1. Lecture 236.837 Fall 2001 Controlling Animation Boundary-Value Problems Shooting Methods Constrained Optimization Robot Control

2. Lecture 23Slide 26.837 Fall 2001 Computer Animation Story (concept, storyboard) Production (modeling, motion, lighting)

3. Lecture 23Slide 36.837 Fall 2001 Motion Keyframing Interpolates motion from “key” positions Perfect control Can be tedious No realism

4. Lecture 23Slide 46.837 Fall 2001 Motion Simulation Solves equations of motion to compute motion Realistic motion Automatic generation Difficult to control

5. Lecture 23Slide 56.837 Fall 2001 Controlling Simulation What initial velocity will cause the hat to land on the coatrack?

6. Lecture 23Slide 66.837 Fall 2001 Production Story dictates the motion. Animator’s must control the behavior.

7. Lecture 23Slide 76.837 Fall 2001 Simulation Function S(t, u) simulator initial velocity nonlinear not continuous

8. Lecture 23Slide 86.837 Fall 2001 How do we compute S(t, u)? Solve an initial-value problem: numerically integrate the first- order differential equation...detect collisions...apply impulses when collisions occur.

9. Lecture 23Slide 96.837 Fall 2001 Boundary-Value Problem A solution to an initial value problem is a motion that is the solution to a differential equation with the specified initial value. A solution to a boundary value must also solve the ordinary differential equation and match the specified boundary values (initial, final, or any other).

10. Lecture 23Slide 106.837 Fall 2001 Boundary-Value Problem What initial velocity will cause the hat to land on the coatrack?

11. Lecture 23Slide 116.837 Fall 2001 Solution by Shooting Shooting Method Pick initial linear and angular velocity u = (v, ω) Simulate with initial value q 0 =(x 0, R 0, u) to compute the simulation function S(t, u) Repeat until S(t, u) matches the other boundary value x 1, R 1 Problem The initial linear and angular velocity are described by 6 parameters. Exploring a parameter space of high-dimension requires many samples: number of points in a uniform grid grows exponentially.

12. Lecture 23Slide 126.837 Fall 2001 Shooting Method We can combat the high dimensional parameter spaces with clever sampling techniques or derivative-based methods. Sampling techniques Importance Sampling Markov Chain Monte Carlo Derivative-based Methods Gradient descent Newton’s root finding method Human assistance (interaction)

13. Lecture 23Slide 136.837 Fall 2001 Interactive Control

14. Lecture 23Slide 146.837 Fall 2001 Interactive Control

15. Lecture 23Slide 156.837 Fall 2001 Interactive Control

16. Lecture 23Slide 166.837 Fall 2001 Interactive Control update

17. Lecture 23Slide 176.837 Fall 2001 More Parameters simulator velocity gravity … surface

18. Lecture 23Slide 186.837 Fall 2001 More Constraints simulator velocity gravity … surface

19. Lecture 23Slide 196.837 Fall 2001 More Bodies simulator velocity gravity … surface

20. Lecture 23Slide 206.837 Fall 2001 Active Characters What about objects that have self-propelling muscles and forces. How can we compute their motion automatically?

21. Lecture 23Slide 216.837 Fall 2001 Constrained Optimization Body, muscle and force degrees of freedom: q(t) Constraints Pose C p Mechanical C m Dynamics C d Objective Function E(q(t))

22. Lecture 23Slide 226.837 Fall 2001 Controller Approach Roboticists address almost identical problems in robot design. How should we control a robot to make it walk or run? Similar techniques have been applied for problems in computer animation. Biomechanics data and observations Feedback controllers designed by hand simulated annealing generate-and-test strategy parameter estimation and parallel optimization techniques

23. Lecture 23Slide 236.837 Fall 2001 Next Time Research in the MIT Computer Graphics Group