1. My Adventure with Inverse Kinematics
Chris Hecker
definition six, inc.

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 Workshop Friday, 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

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. 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 g
p = (x, y)
q = (q1, q2) q2 q1

9. Theory: How to move a guy with the mouse? (cont.)p q2 q1
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. Problem: Analytical IK can’t handle human.
human has lots of DOF
even 2D arm with shoulder has more DOF than needed to reach goal
solving p = f(q) requires certain characteristics q3 g 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. Solution: Analytical IK can’t handle human.p
use Cyclic Coordinate Descent (CCD) to solve arms
iterative, recursive position-space algorithm
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 g q2 q1
p = (x, y)
q = (q1 , q2 , q3)

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

15. Theory: CCD only handles serial chains.p2 p1
coupling between goals
either solvable alone, neither solvable together
how to distribute desired-angle error? g2 q3 g1 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 at branch parents to attain multiple goals and distribute error
q1 in example, imagine p1, p2 pulled 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 root in 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 with respect to the root

23. Solution: Human is not rooted tree.
just type it in
invert tree to new root
Caml is pretty good at this sort 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: q ³ qmin
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-axis aligned planes in C-space
simultaneous solution necessary at branches with child-child limits

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: google & citeseer GDC talks
Paolo Baerlocher
Chris Welman
google & citeseer
GDC talks
David Wu
Ken Perlin
Jeff Lander’s articles & code
Experimental Gameplay Workshop