept • “ways in which system can change” • configuration space • find DOF of system to understand it • control & constrain DOF to control it • examples • physical system, polynomial, etc. • fundamental task of IK is to control degrees of freedom to attain goal p = (x, y) q = (q1, q2)

10. Solution: How to move a guy with the mouse? • analytical IK • invert p = f(q) to get q from p • relatively easy for the 2D arm • inverse trig, some vector algebra • gets complicated as n increases

11. g Problem: Analytical IK can’t handle human. p • human has lots of DOF • even 2D arm with shoulderhas more DOF than neededto reach goal • solving p = f(q) requirescertain characteristics q3 q2 q1 p = (x, y) q = (q1 , q2 , q3)

12. Theory: Analytical IK can’t handle human. p • p = f(q) is a set of nonlinear equations • generally no closed forms • must iterate numerically • must be square, nDOF = ngoal • over-, under-constrained system • “redundant manipulator” • ex. human reaching for point and/or orientation • vast literature on solving f(q) = p q3 q2 q1 p = (x, y) q = (q1 , q2 , q3)

13. g Solution: Analytical IK can’t handle human. p • use Cyclic Coordinate Descent(CCD) to solve arms • iterative, recursive position-spacealgorithm • fast, easy, robust • start from last position • not path independent, but faster • I draw every outside iteration • looks like animation, but it’s not q3 q2 q1 p = (x, y) q = (q1 , q2 , q3)

14. g1 g2 Problem: CCD only handles serial chains. p2 p1 • multiple goals are necessaryfor human • stack up vectors inp=f(q) • coupled at branches q3 q5 q4 q2 q1 p = (x1, y1, x2, y2) q = (q1 , q2 , q3 , q4 , q5)

15. g1 g2 Theory: CCD only handles serial chains. p2 p1 • coupling between goals • either solvable alone,neither solvable together • how to distribute desired-angleerror? q3 q5 q4 q2 q1 p = (x1, y1, x2, y2) q = (q1 , q2 , q3 , q4 , q5)

16. Solution: CCD only handles serial chains. p2 p1 • simply average desired angles atbranch parents to attainmultiple goals and distributeerror • q1 in example, imagine p1, p2pulled in opposite directions q3 q5 q4 q2 q1 p = (x1, y1, x2, y2) q = (q1 , q2 , q3 , q4 , q5)

17. Problem: Drift on multiple goals. • average can’t satisfy all goals • or even prioritize them • I specifically want to drag climber hand with mouse and have other end effectors stay on holds q

18. Theory: Drift on multiple goals. • want a rigid constraint at end effectors • closed loops are hard, CCD doesn’t handle them, nor do common IK algorithms • must solve simultaneously • not hierarchical anymore • closed loops don’t have local behavior q

19. Solution: Drift on multiple goals. • attempt to “Gram-Schmidt” off any movement that would result in error b c a

20. Problem: Gram-Schmidt removes the wrong direction sometimes. • could have tried real Jacobian force mapping Gram-Schmidt

21. Problem: Human is not rooted tree. • Which end effector (hand, foot) is the root? • There is no persistent fixed rootin a rock climber • or walking person • CCD must start at a fixed root

22. Theory: Human is not rooted tree. • switching root is basically a datastructure problem • relative angles must change • joint limits (see below) • behavior will change slightly as well,since CCD isn’t symmetric withrespect to the root

23. Solution: Human is not rooted tree. • just type it in • invert tree to new root • Caml is pretty good at thissort of thing • see last year’s GDC talk • dragging torso is done by inverting to a torso node, then changing the root position • the hands and feet are all end-effectors • hack!

24. Problem: Joint limits • elbow can’t bend backwards • should lend believability and intuition

25. Theory: Joint limits • inequality constraints: q³qmin • abandon all (most) hope of analytical solution • these inequalities are axis aligned planes in configuration space qmax q qmin

26. Solution: Joint limits • CCD makes parent-child limits easy • do min/max clamp in CCD inner iteration • easy, and rest of system compensates • issue: CCD doesn’t “know” about the joint limits • often compensates, but doesn’t know to avoid them

27. Problem: Child-child limits • parent-child not enough with moving root • pelvis example: • left foot root • right foot root • chest root

28. Problem: Child-child limits • parent-child not enough with moving root • pelvis example: • left foot root • right foot root • chest root

29. Problem: Child-child limits • parent-child not enough with moving root • pelvis example: • left foot root • right foot root • chest root ?

30. Theory: Child-child limits • both angles are dependent on each other, coupled • parent is fixed in CCD • constraints are now non-axisaligned planes in C-space • simultaneous solution necessaryat branches with child-childlimits

31. Solution: Child-child limits • Good solution should: • satisfy constraints always • have zero error in goals if possible • have least squares error if not possible • Nonlinear Least Squares with Linear Inequalities: minimize (q1 - d1)2 + (q2 - d2)2 w.r.t. qmin12£ q1 - q2 £qmax12 • lots of literature on NLSLI • I convert to LCP and use Lemke’s algorithm • works great!

32. Accomplishments • We extended CCD to handle: • multiple end effector goals • dynamically changing root node • arbitrary joint limits

33. Problems • drift on multi-goals is still a problem • CCD doesn’t know about joint limits • sometimes non-physical movement • CCD isn’t doing physics, not conserving energy, or anything else for that matter • rooted hierarchies are inconvenient • how to control redundant DOF?

34. Body Knowledge System • system to manage movement and keep it natural • “situation” & “reaction” • ex. joint limits handled in body knowledge • huge rules system • allocates redundant DOF • this is the key to natural movement!

35. Body Knowledge System (cont.) • IK just a rough initial guess for body knowledge system; should be: • physically plausible • no added energy, etc. • easy to layer conflicting goals • able to cope with over-constrained goals • don’t know enough about this yet, but I think it will be more important than IK for good dynamic animation

36. What I Didn’t Talk About • development process stuff • developing on lower dimensional subproblems • using visualizations to help debug math • Mathematica and Matlab for development and debugging • running multiple techniques on top of each other • physics-based techniques I tried

37. What I Didn’t Try • I should have done a weighted average based on error for multi-goals, duh. • even weighted average wouldn’t be rigid • although could make weight very big... • similar to a stiff system • Wrong: “CCD must have fixed root” • now that I understand better,this isn’t actualy true • the root’s position is just more DOF • CCD probably wouldn’t deal with this very well?

38. What I Didn’t Try (cont.) • pseudo-inverse methods • give explicit control over redundant DOF • data-based methods (model-based) • functions as lookup tables • totally ad-hoc methods

39. References • two great theses: • Paolo Baerlocher • Chris Welman • google & citeseer • GDC talks • David Wu • Ken Perlin • Jeff Lander’s articles & code • Experimental Gameplay Workshop