Constrained Dynamics Marq Singer (marq@essentialmath.com)

Constrained Dynamics

Marq Singer (marq@essentialmath.com)

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The Problem

• What are they

• Why do we care

• What are they good for

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The Basics

• Constraint – something that keeps an entity in the system from moving freely

• For our purposes, we will treat each discreet entity as one particle in a system

• Particles can be doors on hinges, bones in a skeleton, points on a piece of cloth, etc.

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Box Constraints

• Simplest case

• Movement constrained within a 2D area

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Box Constraints

P

1000

1000

y

x

P

P1000

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Box Constraints (cont)

• Restrict P to extents of the box

• Recover from violations in position (last valid, rebound, wrap around)

• Simple, yet the basis for the rest of this

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Bead on a Wire

• The Problem: Restrict bead to path

• Solutions: Explicit (parametric)

method Implicit method

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Parametric Constraints

]sin,[cos rx

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Bead on a Wire

• From Baraff, Witkin

N = gradient

f = force

fc = constraint force

f ‘ = f + fc

Nf

cf'f

NNN

Nffc

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Implicit Representation

legal position

legal velocity

legal acceleration

N

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Implicit Representation

Constraint force = gradient vector times scalar

N

x

CN

0)( rxxC

Nfc

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Spring Constraints

• Seems like a reasonable choice for soft body dynamics (cloth)

• In practice, not very useful

• Unstable, quickly explodes

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Stiff Constraints

• A special spring case does work

• Ball and Stick/Tinkertoy

• Particles stay a fixed distance apart

• Basically an infinitely stiff spring

• Simple

• Not as prone to explode

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Cloth Simulation

• Use stiff springs

• Solving constraints by relaxation

• Solve with a linear system

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Cloth Simulation0P

1P

5P

1,0C 2,1C5,0C

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Cloth Simulation

• Forces on our cloth

m

tFtta

tatvttv

tvtpttp

ii

iii

iii

)()(

)()(

)()(

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Cloth Simulation

• Relaxation is simple

• Infinitely rigid springs are stable1. Predetermine Ci distance between particles

2. Apply forces (once per timestep)

3. Calculate for two particles

4. If move each particle half the distance

5. If n = 2, you’re done!

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Relaxation Methods0P

1P1,0C

201

2011,0 )()( yyxx PPPPC

1,001 )( CPPp

0P 1P

1,0C

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Relaxation Methods0P 1P

1,0C

1,0C0P 1P

2)()(,

2)()(:0 1100

ptPttP

ptPttPp

2)()(,

2)()(:0 1100

ptPttP

ptPttPp

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Cloth Simulation• When n > 2, each particle’s movement

influenced by multiple particles• Satisfying one constraint can

invalidate another• Multiple iterations stabilize system

converging to approximate constraints• Forces applied before iterations• Fixed timestep (critical)

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More Cloth Simulation

• Use less rigid constraints

• Vary the constraints in each direction (i.e. horizontal stronger than vertical)

• Warp and weft constraints

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Still More Cloth Simulation• Sheer Springs

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Still More Cloth Simulation• Flex Springs

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Articulated Bodies

• Pin Joints

• Hinges

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Angular Constraints

• Restrict the angle between particles

• Results in a cone-shaped constraint

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1x

2x

Angular Constraints

• Unilateral distance constraint

• Only apply constraint in one direction

10012 xx

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Angular Constraints

• Dot product constraint

• Recovery is a bit more involved

0102 xxxx

1x

2x

0x

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Stick Man

• Uses points and hinges

• Angular (not shown) allow realistic orientations

• Graphic example of why I’m an engineer and not an artist

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Using A Linear System

• Can sum up forces and constraints

• Represent as system of linear equations

• Solve using matrix methods

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Basic Stuff

Systems of linear equations

Where:

A = matrix of coefficients

x = column vector of variables

b = column vector of solutions

bAx

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Basic Stuff• Populating matricies is a bit tricky, see [Boxerman] for a good example

Isolating the ith equation:

i

n

j

jij bxa 1

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Jacobi Iteration

Solve for xi (assume other entries in x unchanged):

ii

ij

kjiji

ki a

xabx

)1(

)(

(Which is basically what we did a few slides back)

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Jacobi Iteration

In matrix form:

bDxULDx kk 1)1(1)( )(

D, -L, -U are subparts of A

D = diagonal

-L = strictly lower triangular

-U = strictly upper triangular

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Jacobi IterationDefinition (diagonal, strictly lower, strictly upper):

A = D - L - U

DLLL

UDLL

UUDL

UUUD

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Gauss-Seidel Iteration

Uses previous results as they are available

ii

ij ij

kjij

kjiji

ki a

xaxabx

)1()(

)(

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Gauss-Seidel Iteration

In matrix form:

)()( )1(1)( bUxLDx kk

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Gauss-Seidel Iteration

• Components depend on previously computed components

• Cannot solve simultaneously (unlike Jacobi)

• Order dependant

• If order changes the components of new iterates change

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Successive Over Relaxation (SOR)

• Gauss-Seidel has convergence problems

• SOR is a modification of Gauss-Seidel

• Add a parameter to G-S

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Successive Over Relaxation (SOR)

• = a Gauss-Seidel iterate

• 0 < • If = 1, simplifies to plain old Gauss-

Seidel

)1()()( )1( ki

ki

ki xxx

x

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Gauss-Seidel Iteration

In matrix form:

bLD

xDULDx kk

1

)1(1)(

)(

])1([)(

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Lots More Math(not covered here)

• I highly recommend [Shewchuk 1994]

• Steepest Descent

• Conjugate Gradient

• Newton’s Method (in some cases)

• Hessian

• Newton variants (Discreet, Quasi, Truncated)

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References

• Boxerman, Eddy and Ascher, Uri, Decomposing Cloth, Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2004)

• Eberly, David, Game Physics, Morgan Kaufmann, 2003.• Jakobsen, Thomas, Advanced Character Physics, Gamasutra Game

Physics Resource Guide • Mathews, John H. and Fink, Kurtis K., Numerical Methods Using Matlab,

4th Edition, Prentice-Hall 2004• Shewchuk, Jonathan Richard, An Introduction to the Conjugate Gradient

Method Without the Agonizing Pain, August 1994. http://www-2.cs.cmu.edu/~jrs/jrspapers.html

• Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modeling, SIGGRAPH 2002.

• Yu, David, The Physics That Brought Cel Damage to Life: A Case Study, GDC 2002