Computer Animation Algorithms and Techniques

1. Computer AnimationAlgorithms and Techniques Dynamics

2. Enforcing Soft and Hard Constraints Energy minimization Space-time constraints

3. Energy minimization

4. Energy minimization Useful Functions Parametric position function P(u,v) Surface normal function, N(u,v) Implicit function I(x)

5. Useful Constraints

6. Spacetime constraints http://www.cs.cmu.edu/~aw/pdf/spacetime.pdf Constrained optimization problem in time and space Constraints – e.g. locating certain points in time and space Non penetration constraints Objective function – e.g. Minimize the amount of force used over time interval Minimize maximum torque

7. Spacetime constraint example Particle position is function of time, x(t) Time-varying force function, f(t) Equation of motion Given f(t), x(t0), - integrate to get x(t)

8. Spacetime constraint example Need to determine f(t) x(t0) = a x(t1) = b Subject to constraints Objective: to minimize total force Objective function:

9. Spacetime constraint example Use discrete x(t), f(t), R, constraints Time derivatives approximated by finite differences R – minimize discrete version subject to constraints

10. Spacetime constraint example Numerical Solution – canonical form Sj – collection of scalar independent variables R(Sj) – objective function to be minimized Ci (Sj) – collection of scalar constraint functions => 0

11. Spacetime constraint example Numerical Solution – example in canonical form Sj – x, y, z components of the xi’s and fi’s R(Sj) – objective function to be minimized Ci (Sj) – components of pi’s, ca, and cb

12. Spacetime constraint example Canonical numerical problem Find Sj that minimizes R(Sj) subject to Ci(Sj) = 0 • Numerical solution method: • Request values of R and Ci for given Sj • Access to derivatives of R and Ci with respect to Sj • Iteratively provides updated values for solution vector Sj

13. Spacetime constraint example Sequential Quadratic Programming (SQP) Computes second-order Newton-Raphson step in R Computes first-order Newton-Raphson step in the Ci’s Projects the first onto the null space of the second to the hyperplane for which all the Ci’s are constant to the first order

14. Spacetime constraint example Sequential Quadratic Programming (SQP) Computes second-order Newton-Raphson step in R Hessian Computes first-order Newton-Raphson step in the Ci’s Jacobian: Plus the first derivative vector:

15. Spacetime constraint example SQP step Solve 2 linear systems in sequence 1 Yields a step that minimizes a 2nd order approx. to R 2 Yields a step that drives linear approx. to Ci’s to zero And projects optimization step Sj to null space of constraint Jacobian

16. Spacetime constraint example Final update: Reaches fixed point: - When Ci’s = 0 - Any further decrease in R violates constraints

17. Spacetime constraint example Note about Linear system solving Large matrices often with spacetime problems Inverting is O(n3) Spacetime problems almost always sparse Over and under constrainted systems easily arise Under constrained: pseudo-inverse Pseudo-inverse for sparse matrix sparse conjugate gradient algorithm: O(n2)

18. Spacetime constraint example Matrix evaluation i=j i=j-1, j+1 otherwise i=j otherwise

19. Spacetime constraint example Matrix evaluation i=j otherwise