10-Collision Response Collision Response Collision Response [Moore - - PowerPoint PPT Presentation

10-Collision Response

Collision Response

Collision Response [Moore and Wilhelms 88]: “automatic suggestions about the motions of objects

immediately following a collision; animation systems using dynamical simulation inherently must respond to collisions automatically and realistically” Intersection/contact information alone (i.e. Yes/no answer to whether objects are colliding) is very useful for interaction, picking, collision avoidance etc. In self-regulating dynamic simulations (e.g. virtual environments) collisions are the basis for interaction between different objects, between users and objects

Problems of Collision Response

In Physically Based Animation, objects need to behave in ways similar to their real-world counterparts

We need to ensure that objects react according to believable dynamics We need to enforce non-interpenetration

Collisions are a source of unpredictable simulation

• discontinuities. As a result they create additional

workload for the simulation mechanism.

Rigid Body Representation

RigidBody Mass = m InertialTensor = IBODY Position = x(t) Orientation = R(t) LinearMomentum = P(t) AngularMomentum = L(t) Force = F(t) Torque = t(t)

Y(t)= Simulation involves finding the change of Y over time:

*Baraff, Witkin, Kass ‘01+

Rigid body geometry

for collision detection, mass distribution: Inertial tensor I, center of mass

                              ) ( ) ( ) ( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t t t t t t t dt d t dt d τ F R ω v L P R x Y

Dynamics Proxy Display Mesh

Integrating Rigid Body State

                              ) ( ) ( ) ( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t t t t t t t dt d t dt d τ F R ω v L P R x Y

P(t) : Linear Momentum = mv(t) L(t) : Angular Momentum = I(t)w(t)

                              ) ( ) ( ) ( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t t t t t t m t t dt d t dt d τ F R ω v w I v R x Y

Spring Based Response

Places a virtual spring at each collision point to push two objects apart Easy to understand and program, applies equally well to rigid and non-rigid bodies Numerical effort increases with violence of collision (smaller time-steps required) A.K.A. Penalty methods

Objects allowed to penetrate Spring force proportional to interpentration

Collision events

Soft Collision

Hard Collision

Impulse

Impulse-based Collision Response

Collision events are discontinuities in the Simulation Loop An instantaneous change of Rigid-Body state Quick local calculation of instantaneous change in velocities is possible at contact points Good solution for rigid body impact but it is difficult to accurately model resting contact and static friction

Rigid Body Collisions

Collisions are a source of discontinuity in the simulation process Particularly in the case of rigid bodies we need to compute an (almost) instantaneous change in state at the time of collision to prevent colliding objects from penetrating

Steady simulation is interrupted by discontinuity due to collision Discontinuity

Physics Pipeline With Collisions

Constraint based methods

Makes a distinction between contact and collisions For objects moving towards each other (negative relative velocities), force applied over time is often not enough to prevent interpenetration Changing nature of constraints can cause high computational demand

mg F

For two objects in contact (relative velocity 0) ensure that the right force is applied to prevent them from interpenetrating

Particle Plane (Impulse response)

• v

v

  

  

V- V+

 

  v J v m ˆ 



.v n n J ˆ j  n J ˆ j 

Change in velocity caused by application of an Impulse in direction normal to plane, and

• f magnitude j

   1 j

• ||

||

v v 



..... (1)

Rigid Body Impulse

Linear component

Similar to particles causing change in velocity inversely proportional to mass as if force was applied at c.o.m.

Angular component

Causes change in angular velocity inversely proportional to moment (determined from inertial tensor) Dependent on position of impulse

m J v  

Effect of an impulse (or for that matter a force) on objects linear and angular momentum is independent x(t) p(t) x(t)

) J (r I ω   

1

) ) ( J x p ( I   

1

Impulsive Torque

But what is the value of J?

Rigid Body Collision

p xb xa

b a

p p p  

C

 

 

rel rel

v v  ) ( ˆ

b a rel

p p n v    

N.B Relative velocity of colliding points on two object is the input into the magnitude of the impulse. How do we get this? May be collocated but pa and pb are separate points on respective objects with different velocities

Aside: Angular Velocity

Angular velocity: rate at which orientation changes Think of it as magnitude w and axis A Instantaneous speed of a point p on circle (radius r) due to its rotation is: Linear Velocity :

) ( ) ( t t  w   A A ω ) ( ) ( ) ( t t t  w   

Speed at which  changes  A w

r t t p ) ( ) ( w   ) ( ) ( ) ( t t t r ω v  

• p

) ) ( ( ) ( ) (

• p

ω v    t t t

Rigid Body Collision

p xb xa

 

 

rel rel

v v  ) ( ˆ

b a rel

p p n v    

) x (p ω v p

b b b b b

     ) x (p ω v p

a a a a a

    

Linear velocity component Linear velocity of point p due to its rotation. See previous slide

Collision Impulse

a a a a

r ω v p   

  



Post-collision velocity should be We want to compute this in terms

• f previous state of object (pre-

collision values)

 

 

a a a

m j v n v ˆ

) ˆ ( n r I ω ω j

a 1 a a a

  

  

 

 

a a 1 a a a a a

j m j r n r I ω n v p      

   

ˆ ˆ   

a a 1 a a a a a

m j r n r I n r ω v               

  

ˆ ˆ

 

a a 1 a a a a

m j r n r I n p p             

  

ˆ ˆ

 

b b 1 b b b b

m j r n r I n p p             

  

ˆ ˆ

Similarly (by Newton 3):

Collision Impulse Magnitude

) ( ˆ

  

 

b a rel

p p n v  

 

 

rel rel

v v 

     

 



n v ˆ

rel

 

a a 1 a a a

m j r n r I n p            

 

ˆ ˆ

 

b b 1 b b b

m j r n r I n p            

 

ˆ ˆ

 

 

 

 

                 

     b b 1 b a a 1 a b a b a rel

m 1 m 1 j r n r I n r n r I n p p n v ˆ ˆ ˆ ˆ ) ( ˆ  

 

 

 

 

                 

    b b 1 b a a 1 a b a rel rel

m 1 m 1 j r n r I n r n r I n v v ˆ ˆ ˆ ˆ 

 

 

 

 

b b 1 b a a 1 a b a rel

m 1 m 1 v 1 j r n r I n r n r I n          

  

ˆ ˆ ˆ ˆ ) ( 

We know that and

Impulse Equations

Impulse magnitude is given by: The actual impulse vector is simply This is applied to the objects as follows:

n J ˆ j 

m J v J p     

) ( ) (

1

J r I ω J r L        

 IMPULSE

t

Linear momentum Angular momentum

 

 

 

 

b b 1 b a a 1 a b a rel

m 1 m 1 v 1 j r n r I n r n r I n          

  

ˆ ˆ ˆ ˆ ) ( 

Collision Impulse Magnitude

Where the variables come from

Constants RB State variables Derived values Contact variables p is the contact point Associated with a contact direction n

 

 

 

 

b b 1 b a a 1 a b a rel

m 1 m 1 v 1 j r n r I n r n r I n          

  

ˆ ˆ ˆ ˆ ) ( 

Collision Response

• We now have a method for 3D Collision response using

impulses

• Given this simple equation we can calculate the

instantaneous change in Rigid Body state quite straightforwardly

• Response is easy but there are some issues to deal with

– Contact Modelling – getting accurate collision points+normals, manifolds etc is still expensive – Interpenetrations often occur and can cause lack of robustness

Non-Interpenetration

Due to discrete timesteps the instant of collision (or point

• f discontinuity) is not obvious.

In most cases collision detection returns the fact that

• bjects have interpenetrated at time t0+t

time = t0+t time = t0

For accurate calculations we need the exact collision time – or at least correct within a certain threshold e

time = tC

Backtracking

Step backwards in simulation time and redo collision detection e.g. Timestep Bisection: a useful (if not reasonable) guess might be t0+t/2 This process is repeated until we achieve a result under some threshold accuracy  Some collision detection approaches attempt to analytically calculate tC ... This can be quite expensive

time = t0+t time = t0+t/2+ t/4 time = t0 time = t0+t/2 a.k.a. Retroactive Detection

Adaptive Timesteps

Reduces wasted computation encountered in bisection by pre- emptively reducing timesteps A conservative approximation is taken as to when discontinuity

• ccurs and simulation

takes place to this point As discontiuity gets closer increasingly small steps are taken

time = t0

t0 + t t0 + 2t …+ 2t + t /2 …+ t/4 …+ t/8

Timestep is halved when the spacetime- bound suggests that the object will interpenetrate in the next step a.k.a. Conservative advancement

Timewarp

[Mirtich 2000] Main problem with previous methods are that they deal with the synchronous evolution of the entire scene Asynchronous processing of individual interactions can greatly improve performance Basic idea is to rollback to time of discontinuity but only for relevant

• bjects that have penetrated and for
• bjects dependent on these

Similar to having several different simulation processes communicating at key points (collisions)

[Mirtich 2000] Brian Mirtich - Timewarp Rigid Body Simulation, ACM Siggraph 2000

Required Reading

• Chris Hecker has a few Gamasutra Articles on the

topic that are relatively easy reading:

– In particular, “Physics Part 4 – The 3rd Dimension” – Available http://chrishecker.com/Rigid_Body_Dynamics

• Baraff Witkin Kass Lecture 2001

– Rigid Body Dynamics Notes – Collision and Contact

Other Useful References

Erik Neumann’s MyPhysics Lab: http://www.myphysicslab.com/collision.html

Papers Referenced in this Lecture

[Moore & Wilhelms] Collision Detection and Response [Mirtich] Timewarp Rigid Body Simulation (http://www.merl.com/papers/docs/TR2000-17.pdf) [Mirtich] Impulse based Rigid Body Simulation

Final Project - Option

You may opt-out of your proposed animation/physics proposal by extending on your rigid body physics engine N.B. This won’t necessarily be easier You must notify me by March 12th that you wish to switch to this option

Rigid body engine (part 1 - basic requirements)

Implement a Rigid Body Engine for Non-convex Meshes, should include as many of the following as possible

Complete working Multi-phase Collision Detection: GJK, BVH , Sweep and Prune Optimized for performance where possible Should support polyhedra, particles, sphere, groundplane File loader for display (mesh + texture) and/or dynamics proxy Center of Mass & Inertial Tensor computation (may be approximate) Non-Euler Integration Quaternions for rotation Visualisation Mode (exposing some of the intermediate variables) Scene definition not hard coded

Rigid body engine (part 2 – extensions)

KEY Investigative Objectives (at least one of the following to be encorporated):

Multiple contact collision response or reasonably stable sequential response [] Constraints [] Friction [] Space-time Bounds / Continuous Collision detection Other equivalent (please specify) Demo + Youtube Video, Proper Documentation and Comments, 4 page report detailing extensions to lab work in weeks 1-7

Multiple point contact resolution

How to deal with multiple contacts? Many Physics tutorials tend to gloss over this point but it is quite an important one. Some Possible Solutions:

Average contact points and solve single net impulse

Not robust - only valid for convex regions

Micro-impulses: approximate an earliest collision and solve each point independently

not a complete solution (we’ll see why soon) and doesn’t guarantee robustness however you could get away with it

Account for influences of each point and solve simultaneously

Multiple Colliding Contacts

Single point collisions between two bodies, A & B At time of collision tc : pA(tc) = pB(tc) = p For any point i in contact between bodies A and B Our assumption for each contacting point on each body:

pi influences pi+n which influences pi

( ) ( ) ( )

i i

A c B c i c

p t p t p t  

C

Multiple contacts

Sequential impulse solution:

Apply impulse at p0

Body moves to the right penetrating into C

Apply impulse at p1

Body moves towards left penetrating into B

Looking at just the linear velocity we require either that…

A must stop A must bounce upwards

Angular velocity:

B C A

p0 p1

B A

p0 p1

B C A

p0 p1

Symmetrical case w=0 A rotates clockwise

Avoiding Interpenetration

Body B must exert a force C on body A at the point of

• contact. Such force must satisfy the following:

1. C acts only at the instant of contact, not before not after 2. C must be a repulsive force. Cannot act like glue between bodies 3. C must prevent the interpenetration of the bodies 4. The system cannot gain kinetic energy from the introduction of this force

Use impulse equation j previously defined j >= 0 at point of contact

j

rel rel

 

 

) ( v v 

( ) ( )

rel c rel c

v t v t 

 

 

• Assume coefficient of restitution relates to

each collision point individually

• Bodies may be pushed away from each
• ther by a third more powerful simultaneous

collision

( ) ( )

rel c rel c

v t v t 

 

 

Rewrite Newton’s restitution law to account for this :

Multiple Contacts

• Newtons law of Restitution for frictionless collisions :
• Doesn’t readily extend to multiple point collisions
• need “looser” assumptions

( ) ( )

rel c rel c

v t v t 

 

 

OR

Using Reformulate post collision velocity equation to account for simultaneous points of contact Remembering Newton’s third law and substituting gives us relative velocity at point i in the normal direction

1 1 1

( )

j j j i i i

n n j j j j

f n I r f r p p n M

            

    

 

i i i

v p p

  

  

 

 

      

r J r I M J v     



) (

1

n ˆ f J A 

N.B. We’re using the symbol f instead of j for impulse magnitude, to avoid confusion with the subscript j

( ) ( )

rel c rel c

v t v t 

 

 

Using: and substituting from (1 )

i

i rel i

b v 



  We get: Or: Which can be expressed as

Aside: Linear Systems of Equations

Often occur in Physics e.g. simultaneous contact LCP, deformable bodies Simultaneous equations of the form: Can be written Ax = b where Outside the scope of this course but if you need to read up on this, see:

“Linear Algebra”, J. Hefferon, available online: http://joshua.smcvt.edu/linearalgebra/

Generic LCP

So Using We have formulated what is known as a Linear Complementarity Problem (LCP) Formally stated:

( ) ( )

rel c rel c

v t v t 

 

 

( )

c

f t 

( )( ( ) ( ))

c rel c rel c

f t v t v t 

 

 

Given a known vector b and a known matrix the problem is to find a vector such that

w Af b   

f 

• r to show that no such vector exists.

So we solve the system The vector f in our case is simply the list of unknown impulse magnitudes f which is to be solved for Last formulation fed into LCP solver and solved for fi Calculated impulse magnitudes then applied to rigid body states to perturb initial non collision motion Note, impulses applied so as to change rigid body momentums

Simultaneous Contact Resolution

Collision Detection: Calculate all contact points for current rigid body state Contact points and rigid body states are used to compute the matrix A and vector d LCP Solver computes f and d. We are guaranteed no- interpenetration since d>= 0 Post-impulse velocities calculated from f and normal vectors at collision points ODE solver computes new positions and orientations of rigid bodies

Simultaneous Contact Resolution

Disadvantages:

Specialised LCP solver needed Addition of friction causes A matrix to become non Positive Semi Definite (PSD) (a solution for the LCP may not exist)

Advantages:

For frictionless contacts/collisions the A matrix is always PSD so a solution will always exists and is solvable within theoretical polynomial time LCPs are closely related to Quadratic Programming (QP) problems so it is possible to solve a PSD LCP with well established QP methods

LCP Solvers

LCP is a well studied problem for discontinuous relationship between two variables e.g. contact conditions between rigid bodies. Four main solutions:

Newton Methods

Quadratic convergence, O(n3) work per iteration

Interior point methods

Quadratic convergence

Direct methods (pivoting methods)

Work per iteration O(n2) Lemke O(n3)

Iterative methods

Iterative Solvers

Current trend is to use iterative methods (a.k.a projection methods) for solving LCPs

Used in Novodex, ODE, IO Interactive

trade off between accuracy and speed

Linear convergence Work per iteration O(n), if upper bound is placed on iterations Worse convergence but can 5-10 iterations

Strategy for LCPs: From Ax = b introduce a splitting

A = M – N so that we have (M-N) x = b and formulate as

Where the projection operator written (x)+ :

 

  

  ) (

1 1 k k

Nx b M x

) ), , max(min(

j lo hi j j

j

x x x x 



See [Murty]

LCP Solvers

PATH from CPNET by Michael Ferris is robust Newton based Solver:

http://www.cs.wisc.edu/cpnet/cpnetsoftware/

David Eberley “Game Physics” has an LCP solver implementation based on Lemke’s algorithm (pivoting method) ODE contains a solver using Successive-Order-Relaxation LCP Solver, Dantzig Pivoting algorithm and Conjugate Gradient

Related problem: Contact Forces

Forces between objects in physical contact but not colliding i.e. vREL >= 0 Resting contact: objects touching but not moving towards each other Accelerating towards each

• ther but kept in check by

an equal and opposite reaction force

Contact Forces

To avoid interpenetration when acceleration of body A is into body B, B must exert a contact force on Body A:

C must prevent interpenetration of the bodies C must be a repulsive force – cannot act like glue between

• bjects

C must become zero when bodies separate

Once again various solutions:

Springs at contacts, Impulses (microimpulses), Analytical solution, LCP

References

Katta G. Murty. Linear Complementarity, Linear and Nonlinear Programming. Helderman-Verlag, 1988.

Web version: http://ioe.engin.umich.edu/people/fac/books/murty/linear_complementarity_webbook/

Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone. The Linear Complementarity Problem. Academic Press, Inc., 1992. David Eberly “Game Physics” Chapter 7 Linear Complementarity and Mathematical Programming Chris Hecker, “The Mixed Linear Complementarity Problem:

http://chrishecker.com/The_Mixed_Linear_Complementarity_Problem

Thanh Giang “Complementarity Based Multiple Point Collision Resolution”

http://isg.cs.tcd.ie/cosulliv/Pubs/GiangEgirl03.pdf

Kenny Erleben. Stable, Robust, and Versatile Multibody Dynamics Animation. PhD thesis, Department of Computer Science, University of Copenhagen (DIKU), Universitetsparken 1, DK-2100 Copenhagen, Denmark, March 2005. ftp://ftp.diku.dk/diku/image/publications/erleben.050401.pdf. Contact Online Demo and tutorial

http://www.myphysicslab.com/contact.html