Simulation and Animation

computer graphics & visualization

Simulation and Animation

Rigid Body Simulation


Simulation and Animation – SS07Jens Krüger – Computer Graphics and Visualization Group

The next few lectures…

• Point Dynamics • Rigid Bodies • Soft Bodies



Comming up…• P

oint Dynamics• no extend• only position

• Rigid Bodies

• extended

• positioned & oriented

• „a set of points“



Dynamic simulation • Movement of point masses, rigid bodies,

systems of point masses etc. with respect to– Forces – Body charcteristics (mass, shape)– Derivation of accelerations from properties and

physical laws

• Dynamic of point masses• Dynamic of rigid bodies



Dynamic simulation• Newtons Axioms

– Without external forces, a bodymoves uniformly (Inertia)

– An external force F applied to amass m results in an acceleration a:F = ma

– Actio = reactio



Now…• P

oint Dynamics• no extend• no orientation

• Rigid Bodies

• Extended

• Oriented




Point dynamics• Dynamic simulation of particles

– Position r, mass m, velocity v,acceleration a, but no extend

– Forces F act on particles

F,a,v,r are 3D vectors!!!

2

F Fv(t dt) v(t) dt v(t dt) v(t) dtm m

1 F 1r(t dt) r(t) v(t)dt dt r(t) v(t) v t dt dt2 m 2

)()( tt rv )()( tt va )(tamF



Point dynamics• Dynamic simulation of particles

– Multiple forces may acton particles

• Forces are added by vector addition

– F is usually a function of time– Mass might change as well

• Change of momentum (Impuls) with change of time

dtmvdF )(



Point dynamics• Particles have no internal structure

• 3 DOF = degrees of freedom (position only)• Direct kinematic: from a v r• Indirect kinematic:

from r and boundary conditions v a



Point dynamics• Relation between force F

and acceleration a

• Direct dynamics

mtt )()( Fa Point mass

0

0

0

0

)()()()(

)()()()(

)()(

rvrvr

vavav

Fa

tdtttt

tdtttt

mtt

t

t

t

t



Point dynamics• Important Forces

– Gravity

– Hooke's Law

– Friction

F mg

F ku

F v



Point dynamics • Linear momentum (Impuls)

– Force F act on center of mass

• Force

• Conservation of momentum

• Example: elastic push

vp m

pF

0p



Point dynamics• Angular momentum (Drall)

• Torque (Drehmoment)

• Conservation of angular momentum

)( rrprL m r

p

dT L (r p) r Fdt

0 FrL

Moment of inertia: I = mr2



Point dynamics• Angular velocity of a point (rate at which the point is rotating):

magnitude of change

• Notation:

0

||0000

rωrωrωrωr

0r||0r

0r

00

0mit **

xy

xz

yz

aaaa

aaababa

ω



Point dynamics• Analog for rotation matrix R - it changes under angular velocity

• Aggregate movement of a body point– (r0/r(t): position in local/world space coordinate system)

RωωωωR *)()()(

zz

zy

zx

yz

yy

yx

xz

xy

xx

RRR

tRRR

tRRR

t

)()()()(

)()()(

)()()()(

0*

0*

ttttttt

tttt

CMCM

CMCMCM

CM

rrωvrrrRωv

rRωvr

CMtt rrRr 0)()(



Point dynamics• Relation between angular momentum

and angular velocity

• Inertia tensor I (Trägheitstensor)

L r p r (mr) mr r mr (ω r)Iω

2 2

2 2

2 2

m(r x ) m yx m zxI m xy m(r y ) m zy

m xz m yz m(r z )

Symmetrictensor

Example:Skater



Point dynamicsSummary

Translation RotationPosition r Angle velocity v Angular velocity Momentum p Angular momentum LForce F Torque T



Next…• P

oint Dynamics• no extend• no orientation

• Rigid Bodies

• Extended

• Oriented




Rigid body simulation• Idea

– Combination of many small particles to a rigid body– Bodies that do not deform – they are stiff– They do not penetrate– They bounce back if they collide– Rigid convex polyhedra of constant density– 6 DOF instead of 3n DOF (for n particles)

• Distinguish between – Movement of center of mass (CM)– Rotation around (CM)



Rigid body simulation• Mass M

in continuous case

imM

small, discrete mass points

3)( dVMV r Volume integral over

entire body

Mass density (= specific weight = Mass/Volume)



Rigid body simulation• CM (center of mass)

in continuous case

Mtm

t ii)(

)(CM

rr

3

VCM

(r, t) r(t)dVr (t)

M



Rigid body simulation• Dynamic simulation of rigid bodies

– Motion consist of translational and angular component

– Velocity v(t) is rate of change of position r(t) over time

• v(t) = r´(t) • v(t) is linear velocity at center of mass

– Bodies also have a spin• About an axis (vector) through the center of mass• Magnitude of the vector defines how fast the body is

spinning



Rigid body simulation• Translation and Rotation:

• Rotation R: – 3*3 Matrix– Redundancies– only 3 DOF

0)()()( rRrr ttt CM

CMlocal/fixedcoordinate system



Rigid body simulation

Translation Rotation



Rigid body simulation• Momentum

• Movement of CM

• Force Fext is sum of all external forces

CM CMp (t) Mr (t)

CM CM CM extp (t) Mr (t) Mv (t) F (t)

)()( iext,ext tt FF

CM CMv (t) r (t)

R

Fext,i

Fext



Rigid body simulation• Angular momentum

• Inertia tensor I

ωrωr

rrprL

I

m

m

iii

iiiii

))((

)(

2 2i i i i i i i i i



m (r x ) m y x m z xI m x y m (r y ) m z y

m x z m z x m (r z )



Rigid body simulation• Angular momentum

• Inertia tensor I in continuous case

ωL I

2 3jk jk j kV

I (r) (r r r )dV

Kronecker-symbol

x,y,z coordinates



• Translation vs. RotationMass Inertia Moment (Trägheitsmoment)

m Velocity Angular Velocity

v =dr/dt = d/dtMomentum Angular Momentum

p = mv L = I = r x pForce Torque

F = dp/dt T = r x F = dL/dt Kinetc Energy Kinetic Energy

E = ½ mv2 E = ½ I2


2

VI (r)r dr



Rigid body simulation• Properties of the Inertia tensor

– Diagonal elements are moments of inertia with respect to coordinate axes

– It is symmetric and real• Has three principal axis (eigenvectors)

• Eigenvectors are orthogonal: directions of inertia (Hauptträgheitsachsen)

• Eigenvalues are real: moments of inertia (Hauptträgheitsmomente)

i i i

i i i i i i

Ir I r

(I diag(I , I , I ))r 0 I diag(I , I , I ) 0 i 1,2,3



Rigid body simulation• Inertia tensor

– In this (directions of inertia) coordinate system, I can be diagonalized by RIRT, where R is a rotation matrix:

• Inertia moment (scalar) for rotation around axis n (normalized)

3

2

1

000000

II

II

Tik i kn

I n I n I n n



Rigid body simulation• Torque of single „body element“

• Total torque

• Important: starting point of force

• Equations of motion for rotation(Euler equations for fixed coordinate system)

iii ttt FrrT ))()(()( CM

ii ttt FrrT ))()(()( CM

TL



Rigid body simulation• State vector of a rigid body

• Constants:– Inertia tensor IKS

– Mass M

)()(

)()(

)(CM

CM

tt

tt

t

LpRr

X

Position

Orientation (rotation matrix)

Impuls

Angular momentum



Rigid body simulation• Derived variables

CMCM

TKS

1 1 TKS

1

p (t)v (t)M

I(t) R(t)I R(t)

I (t) R(t)I R(t)

ω(t) I(t) L(t)



Rigid body simulation• Equations of motion

)()(

)()()(

)()(

)()(

)(ext

*CM

CM

CM

tt

ttt

tt

tt

dtdt

TFRω

v

LpRr

X



Rigid body simulation• System of ordinary partial differential equations• Initial boundary problem• Structure

• In general, numeric solution (Integration)– Explicit solve: Euler, Runge-Kutta– Implicit solver

00 )())(,()(

xxxfx

tttt



Numerical Integration• Initial value problem:

• Simple approach: Euler

• Derivation: Taylor expansion

• First order scheme• Higher accuracy with smaller step size

),()( tt xfx

),()()( ttttt xfxx

)()()()( 2tOttttt xxx



Numerical Integration• Problems of Euler-Scheme

• Inaccurate

• Unstable

• Example:kt

f (x, t) kx

x(t) e

Divergenz für t > 2/k



Numerical Integration• Midpoint method:

1. Euler-Step

2. Evaluation of f at midpoint

3. Step with value at midpoint

• Second order scheme

),( tt xfx

2

,2mid

ttxxff

mid)( fx ttt



Numerical Integration• Fourth order Runge-Kutta

• Adaptive step size control

54321

34

23

12

1

6336)(

,2

,2

2,

2

),(

tOtt

ttt

ttt

ttt

tt

kkkkxx

kxfk

kxfk

kxfk

xfk



Numerical Integration• So far: Explicit techniques• Stable integration by means of implicit integration

schemes• Implicit Euler-Step

• „rewind“ the explicit Euler-Step• Taylor-expansion around t + t instead of t• Solving the non-linear system of equations for

x(t + t)

ttttftttt ),()()( xxx



Rigid body simulation• Demo



Rigid body simulation• Summary

Torque TForces Fext

Angular moment LImpuls pCM

Angular velocity Velocity vCM

Orientation R Position rCM

RotationTranslation

Inertia tensor Mass M

Simulation and Animation

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