1cs426-winter-2008 Notes  SIGGRAPH crunch time - my apologies :-)

2cs426-winter-2008 IK issues 2  Target impossible (or just barely possible) --- not enough degrees of freedom Try to get as close as possible to the target in some sense Also penalize bad joint angles, huge differences from initial pose  So if pose is only just possible, via some weird unrealistic angles, we don’t do it

3cs426-winter-2008 How do you do IK?  Simple skeletons: analytic solutions may exist Look up formulas from papers Not fun, not flexible  More generally: numerical solution of optimization problem Minimize difference between actual end- effector pose and the target Possibly include constraints  E.g. joint angle limits, collisions Possibly penalize awkward or jerky poses

4cs426-winter-2008 IK as optimization  Unconstrained optimization problem:  In our case: theta is a vector of all the joint angles f(.) is a measure of the distance between the pose of the end-effector and the target (plus optionally penalty term on bad poses)  Start from an initial guess, use algorithms to steadily make better guesses until close enough

5cs426-winter-2008 Line Search Approach  Say we have guess  k  Pick a “direction” d k to search along  Look for new guess as  k+1 =  k +  d k  Line search: figure out a scalar  (the “step size”) that will make f(  k+1 ) smaller than f(  k )  Assuming d k is scaled properly, simplest algorithm: Check  =1: if it’s an improvement stop Otherwise halve  and try again  If d k is “downhill” d k  f<0 then guaranteed to improve if  is small enough

6cs426-winter-2008 Picking the direction  Cyclic Coordinate Descent: in each direction, pick a different axis to move along (adjust just one angle at a time) Problem 1: scaling isn’t clear Problem 2: extremely inefficient, might not converge at all  Steepest Descent: use the direction that f(.) is decreasing most rapidly in: -  f Problem 1: scaling isn’t clear Problem 2: inefficient for difficult f

7cs426-winter-2008 More on Steepest Descent  Typically we have a nonlinear least-squares problem:  In this case, -  f=J T (x target -x(  )) where J=∂x/∂  is the Jacobian  Then sometimes this is called the “Jacobian method”

8cs426-winter-2008 Gauss-Newton  Can approximate the nonlinear term with a linear function (Taylor series with the Jacobian!) and get a linear least squares problem: Gauss-Newton  Much more efficient, though we need to solve a (small) linear system to get the direction

9cs426-winter-2008 Evaluating Jacobians  Simplest approach in code: numerically approximate with a finite difference  Can also work out derivative analytically by hand (a little painful)

10cs426-winter-2008 When to stop  In our case, absolute minimum of f(.) is zero: stop when it’s smaller than some tolerance  It might be impossible to get to zero, but at the minimum  f=0: stop when |  f| is small enough  Or give up when maximum number of iterations reached

11cs426-winter-2008 Character Rigging  A “rig” is a model together with a UI for posing it  At its simplest, a skeleton with joint angles available for motion curves  May simplify DOF by enforcing relationships between joints E.g. hand and fingers  May define standard poses (especially for facial expressions!) that can be mixed together Then can set sliders to, say, 70% happy, 20% surprised, … Take weighted linear combination of pose angles

12cs426-winter-2008 What’s left?  We now have the basics of animation  Plan for rest of course: Rendering animations (Semi-)automatic animation  Dynamics for rigid bodies  Particle systems  Skinning, morphing, blending  Motion capture  Motion control