1. Spring 20061 Rigid Body Simulation

2. Spring 20062 Contents Unconstrained Collision Contact Resting Contact

3. Spring 20063 Review Particle Dynamics State vector for a single particle: System of n particles: Equation of Motion

4. Spring 20064 Rigid Body Concepts

5. Spring 20065 Rotational Matrix Direction of the x, y, and z axes of the rigid body in world space at time t.

6. Spring 20066 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 20067 Rotate a Vector

8. Spring 20068 = = Change of R(t)

9. Spring 20069 Rigid Body as N particles Coordinate in body space

10. Spring 200610 Center of Mass World space coordinate Body space coord.

11. Spring 200611 Force and Torque Total force Total torque

12. Spring 200612 Linear Momentum Single particle Rigid body as particles

13. Spring 200613 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.

14. Spring 200614 Inertia Tensor

15. Spring 200615 Inertia Tensor

16. Spring 200616 [Moment of Inertia (ref)]ref Moment of inertia

17. Spring 200617 Table: Moment of Inertia

18. Spring 200618 Equation of Motion (3x3)

19. Spring 200619 Implementation (3x3)

20. Spring 200620 Equation of Motion (quaternion) 3×3 matrix quaternion

21. Spring 200621 Implementation (quaternion)

22. Spring 200622 Non-Penetration Constraints

23. Spring 200623 Collision Detection

24. Spring 200624 Colliding Contact

25. Spring 200625 Collision Relative velocity Only consider v rel < 0 Impulse J : J

26. Spring 200626 Impulse Calculation [See notes for details]

27. Spring 200627 Impulse Calculation For things don ’ t move (wall, floor):

28. Spring 200628 Uniform Force Field Such as gravity acting on center of mass No effect on angular momentum

29. Spring 200629 Resting Contact: See Notes

30. Spring 200630 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

31. Spring 200631 x y 5 3 Three walls