1. Spacetime Constraints Witkin & Kass Siggraph 1988

2. Overview • Unlike common Newtonian dynamic simulation, the due driving force is unknown • Specify the high-level spacetime constraint and let the optimization solve for the position and force unknowns by minimizing the “energy consumption”

4. Lamp on floor

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

6. Discretize continuous function Discretize unknown function x(t) and f(t) as: x1, x2, …xi, … xn-1, xn f1, f2, …fi, … fn-1, fn Our goal is to solve these discretized 2n values… x1xn satisfies goals while optimizing f1fn Next step is to discretize our motion equation and object equation. i 1 n

7. Difference Formula h h xi - 0.5 xi + 0.5 xi - 1 xi xi + 1 Backward Forward Central Central

8. Discretized Function Motion equation: x x4, f4 x3, f3 x2, f2 Boundary Conditions: x1, f1 t Object Function: When does R have minimum value?

9. Generalize Our Notation Unknown vector: S = (S1, S2, …Sn) x x4, f4 Constraint Functions: Ci(S) = 0 x3, f3 x2, f2 x1, f1 Minimize Object Function R(S): t S = (x1, x2, x3, x4, f1, f2, f3, f4)

10. Sequential Quadratic Programming (SQP) Step One Pick a guess S0, evaluate Most likely Taylor series expansion of function f(x) at point a is: Similarly, we have: Set equal to 0 Omit Sa is the change to S0 that makes derivative equal to 0

11. SQP Step Two Now we got S1’, evaluate our constraints Ci(S1’), 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 Ci(S1’) ≠ 0, let’s apply Taylor series expansion on the constraint function Ci(S) at point S1’ : Omit Set equal to 0 Sb is the change to S0 that makes derivative equal to 0 Then we will continue with step one and step two until we got a solution Sn which minimizes our object function and also satisfies our constraints. S0  S1’ S1  S2’  S2  …  Sn

12. Graphical Explanation of SQP C(S) S1 S2’ S2 S S0 S1’

14. Homework Spacetime Particle (2D version)

15. Discretize unknown function x(t) and f(t) as: x1, x2, …xi, … xn-1, xn f1, f2, …fi, … fn-1, fn Our goal is to solve these discretized 2n values… x1xn satisfies goals while optimizing f1fn i 1 n Discretize, (xi,fi) as variables, minimize sum of fi.fi Formulate as constrained optimization.