collision response jim van verth ([email protected])

Collision Response

Jim Van Verth ([email protected])

Essential Math for Games

Collision Response

• Physical response to collision

• Two main cases Colliding (opposing velocities)

• Bouncing

Resting (orthogonal velocities)• Rolling, sliding


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


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


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



• 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



• 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



• Ex: spheres

• Spheres sweep out capsules, intersect?



• Or: relative velocity• Compare sphere vs. capsule

• Sphere centers Pa, Pb; radii ra, rb

• Relative velocity v = va- vb

Sa

Sb



• Idea: determine point(s) on velocity line distance ra+rb from 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



• 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



• Could be 1 solution Just touching, radical = 0

• Could be no solutions Never touch, radical is imaginary

• Pick solution with smallest t 0, 1


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


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



• Do simple case – two spheres a & b incoming velocities va,vb

collision normal n want to compute impulse magnitude f

n

va

vb



• Compute relative velocity vab

n

va

vb

vab



• Compute velocity along normal Use dot product to project onto normal

n

vab

vn



• Outgoing velocity dependant on coefficient of restitution

= 1: pure elastic collision (superballs) = 0: pure non-elastic collision (clay)

or



• Also need to conserve momentum

or

and

or



• Combining last two slides (with a wave of my hand) gives



• Compute final velocities

n

va-

vb- vb

+

va+


Angular Collision Response

• Like linear, but include angular velocity

• Compute velocity at collision point

vb

va

ra rba

b



• Need to conserve angular momentum

and

or

and

or



• Final impulse equation (more waving of hands)



• Compute new angular momentum

remember, sim uses angular momentum

• Then angular velocity from momentum


Multiple Colliding Contacts

• 3-body problem (kinda)

• Do one at a time, will end up with penetration

• What to do?



• One solution: Do independently Handle penetration as part of collision

resolution

• Another solution: Generate constraint forces Use relaxation techniques



• 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:



• 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



• 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


Resting Contacts

• Mirtich & Canny use micro-impulses, simulate molecular micro-collisions

• Works with current impulse system


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 simg


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


Resting Contact

• Can build expression for :


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


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


Resting Contact

• Things get a lot more complicated if both move, and angular terms

• See Eberly for the full details


Multiple Resting Contacts

• Can have stacks of stuff

• What to do then?


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


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


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


Putting It Together

1. Compute forces, torques on objects

2. Determine collisions

3. Adjust velocities, forces, torques based on collision

4. Predict future collision

5. Step ahead to future collision (if any) or fixed step (if not)


Recap

• Try to keep objects non-penetrating

• Colliding objects get impulses

• Resting objects either micro-impulses or constraint forces

• Multiple contacts hard, but manageable


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.


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.

http://portal.acm.org/citation.cfm?id=199436

collision response jim van verth ([email protected])

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