1. Spacetime Constraints Andrew Witkin Michael Kass Presenter: Fayun Luo

2. Outline Introduction and Motivation A Particle Example SQP Method Extension to Complex Models Discussion A Tiny Movie Demo

3. Early Computer Simulations Question: can we generate those motion automatically? The first computer generated simulation— Pixar’s Luxo, Jr. 1986 Yes, by adding physics in the simulation.

4. Physically-based Approaches Solving Initial Value Problems V0V0 x0x0 x1x1

5. Physically-based Approaches Constraints force methods A man executing a kick The same kick on a frictionless floor

6. Physically-based Approaches Spacetime Constraints Basic idea is solve for the character’s motion and varying force over the entire time. Newtonian physics Object Function

7. Roadmap

8. Problem Statement Governing Equation (Motion Equation): Object Function (Energy Consumption): Boundary Conditions: g f(t)

9. Discretize continuous function 1 n i Discretize unknown function x(t) and f(t) as: x 1, x 2, …x i, … x n-1, x n f 1, f 2, …f i, … f n-1, f n Our goal is to solve these discretized 2n values. Next step is to dicretize our motion equation and object equation.

10. Difference Formula x i - 1 xixi x i + 1 x i - 0.5 x i + 0.5 hh BackwardForwardMiddle

11. Discretized Function t x x 1, f 1 x 2, f 2 x 3, f 3 x 4, f 4 Motion equation: Boundary Conditions: Object Function:When does R have minimum value?

12. Roadmap

13. Generalize Our Notation t x x 1, f 1 x 2, f 2 x 3, f 3 x 4, f 4 Unknown vector: S = (S 1, S 2, …S n ) Constraint Functions: C i (S) = 0 Object Function R(S): S = (x1, x2, x3, x4, f1, f2, f3, f4)

14. SQP Step One Pick a guess S 0, evaluate Taylor series expansion of function f(x) at point a is: Most likely Similarly, we have: = 0 Omit

15. SQP Step Two Now we got S 1 ’, evaluate our constraints C i (S 1 ’ ), if equal to 0, we are done but most likely it will not evaluate to 0 in the first several steps. So, let’s say C i (S 1 ’ ) ≠ 0, let’s apply Taylor series expansion on the constraint function C i (S) at point S 1 ’ : Then we will continue with step one and step two until we got a solution S n which minimize our object function and also satisfy our constraints. = 0 Omit S 0  S 1 ’  S 1  S 2 ’  S 2  …  S n

16. Graphical Explanation of SQP S0S0 S1’S1’ S1S1 S2’S2’ S2S2 C(S) S

17. Roadmap

18. Difficulties Set up the motion equations Define the objective equation Evaluate the derivatives Fit them into our SQP solver

19. Derive Motion Equation Use Lagrangian Dynamics to derive our motion equations dynamically: T – Kinetic Energy q – Generalized Coordinates Q – Generalized Forces

20. The Authors’ Automatic System Graphical User Interface Function Boxes Dynamic System SQP Solver T, Q, qJ RH

21. Roadmap

22. Lagrangian Dynamics Define T for complex system is still too much work.

23. Define Objective Functions Walking on hot coals Walking on eggs Carrying a bowl of hot soup Pursued by a bear Define appropriate objective functions may be extremely difficult:

24. The Author’s Automatic System Symbolic Analysis is really complex, especially for complex system. The state of art symbolic analysis tool is Matlab, Maple. The author’s automatic system may work for some relative simple system.

25. Local Optimization vs Global Optimization S R S0S0 S*S* S*S*

26. Roadmap

27. References Pixar, Luxo, Jr. 1986 Ronen Barzel, et al. Dynamic Constraints. Siggraph 1987 Paul Issacs and Michael Cohen, Controlling Dynamic Simulation with Kinematic Constraints, Proc. Siggraph 1987 David C. Brogan, et al. Spacetime Constraints for Biomechanical Movements. Applied Modeling and Simulation, 2002 Phillip Gill, et al. Practical Optimization, Academic Press, New York, NY, 1981