Jim Van Verth (jim@essentialmath)

1. Collision Response Jim Van Verth (jim@essentialmath.com)

2. Collision Response • Physical response to collision • Two main cases • Colliding (opposing velocities) • Bouncing • Resting (orthogonal velocities) • Rolling, sliding Essential Math for Games

3. Contact Point • Need location of point of collision and normal to surface at actual time of collision • We might not have this • Objects could be interpenetrating Essential Math for Games

4. Determine Collision Time • Do binary search • Back off simulation half of previous time step • Still penetrating? Back off ¼ time step • No collision? Move forward ¼ time step • Continue until time step too small or exact collision found • Slow, may not find exact point Essential Math for Games

5. Determining Collision Time • Or just fake it • Determine penetration distance • Push objects away until they just touch • Can use mass, velocities to adjust how much each moves (if’n you feel fancy) • Problem: can cause a new collision this way Essential Math for Games

6. Determining Collision Time • Alternative: push apart a little, use collision response to do the rest • Assumes collision response can handle penetration • Apply force (penalty force) to push them apart Essential Math for Games

7. Determining Collision Time • Or use predictive method to determine time of impact (TOI) • Integrate up to TOI • Do collision response • Find next TOI, integrate, respond, repeat until done Essential Math for Games

8. Determining Collision Time • Ex: spheres • Spheres sweep out capsules, intersect? Essential Math for Games

9. Determining Collision Time • Or: relative velocity • Compare sphere vs. capsule • Sphere centers Pa, Pb; radii ra, rb • Relative velocity v = va- vb Sa Sb Essential Math for Games

10. Determining Collision Time • Idea: determine point(s) on velocity line distance ra+rbfrom Pb • Or: ||Pb– (Pa+vt)|| = ra+rb • Or: (w – vt)•(w – vt) – (ra+rb)2 = 0; w = Pb– Pa • Quadratic: solve for t Sa Sb Essential Math for Games

11. Determining Collision Time • Expand (w – vt)•(w – vt) – (ra+rb)2 = 0 • Get (v•v)t2– 2(v•w)t + w•w –(ra+rb)2 = 0 • Use quadratic equation, get Essential Math for Games

12. Determining Collision Time • Could be 1 solution • Just touching, radical = 0 • Could be no solutions • Never touch, radical is imaginary • Pick solution with smallest t 0, 1 Essential Math for Games

13. Have Collision, Will Travel • Have normal n (from object A to object B), collision point P, velocities va, vb • Three cases: • Separating • Relative velocity along normal < 0 • Colliding • Relative velocity along normal > 0 • Resting • Relative velocity along normal = 0 Essential Math for Games

14. Linear Collision Response • Have normal n, collision point p, velocities v1, v2 • How to respond? • Idea: collision is discontinuity in velocity • Generate impulse along collision normal – modify velocities • How much depends on incoming velocity, masses of objects Essential Math for Games

15. Linear Collision Response • Do simple case – two spheres a & b • incoming velocities va,vb • collision normal n • want to compute impulse magnitude f va n vb Essential Math for Games

16. Linear Collision Response • Compute relative velocity vab vab va n vb Essential Math for Games

17. Linear Collision Response • Compute velocity along normal • Use dot product to project onto normal vab n vn Essential Math for Games

18. Linear Collision Response • Outgoing velocity dependant on coefficient of restitution  •  = 1: pure elastic collision (superballs) •  = 0: pure non-elastic collision (clay) or Essential Math for Games

19. Linear Collision Response • Also need to conserve momentum or and or Essential Math for Games

20. Linear Collision Response • Combining last two slides (with a wave of my hand) gives Essential Math for Games

21. Linear Collision Response • Compute final velocities va- va+ n vb- vb+ Essential Math for Games

22. Angular Collision Response • Like linear, but include angular velocity • Compute velocity at collision point va b ra rb a vb Essential Math for Games

23. Angular Collision Response • Need to conserve angular momentum or and and or Essential Math for Games

24. Angular Collision Response • Final impulse equation (more waving of hands) Essential Math for Games

25. Angular Collision Response • Compute new angular momentum • remember, sim uses angular momentum • Then angular velocity from momentum Essential Math for Games

26. Multiple Colliding Contacts • 3-body problem (kinda) • Do one at a time, will end up with penetration • What to do? Essential Math for Games

27. Multiple Colliding Contacts • One solution: • Do independently • Handle penetration as part of collision resolution • Another solution: • Generate constraint forces • Use relaxation techniques Essential Math for Games

28. Multiple Colliding Contacts • Crunchy math solution • Build systems of equations where • , normal component of rel. vel. before & after • ai,j describes how body j affects impulse on body i • Get matrix equation: Essential Math for Games

29. Multiple Colliding Contacts • Results coupled – want “as close as possible” • If (separating) want close to • If (colliding) want close to • Represent difference by new vector b • If then • If then Essential Math for Games

30. Multiple Colliding Contacts • Now want to minimize length of vector • Impulses are non-positive so want • Also want • Quadratic programming problem • Can convert to Linear Complementary Problem (LCP) – use Lemke’s Algorithm Essential Math for Games

31. Resting Contacts • Mirtich & Canny use micro-impulses, simulate molecular micro-collisions • Works with current impulse system Essential Math for Games

32. Resting Contacts • Impulses work, but can be kind of jittery • Forces generate velocity into object • Counteract in next frame – but still move a bit • One solution: once close to rest, turn off sim g Essential Math for Games

33. Resting Contact • Velocity along normal is 0, so look at forces along normal • If will push together, counteract with new constraint force C = gn,g < 0 • Want acceleration along normal • If will eventually push apart, want g = 0 • So want: ≤0, g≤ 0, g = 0 Essential Math for Games

34. Resting Contact • Can build expression for : Essential Math for Games

35. Resting Contact • Expand out, end up with expression like • Ag repr. acceleration due to response • b repr. acceleration due to other forces • Remember, want ≤ 0, g≤ 0, g = 0 • If b > 0, then g = -b/A, = 0 • If b≤ 0, then g = 0, = b Essential Math for Games

36. Resting Contact • Simple example: assume object on ground, only linear forces • If b > 0, then • Counteracts force along normal • If b≤ 0, then C = 0 Essential Math for Games

37. Resting Contact • Things get a lot more complicated if both move, and angular terms • See Eberly for the full details Essential Math for Games

38. Multiple Resting Contacts • Can have stacks of stuff • What to do then? Essential Math for Games

39. Multiple Resting Contacts • Build systems of equations where • Get matrix equation where , , • A is the same as with colliding contacts! • Another LCP problem – use Lemke’s Essential Math for Games

40. Resting Contacts • Resting contact is a hard problem • Very susceptible to floating point error, jittering • Don’t do more than application needs • Don’t get discouraged Essential Math for Games

41. Contact Friction • Generate force opposed to velocity • For contact: tangential relative velocity • Magnitude is relative to normal force • If using impulses, can ignore normal force, use constant Essential Math for Games

42. Putting It Together • Compute forces, torques on objects • Determine collisions • Adjust velocities, forces, torques based on collision • Predict future collision • Step ahead to future collision (if any) or fixed step (if not) Essential Math for Games

43. Recap • Try to keep objects non-penetrating • Colliding objects get impulses • Resting objects either micro-impulses or constraint forces • Multiple contacts hard, but manageable Essential Math for Games

44. References • Hecker, Chris, “Behind the Screen,” Game Developer, Miller Freeman, San Francisco, Dec. 1996-Jun. 1997. • Lander, Jeff, “Graphic Content,” Game Developer, Miller Freeman, San Francisco, Jan., Mar., Sep. 1999. • Witkin, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modelling, SIGGRAPH 2002. Essential Math for Games

45. References • Mirtich, Brian and John Canny, “Impulse-Based Simulation of Rigid Bodies”, Proceedings of 1995 Symposium on Interactive 3D Graphics, April 1995. (available online) • Eberly, David H., Game Physics, Morgan Kaufmann, 2004. Essential Math for Games