1. Physics-Based Simulation: Graphics and Robotics Chand T. John

2. Forward Dynamic Simulation Problem: Determine the motion of a mechanical system generated by a set of forces or control values. Challenges: - Contact/collision - Large number of bodies - Drift - Control of end result - Creating natural motion - High-level motion control - Being efficient and accurate Update State Time Step

3. Forward Dynamic Simulation Forward dynamic simulation is needed in: Computer graphics Robotics Biomechanics

4. Bridging Biomechanics & Graphics Graphics has: - Fast algorithms for making visually realistic movies of complicated scenes Biomechanics has: - A need for physically realistic simulations - Real science to back up dynamic models Biomechanics goals: Physically realistic Make clinically relevant conclusions Meaningfulness > accuracy > efficiency Graphics goals: Visually realistic Make cool SIGGRAPH movies Accuracy > efficiency > meaningfulness

5. Constraining Mechanical Systems Use fewer coordinates Add constraint forces Enforce constraints after each time step Optimization Control

6. Robot Controller Simulation 22 33 (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) (x 3, y 3, z 3 ) Reduced/generalized/joint-space/link coordinates Maximal/redundant/operational-space/task-space/absolute coordinates

7. Recursive Newton-Euler Algorithm v0v0 a0a0 v1v1 a1a1 v2v2 a2a2 v3v3 a3a3 Propagate velocities outward Propagate accelerations outward

8. Recursive Newton-Euler Algorithm f0f0 f1f1 f2f2 f3f3 Propagate forces inward O(N) algorithm for inverse dynamics!

9. Composite Rigid-Body Algorithm Link i This part of the robot is in static equilibrium. This part of the robot is a (composite) rigid body. CiCi When all velocity and acceleration-independent forces are zero, M i is the force causing unit acceleration  i to the robot.

10. Composite Rigid-Body Algorithm CiCi If the robot is given an acceleration of  i : all of C i will act like a single rigid body with acceleration h i, and the rest of the robot stays in static equilibrium. Let f i C = force needed to induce acceleration h i on C i.

11. Composite Rigid-Body Algorithm M ji = h j T f i C for parents j of i M ji = 0 for other j not in C i Fill in symmetric values M ij Inertia I i C of C i = sum of inertias I j f i C = I i C h j f i C (j) = f i C for all j in C i f i C = 0 for all other j Implement in joint space Each parent of link i has force f i C h j T I i C h i if i in C j h j T I j C h i if j in C i 0 otherwise M ji = O(N 3 ), fast for small N IiCIiC

12. Articulated-Body Algorithm a =  f + b Invert f = I A a + p A I A =  -1 p A = -I A b Invertible if body unconstrained Key observation: Acceleration a of a body is always an affine function of applied force f.

13. Articulated-Body Algorithm Propagate I i A, p i A inward. 3 equations, 3 unknowns: a i = a (i) + h i ’q i ’ + h i q i ’’ f i J = I i A a i + p i A  i = h i T f i J Solve for q i ’’ Evaluate a i Propagate a i outward. O(N), faster than CRBA for N > 9 IiApiAIiApiA aiai

14. Adaptive Dynamics Simplification Based on: –Featherstone’s O(log N) divide-and-conquer algorithm –For forward dynamics on O(N) parallel processors Closed loops not supported yet Algorithm: –User picks number of DOFs desired –In each time step: Algorithm picks active joints Forward dynamics steps forward in time http://gamma.cs.unc.edu/AD/

15. Dealing with Closed Loops Closed loop!

16. Dealing with Closed Loops Compute spanning tree Compute dynamics for tree Mimic loop with constraint forces M L LTLT 0 q’’ -  – C +  a c = Equation of motion Constraint forces

17. Lagrange Multiplier Constraints Reduced-coordinate methods are fast Multiplier methods (with maximal coordinates): –Have a drift problem –Involve solving a matrix equation But multiplier methods: –Do not require parameterization –Allow use of nonholonomic constraints Limited branching  sparse matrix Baraff gives O(N) multiplier method for computing constraint forces for dynamics

18. Lagrange Multiplier Constraints This is a system with 127 constraints. Each sphere is a 3 DOF constraint between two rigid bodies. There are 381 Lagrange multipliers.

19. Post-Stabilization with DAEs http://www.cs.ubc.ca/nest/scv/demos/Slider/slider.html Constraint manifold State at time t State at time t + 1 After projection

20. Spacetime Constraints Given: –Equation of motion –Objective function to minimize –Spatial constraints to meet Computes: –Actuation force function by constrained optimization –Position function by integrating equation of motion High-energy motions modeled well Low-energy motions require finer physics model http://www.cs.cmu.edu/~aw/

21. Motion Transformation First physics-based mocap editing method Steps: –Map input motion onto simplified character –Find spacetime optimization problem most closely matching simplified character motion –Alter parameters, constraints, etc. –Map new motion onto original to get final animation High-energy works well; low-energy not so well No comparison to real actor’s motion physics http://www.cs.washington.edu/homes/zoran/sigg99/

22. Physics-Based Motion Style Tasks represented as constraints C E(X;  ) = total torque from muscle forces X describes a motion sequence  is a vector of style parameters Given motion capture X T and C, minimize E(X T ;  ) to compute style  With new constraints C’, minimize E(X;  ) to compute new motion X http://grail.cs.washington.edu/projects/charanim/phys-style.html

23. Neuromuscular Perturbation

24. Proportional-Derivative Control Rachel will talk about her work on PD control and human motion next week. m kvkv kpkp mx’’ + k v x’ + k p x = 0 DerivativeProportional Examples: Cruise control, CMC

25. Task-Level Control Vince will talk about task-level control of shoulder models two weeks from now.

26. Summary Reduced coordinates –Composite rigid-body algorithm, O(N 3 ) –Articulated-body algorithm, O(N) –Adaptive dynamics simplification Constraint force application –Closed-loop systems –Lagrange multiplier constraints Projection onto constraint manifold –Post-stabilization with DAEs Optimization –Spacetime constraints –Motion transformation –Physics-based motion style Control –Neuromuscular perturbation –Proportional-derivative control –Task-level control