1. UNC Chapel Hill M. C. Lin Announcements Homework #1 is due on Tuesday, 2/10/09 Reading: SIGGRAPH 2001 Course Notes on Physically-based Modeling

2. UNC Chapel Hill M. C. Lin Disclaimer The following slides reuse materials from SIGGRAPH 2001 Course Notes on Physically-based Modeling (copyright  2001 by Andrew Witkin at Pixar).

3. UNC Chapel Hill M. C. Lin A Bead on a Wire Desired Behavior –The bead can slide freely along the circle. –It can never come off no matter how hard we pull. So, how do we make this happen?

4. UNC Chapel Hill M. C. Lin Penalty Constraints How about using a spring to hold the bead to the wire? Problem –Weak springs  sloppy constraints –Strong springs  instability

5. UNC Chapel Hill M. C. Lin Basic Ideas Convert each constraint into a force imposed on a particle (system) Use principle of virtual work – constraint forces do not add or remove energy Solve the constraints using Lagrange multipliers, ’ s –For particle systems, need to use the derivative matrix, J, or the Jacobian Matrix.

6. UNC Chapel Hill M. C. Lin Geometric Interpretation

7. UNC Chapel Hill M. C. Lin Basic Formulation (F = ma) Curvature (k) has to match k depends on both a & v : –The faster the bead is going, the faster it has to turn Calculate f c to yield a legal combination of a & v

8. UNC Chapel Hill M. C. Lin Implicit Representation of Constraints

9. UNC Chapel Hill M. C. Lin Maintaining Constraints

10. UNC Chapel Hill M. C. Lin Constraint Gradient

11. UNC Chapel Hill M. C. Lin Constraint Forces

12. UNC Chapel Hill M. C. Lin Derivation

13. UNC Chapel Hill M. C. Lin Example: A Bead on a Wire

14. UNC Chapel Hill M. C. Lin Drift and Feedback

15. UNC Chapel Hill M. C. Lin Constrained Particle Systems

16. UNC Chapel Hill M. C. Lin Compact Notation

17. UNC Chapel Hill M. C. Lin Particle System Constraint Equations

18. UNC Chapel Hill M. C. Lin Implementations A global matrix equation Matrix block structure with sparsity Each constraint adds its own piece to the equation

19. UNC Chapel Hill M. C. Lin Mathematical Formulation

20. UNC Chapel Hill M. C. Lin Take a Closer Look

21. UNC Chapel Hill M. C. Lin Constraint Structure

22. UNC Chapel Hill M. C. Lin Constrained Particle Systems

23. UNC Chapel Hill M. C. Lin Other Modification

24. UNC Chapel Hill M. C. Lin Constraint Force Evaluation After computing ordinary forces:

25. UNC Chapel Hill M. C. Lin Parametric Representation of Constraints

26. UNC Chapel Hill M. C. Lin Parametric Constraints

27. UNC Chapel Hill M. C. Lin Parametric bead-on-wire (F = mv)

28. UNC Chapel Hill M. C. Lin Some Simplification……

29. UNC Chapel Hill M. C. Lin Lagrangian Dynamics Advantages –Fewer DOF’s –Constraints are always met Disadvantages –Difficult to formulate constraints –Hard to combine constraints