kinematics for animation - inria · direct and inverse kinematics in animation ! fast numerical...
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KINEMATICS FOR ANIMATION Rémi Ronfard, Animation, M2R MOSIG
Direct and inverse kinematics in animation
¨ Fast numerical methods for inverse kinematics by Bill Baxter
¨ My Adventures with Inverse Kinematics by Chris Hecker
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
Fast numerical methods for inverse kinematics by Bill Baxter
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!
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
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.
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.
Talk Structure (linear with occasional branches) ¨ ... ¨ attempted solution ¨ problem statement and discussion ¨ related theory ¨ attempted solution ¨ problem statement and discussion ¨ related theory ¨ ...
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
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
θ2
θ1
p = (x, y) q = (θ1, θ2)
g
Theory: How to move a guy with the mouse? (cont.) ¨ degrees of freedom (DOF)
n incredibly important concept
¨ “ways in which system can change” n configuration space
¨ find DOF of system to understand it ¨ control & constrain DOF to control it ¨ examples
n physical system, polynomial, etc.
¨ fundamental task of IK is to control degrees of freedom to attain goal
p
θ2
θ1
p = (x, y) q = (θ1, θ2)
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
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
p
θ2
θ1
θ3
p = (x, y) q = (θ1 , θ2 , θ3)
g
Theory: Analytical IK can’t handle human. ¨ 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
p
θ2
θ1
θ3
p = (x, y) q = (θ1 , θ2 , θ3)
Solution: Analytical IK can’t handle human. ¨ use Cyclic Coordinate Descent
(CCD) to solve arms ¤ iterative, recursive position-space
algorithm ¤ fast, easy, robust ¤ start from last position
n not path independent, but faster ¤ I draw every outside iteration
n looks like animation, but it’s not
p
θ2
θ1
θ3
p = (x, y) q = (θ1 , θ2 , θ3)
g
Problem: CCD only handles serial chains.
¨ multiple goals are necessary for human ¤ stack up vectors in
p=f(q) ¤ coupled at branches
p1
θ2
θ1
θ3
p = (x1, y1, x2, y2) q = (θ1 , θ2 , θ3 , θ4 , θ5)
θ4
θ5
p2
g1 g2
Theory: CCD only handles serial chains.
¨ coupling between goals ¨ either solvable alone,
neither solvable together ¨ how to distribute desired-angle
error?
p1
θ2
θ1
θ3
p = (x1, y1, x2, y2) q = (θ1 , θ2 , θ3 , θ4 , θ5)
θ4
θ5
p2
g1 g2
Solution: CCD only handles serial chains.
¨ simply average desired angles at branch parents to attain multiple goals and distribute error ¤ θ1 in example, imagine p1, p2
pulled in opposite directions
p1
θ2
θ1
θ3
p = (x1, y1, x2, y2) q = (θ1 , θ2 , θ3 , θ4 , θ5)
θ4
θ5
p2
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
θ
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 θ
Solution: Drift on multiple goals. ¨ attempt to “Gram-Schmidt” off any movement that
would result in error
a
b c
Problem: Gram-Schmidt removes the wrong direction sometimes.
¨ could have tried real Jacobian force mapping Gram-Schmidt
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
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
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!
Problem: Joint limits ¨ elbow can’t bend backwards ¨ should lend believability and intuition
Theory: Joint limits ¨ inequality constraints: θ ≥ θmin
¤ abandon all (most) hope of analytical solution ¤ these inequalities are axis aligned planes in
configuration space
θmax
θmin
θ
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
Problem: Child-child limits ¨ parent-child not enough with moving root ¨ pelvis example:
¤ left foot root ¤ right foot root ¤ chest root
Problem: Child-child limits ¨ parent-child not enough with moving root ¨ pelvis example:
¤ left foot root ¤ right foot root ¤ chest root
Problem: Child-child limits ¨ parent-child not enough with moving root ¨ pelvis example:
¤ left foot root ¤ right foot root ¤ chest root
?
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
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 (θ1 - d1)2 + (θ2 - d2)2 w.r.t. θmin12 ≤ θ1 - θ2 ≤ θmax12
¤ lots of literature on NLSLI
¤ I convert to LCP and use Lemke’s algorithm
¤ works great!
Accomplishments ¨ We extended CCD to handle:
¤ multiple end effector goals ¤ dynamically changing root node ¤ arbitrary joint limits
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?
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!
Body Knowledge System (cont.) ¨ IK just a rough initial guess for body knowledge system;
should be: ¤ physically plausible
n 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
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
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
n although could make weight very big... n similar to a stiff system
¨ Wrong: “CCD must have fixed root” ¤ now that I understand better,
this isn’t actualy true n the root’s position is just more DOF n CCD probably wouldn’t deal with this very well?
What I Didn’t Try (cont.) ¨ pseudo-inverse methods
n give explicit control over redundant DOF
¨ data-based methods (model-based) n functions as lookup tables
¨ totally ad-hoc methods