1. UNC Chapel Hill S. Redon - M. C. Lin Rigid body dynamics II Solving the dynamics problems

2. UNC Chapel Hill S. Redon - M. C. Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion

3. UNC Chapel Hill S. Redon - M. C. Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion

4. UNC Chapel Hill S. Redon - M. C. Lin Algorithm Overview

5. UNC Chapel Hill S. Redon - M. C. Lin Algorithm Overview Two modules –Collision detection –Dynamics Calculator Two sub-modules for the dynamics calculator –Constrained motion computation (accelerations/forces) –Collision response computation(velocities/impulses)

6. UNC Chapel Hill S. Redon - M. C. Lin Two modules –Collision detection –Dynamics Calculator Two sub-modules for the dynamics calculator –Constrained motion computation (accelerations/forces) –Collision response computation(velocities/impulses) Two kinds of constraints –Unilateral constraints (non-penetration constraints…) –Bilateral constraints (hinges, joints…) Algorithm Overview

7. UNC Chapel Hill S. Redon - M. C. Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion

8. UNC Chapel Hill S. Redon - M. C. Lin Constrained accelerations Solving unilateral constraints is enough When v rel ~= 0 -- have resting contact All resting contact forces must be computed and applied together because they can influence one another

9. UNC Chapel Hill S. Redon - M. C. Lin Constrained accelerations

10. UNC Chapel Hill S. Redon - M. C. Lin Here we only deal with frictionless problems Two different approaches –Contact-space : the unknowns are located at the contact points –Motion-space : the unknowns are the object motions Constrained accelerations

11. UNC Chapel Hill S. Redon - M. C. Lin Contact-space approach –Inter-penetration must be prevented –Forces can only be repulsive –Forces should become zero when the bodies start to separate –Normal accelerations depend linearly on normal forces This is a Linear Complementarity Problem Constrained accelerations

12. UNC Chapel Hill S. Redon - M. C. Lin Motion-space approach –The unknowns are the objects’ accelerations Gauss’ principle of least contraints –“The objects’ constrained accelerations are the closest possible accelerations to their unconstrained ones” Constrained accelerations

13. UNC Chapel Hill S. Redon - M. C. Lin Formally, the accelerations minimize the distance over the set of possible accelerations a is the acceleration of the system M is the mass matrix of the system Constrained accelerations

14. UNC Chapel Hill S. Redon - M. C. Lin The set of possible accelerations is obtained from the non-penetration constraints This is a Projection problem Constrained accelerations

15. UNC Chapel Hill S. Redon - M. C. Lin Example with a particle Constrained accelerations The particle’s unconstrained acceleration is projected on the set of possible accelerations (above the ground)

16. UNC Chapel Hill S. Redon - M. C. Lin Both formulations are mathematically equivalent The motion space approach has several algorithmic advantages –J is better conditionned than A –J is always sparse, A can be dense –less storage required –no redundant computations Constrained accelerations

17. UNC Chapel Hill S. Redon - M. C. Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion

18. UNC Chapel Hill S. Redon - M. C. Lin We consider -one- contact point only The problem is formulated in the collision coordinate system Computing a frictional impulse -j +j

19. UNC Chapel Hill S. Redon - M. C. Lin Notations –v : the contact point velocity of body 1 relative to the contact point velocity of body 2 –v z : the normal component of v –v t : the tangential component of v – : a unit vector in the direction of v t –f z and f t : the normal and tangential (frictional) components of force exerted by body 2 on body 1, respectively. Computing a frictional impulse

20. UNC Chapel Hill S. Redon - M. C. Lin When two real bodies collide there is a period of deformation during which elastic energy is stored in the bodies followed by a period of restitution during which some of this energy is returned as kinetic energy and the bodies rebound of each other. Computing a frictional impulse

21. UNC Chapel Hill S. Redon - M. C. Lin The collision occurs over a very small period of time: 0  t mc  t f. t mc is the time of maximum compression v z is the relative normal velocity. (We used v rel before) Computing a frictional impulse vzvz

22. UNC Chapel Hill S. Redon - M. C. Lin j z is the impulse magnitude in the normal direction. W z is the work done in the normal direction. jz jz Computing a frictional impulse

23. UNC Chapel Hill S. Redon - M. C. Lin v - =v(0), v 0 =v(t mc ), v + =v(t f ), v rel =v z Newton’s Empirical Impact Law: Poisson’s Hypothesis: Stronge’s Hypothesis: –Energy of the bodies does not increase when friction present Computing a frictional impulse

24. UNC Chapel Hill S. Redon - M. C. Lin Sliding (dynamic) friction Dry (static) friction Assume no rolling friction Computing a frictional impulse

25. UNC Chapel Hill S. Redon - M. C. Lin where: r = (p-x) is the vector from the center of mass to the contact point Computing a frictional impulse

26. UNC Chapel Hill S. Redon - M. C. Lin The K Matrix K is constant over the course of the collision, symmetric, and positive definite

27. UNC Chapel Hill S. Redon - M. C. Lin Collision Functions Change variables from t to something else that is monotonically increasing during the collision: Let the duration of the collision  0. The functions v, j, W, all evolve over the compression and the restitution phases with respect to .

28. UNC Chapel Hill S. Redon - M. C. Lin Collision Functions We only need to evolve v x, v y, v z, and W z directly. The other variables can be computed from the results. (for example, j can be obtained by inverting K)

29. UNC Chapel Hill S. Redon - M. C. Lin Sliding or Sticking? Sliding occurs when the relative tangential velocity –Use the friction equation to formulate Sticking occurs otherwise –Is it stable or instable? –Which direction does the instability get resolved?

30. UNC Chapel Hill S. Redon - M. C. Lin Sliding Formulation For the compression phase, use – is the relative normal velocity at the start of the collision (we know this) –At the end of the compression phase, For the restitution phase, use – is the amount of work that has been done in the compression phase –From Stronge’s hypothesis, we know that

31. UNC Chapel Hill S. Redon - M. C. Lin Sliding Formulation Compression phase equations are:

32. UNC Chapel Hill S. Redon - M. C. Lin Sliding Formulation Restitution phase equations are:

33. UNC Chapel Hill S. Redon - M. C. Lin Sliding Formulation where the sliding vector is:

34. UNC Chapel Hill S. Redon - M. C. Lin Sliding Formulation These equations are based on the sliding mode Sometimes, sticking can occur during the integration

35. UNC Chapel Hill S. Redon - M. C. Lin Sticking Formulation

36. UNC Chapel Hill S. Redon - M. C. Lin Sticking Formulation Stable if –This means that static friction takes over for the rest of the collision and v x and v y remain 0 If instable, then in which direction do v x and v y leave the origin of the v x, v y plane? –There is an equation in terms of the elements of K which yields 4 roots. Of the 4 only 1 corresponds to a diverging ray – a valid direction for leaving instable sticking.

37. UNC Chapel Hill S. Redon - M. C. Lin Impulse Based Experiment Platter rotating with high velocity with a ball sitting on it. Two classical models predict different behaviors for the ball. Experiment and impulse-based dynamics agree in that the ball rolls in circles of increasing radii until it rolls off the platter. Correct macroscopic behavior is demonstrated using the impulse-based contact model.

38. UNC Chapel Hill S. Redon - M. C. Lin Outline Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion

39. UNC Chapel Hill S. Redon - M. C. Lin Systems can be classified according to the frequency at which the dynamics calculator has to solve the dynamics sub-problems Extensions - Discussion

40. UNC Chapel Hill S. Redon - M. C. Lin Systems can be classified according to the frequency at which the dynamics calculator has to solve the dynamics sub-problems It is tempting to generalize the solutions (fame !) –Lasting non-penetration constraints can be viewed as trains of micro-collisions, resolved by impulses –The LCP / projection problems can be applied to velocities and impulses Extensions - Discussion

41. UNC Chapel Hill S. Redon - M. C. Lin Problems with micro-collisions –creeping : a block on a ramp can’t be stabilized Extensions - Discussion

42. UNC Chapel Hill S. Redon - M. C. Lin Problems with micro-collisions –creeping : a block on a ramp can’t be stabilized Extensions - Discussion

43. UNC Chapel Hill S. Redon - M. C. Lin Problems with micro-collisions –creeping : a block on a ramp can’t be stabilized –A hybrid system is required to handle bilateral constraints (non-trivial) Extensions - Discussion

44. UNC Chapel Hill S. Redon - M. C. Lin Problems with micro-collisions –creeping : a block on a ramp can’t be stabilized –A hybrid system is required to handle bilateral constraints (non-trivial) –Objects stacks can’t be handled for more than three objects (in 1996), because numerous micro-collisions cause the simulation to grind to a halt Extensions - Discussion

45. UNC Chapel Hill S. Redon - M. C. Lin Extending the LCP –accelerations are replaced by velocities –forces are replaced by impulses –constraints are expressed on velocities and forces Problem : –constraints are expressed on velocities and forces (!) This can add energy to the system –Integrating Stronge’s hypothesis in this formulation ? Extensions - Discussion

46. UNC Chapel Hill S. Redon - M. C. Lin Extending the projection problem –accelerations are replaced by velocities –constraints are expressed on velocities and forces Extensions - Discussion

47. UNC Chapel Hill S. Redon - M. C. Lin Extending the projection problem –accelerations are replaced by velocities –constraints are expressed on velocities Problem : –constraints are expressed on velocities (!) –This can add energy to the system Extensions - Discussion