1. Spring 20131 Rigid Body Simulation

2. Spring 20132 Contents Unconstrained Collision Contact Resting Contact

3. Spring 20133 Review Particle Dynamics State vector for a single particle: System of n particles: Equation of Motion Solved by ODE Solvers (Euler, RK4, etc.)

4. Rigid Body Concepts Body space –Origin: center of mass p 0 : an arbitrary point on the rigid body, in body space. –Its world space location p(t) Spatial variables of the rigid body: 3-by-3 rotation matrix R(t) and x(t) 4 Spring 2013

5. 5 Rotational Matrix Direction of the x, y, and z axes of the rigid body in world space at time t.

6. Spring 20136 Velocity Linear velocity Angular veclocity Spin:  (t) How are R(t) and  (t) related? Columns of dR(t)/dt: describe the velocity with which the x, y, and z axes are being transformed

7. Spring 20137 Rotate a Vector

8. Spring 20138 = = Change of R(t)

9. Spring 20139 Rigid Body as N particles Coordinate in body space

10. Spring 201310 Center of Mass World space coordinate Body space coord.

11. Spring 2013 11 Force and Torque Total force Total torque

12. Uniform Force Field No effect on the angular momentum 12 Spring 2013

13. 13 Linear Momentum Single particle Rigid body as particles

14. Spring 201314 Angular Momentum I(t) — inertia tensor, a 3  3 matrix, describes how the mass in a body is distributed relative to the center of mass I(t) depends on the orientation of the body, but not the translation.

15. Spring 201315 Inertia Tensor

16. Spring 201316 Inertia Tensor

17. Spring 201317 [Moment of Inertia (ref)]ref Moment of inertia

18. Spring 201318 Table: Moment of Inertia

19. Inertia Tensor Table (ref)ref Solid sphere Hollow sphere Solid ellipsoid 19 Spring 2013

20. The Football in Flight (ref)ref Gravity does not exert torque Angular momentum stays the same 20 Spring 2013

21. 21 Equation of Motion (3x3)

22. Spring 201322 Implementation (3x3)

23. Using Quaternion quaternion multiplication Unit quaternion as rotation Equation of motion quaternion derivative 23 Spring 2013 Later …

24. Spring 201324 Equation of Motion (quaternion) 3×3 matrix quaternion

25. Spring 201325 Implementation (quaternion)

26. Computing Qdot 26 Incremental rotation represented in quaternion: Spring 2013

27. 27 Spring 2013

28. 28 Non-Penetration Constraints

29. Spring 201329 Collision Detection (Particle)

30. Spring 201330 Colliding Contact

31. Spring 201331 Collision Relative velocity Only consider v rel < 0 Impulse J : J

32. Spring 201332 Impulse Calculation [See notes for details]

33. Spring 201333 Impulse Calculation For things don ’ t move (wall, floor):

34. Spring 201334 Resting Contact

35. Spring 201335 Solve the contact forces f i so that for Relative displacement at contact point i:

36. Spring 201336 In contact at t 0 : and we want: We need: Similar logic, we want:

37. Solving Contact Forces Spring 201337 First constraint on contact forces f i : Second constraint on f i (repulsive): Third constraint (no force when contact breaks) Quadric Programming (or Linear Complementarity Program to be exact)

38. Details (appendix D) Spring 201338

39. Spring 201339 Exercise Implement a rigid block falling on a floor under gravity x y 5 3 thickness: 2 M = 6 Moments of inertia Ixx = (3 2 +2 2 )M/12 Iyy = (5 2 +2 2 )M/12 Izz = (3 2 +5 2 )M/12 Inertia tensor

40. Spring 201340 x y 5 3 Three walls