1. Handling of Resting Contact
Global vs. Local Methods
Constraint-based vs. Impulse-based
Colliding Contacts
Resting Contacts
Force application
Friction
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M. C. Lin

2. Impulse vs. Constraint Impulse-based dynamics (local)
Faster
Simpler
No explicit contact constraints
Constraint-based dynamics (global)
Must declare each contact to be a resting contact or a colliding contact
UNC Chapel Hill
M. C. Lin

3. Impulse vs. Constraint Impulse-based dynamics (local)
Constraint-based dynamics (global)
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M. C. Lin

4. Impulse vs. Constraint
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M. C. Lin

5. Resting Contact Response
When vrel ~= 0 -- have resting contact
All resting contact forces must be computed and applied together because they can influence one another
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M. C. Lin

6. Resting Contact Response
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M. C. Lin

7. Resting Contact Response
The forces at each contact must satisfy three criteria
Prevent inter-penetration:
Repulsive -- we do not want the objects to be glued together:
Should become zero when the bodies start to separate:
To implement hinges and pin joints:
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M. C. Lin

8. Resting Contact Response
We can formulate using LCP:
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M. C. Lin

9. Resting Contact Response
Need to solve a quadratic program to solve for the fi’s
A is symmetric positive semi-definite (PSD) making the solution practically possible
There is an iterative method to solve for without using a quadratic program [3]
UNC Chapel Hill
M. C. Lin

10. Global / Constraint-Based: Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2 if Relative_Normal_Velocity(A,B) < - then
3 Cs = Contacts(A,B);
4 Apply_Impulses(A,B,Cs);
5 else if Relative_Normal_Velocity(A,B) <  then
6 Flag_Pair(A,B);
7 Solve_For_Forces(flagged pairs);
8 Apply_Forces(flagged pairs);
UNC Chapel Hill
M. C. Lin

11. Local / Impluse-Based: Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2 if Relative_Normal_Velocity(A,B) < - then
3 Cs = Contacts(A,B);
4 Apply_Impulses(A,B,Cs);
5 else if Relative_Normal_Velocity(A,B) <  then
6 Compute_Restitution(A,B);
7 Cs = Contacts(A,B);
8 Apply_Impulses(A,B,Cs);
UNC Chapel Hill
M. C. Lin

12. Algorithm Overview 0 Initialize(); 1 for t = 0; t < tf; t += h do
2 Read_State_From_Bodies(S);
3 Compute_Time_Step(S,t,h);
4 Compute_New_Body_States(S,t,h);
5 Write_State_To_Bodies(S);
6 Zero_Forces();
7 Apply_Env_Forces();
8 Apply_BB_Forces();
UNC Chapel Hill
M. C. Lin

13. Global / Constraint-Based: Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2 if Relative_Normal_Velocity(A,B) < - then
// Colliding Contact
3 Cs = Contacts(A,B);
4 Apply_Impulses(A,B,Cs);
5 else if Relative_Normal_Velocity(A,B) <  then
// Resting Contact
Flag_Pair(A,B);
7 Solve_For_Forces(flagged pairs);
8 Apply_Forces(flagged pairs);
UNC Chapel Hill
M. C. Lin

14. Local / Impulse-Based: Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2 if Relative_Normal_Velocity(A,B) < - then
// Normal collision
3 Cs = Contacts(A,B);
4 Apply_Impulses(A,B,Cs);
5 else if Relative_Normal_Velocity(A,B) <  then
// Micro-collision
Compute_Restitution(A,B);
Cs = Contacts(A,B);
8 Apply_Impulses(A,B,Cs);
UNC Chapel Hill
M. C. Lin

15. Compute_Restitution(A,B)
є = f( Distance(A,B) )
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M. C. Lin

16. Frictional Forces Extension
Constraint-based dynamics
Reformulate constraints and solve
No more on this here
Impulse-based dynamics
Must not add energy to the system in the presence of friction so we have to reformulate the impulse to be applied
UNC Chapel Hill
M. C. Lin

17. Collision Coordinate System
p is the applied impulse. We use j because P is for linear momentum
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M. C. Lin

18. Impulse Reformulation
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.
UNC Chapel Hill
M. C. Lin

19. Impulse Reformulation
The collision is instantaneous but we can assume that it occurs over a very small period of time: 0  tmc  tf.
tmc is the time of maximum compression
uz is the relative normal velocity. We used vrel before. From now on we will use vz.
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M. C. Lin

20. Impulse Reformulation
pz is the impulse magnitude in the normal direction. We used j before. From now on we will use jz.
Wz is the work done in the normal direction.
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M. C. Lin

21. Impulse Reformulation
v-=v(0), v0=v(tmc), v+=v(tf), vrel=vz
Newton’s Empirical Impact Law:
Poisson’s Hypothesis:
Stronge’s Hypothesis:
Energy of the bodies does not increase when friction present
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M. C. Lin

22. Friction Formulae Assume the Coulomb friction law:
At some instant during a collision between bodies 1 and 2, let v be the contact point velocity of body 1 relative to the contact point velocity of body 2. Let vt be the tangential component of v and let be a unit vector in the direction of vt. Let fz and ft be the normal and tangential (frictional) components of force exerted by body 2 on body 1, respectively.
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M. C. Lin

23. Friction Formulae Sliding (dynamic) friction Dry (static) friction
Assume no rolling friction
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M. C. Lin

24. Impulse with Friction
Recall that the impulse looked like this for frictionless collisions:
Recall also that vz = j/m
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M. C. Lin

25. Impulse with Friction where:
r = (p-x) is the vector from the center of mass to the contact point
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M. C. Lin

26. The K Matrix
K is constant over the course of the collision, nonsingular, symmetric, and positive definite
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M. C. Lin

27. 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 .
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M. C. Lin

28. Collision Functions
We only need to evolve vx, vy, vz, and Wz directly. The other variables can be computed from the results.
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M. C. Lin

29. 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?
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M. C. Lin

30. 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
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M. C. Lin

31. Sliding Formulation Compression phase equations are: UNC Chapel Hill
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32. Sliding Formulation Restitution phase equations are: UNC Chapel Hill
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33. Sliding Formulation where the sliding vector is: UNC Chapel Hill
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34. Sliding Formulation
Notice that there is a problem at the point of maximum compression because vz = 0:
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35. Sliding Formulation
Let us integrate using vz a bit into the restitution phase (extension integration) so that we never divide by 0.
But we don’t want to integrate too far! Otherwise we exceed the amount of work that is to be done in the restitution phase.
We are safe if we stop at:
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M. C. Lin

36. Sliding Formulation
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37. Sliding Formulation
There is another problem if the tangential velocity becomes 0 because the equations that we have derived were based on which no longer holds.
This brings us to the sticking formulation
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M. C. Lin

38. Sticking Formulation
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M. C. Lin

39. Sticking Formulation Stable if
This means that static friction takes over for the rest of the collision and vx and vy remain 0
If instable, then in which direction do vx and vy leave the origin of the vx, vy 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.
UNC Chapel Hill
M. C. Lin

40. Sticking Formulation
If sticking occurs, then the remainder of the collision may be integrated analytically due to the existence of closed form solutions to the resulting simplified equations.
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M. C. Lin

41. Static Contacts
What about a block on a ramp when the static friction force is greater than gravity? Will it not creep down?
The solution is to create another type of micro-collision which reverses the initial relative velocity. It is applied if it lies within the friction “cone” otherwise the standard collision integration is performed.
UNC Chapel Hill
M. C. Lin

42. 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.
UNC Chapel Hill
M. C. Lin