Simulation and Animation
Rigid Body Simulation
computer graphics & visualization
The next few lectures…
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Comming up…
Point Dynamics
• no extend
• only position
Rigid Bodies
• extended
• positioned & oriented
• „a set of points“
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
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
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Dynamic simulation
• Newtons Axioms
– Without external forces, a body
moves uniformly (Inertia)
– An external force F applied to a
mass m results in an acceleration a:
F = ma
– Actio = reactio
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Now…
Point Dynamics
• no extend
• no orientation
Rigid Bodies
• Extended
• Oriented
• „a set of points“
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Dynamic simulation of particles
– Position r, mass m, velocity v,
acceleration a, but no extend
– Forces F act on particles
v(t )  r(t )
a(t )  v (t ) F  m  a(t )
F,a,v,r are 3D vectors!!!
v(t  dt)  v(t) 
F
dt
m
v(t  dt)  v(t) 
F
dt
m
1 F 2
1
r(t  dt)  r(t)  v(t)dt 
dt  r(t)   v(t)  v  t  dt  dt 
2m
2
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Dynamic simulation of particles
– Multiple forces may act
on 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
d (mv)
F
dt
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
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
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Relation between force F
and acceleration a
a(t ) 
F(t )
m
Point mass
• Direct dynamics
a(t ) 
F(t )
m
t
v (t )  a(t )

v(t )   a(t ) dt   v 0
t0
t
r(t )  v(t )

r (t )   v(t ) dt   r0
t0
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Important Forces
– Gravity
F  mg
– Hooke's Law
F  ku
– Friction
F  v
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Linear momentum (Impuls)
– Force F act on center of mass
p  mv
• Force
F  p
• Conservation of momentum
p  0
• Example: elastic push
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Angular momentum (Drall)
p
L  r  p  r  (mr )
• Torque (Drehmoment)
TL
r
Moment of inertia: I = mr2
d
(r  p)  r  F
dt
• Conservation of angular momentum
L  r  F  0
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Angular velocity of a point (rate at which the point is rotating):
r0  ω  r0  ω  r0  ω  r0||
 ω  r0
ω
magnitude of change


r0
• Notation:
r0||
a  b  a *b
mit
 0

*
a   az
 a
 y
 az
0
ax
ay 

 ax 
0 
r0
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Analog for rotation matrix R - it changes under angular velocity

 Ryx 
 Rxx 
 Rzx  







R   ω(t )   Rxy  ω(t )   Ryy  ω(t )   Rzy    ω *R

R 
R 
 R  
 xz 
 zz  
 yz 

• Aggregate movement of a body point
– (r0/r(t): position in local/world space coordinate system)
r(t )  R(t )r0  rCM
r (t )  v CM (t )  ω(t )* R (t )r0
 v CM (t )  ω(t )* R (t )r0  rCM  rCM 
 v CM (t )  ω(t )  r (t )  rCM (t ) 
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
• Relation between angular momentum
and angular velocity
L  r  p  r  (mr)  mr  r  mr  (ω  r)
 Iω
• Inertia tensor I (Trägheitstensor)
 m(r 2  x 2 )
m yx
m zx 


I    m xy
m(r 2  y 2 )
m zy 
2
2 
  m xz

m
yz
m(r

z
)

Example:
Skater
Symmetric
tensor
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Point dynamics
Summary
Translation
Rotation
Position r
Angle 
velocity v
Angular velocity 
Momentum p
Angular momentum L
Force F
Torque T
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Next…
Point Dynamics
• no extend
• no orientation
Rigid Bodies
• Extended
• Oriented
• „a set of points“
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
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)
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Mass M
M   mi
small, discrete mass points
in continuous case
M    (r ) dV 3
V
Volume integral over
entire body
Mass density (= specific weight
= Mass/Volume)
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• CM (center of mass)
 m r (t )
rCM (t ) 
i i
M
in continuous case
rCM (t) 

V
(r, t) r(t) dV3
M
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
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
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Translation and Rotation:
r(t )  rCM (t )  R(t )r0
CM
local/fixed
coordinate system
• Rotation R:
– 3*3 Matrix
– Redundancies
– only 3 DOF
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
Translation
Rotation
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Momentum
pCM (t)  MrCM (t)
• Movement of CM
vCM (t)  rCM (t)
pCM (t)  MrCM (t)  MvCM (t)  Fext (t)
• Force Fext is sum of all external forces
Fext (t )  Fext,i (t )
Fext,i
R
Fext
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Angular momentum
L   ri  p i   ri  (mi ri )
  mi (ri  (ω  ri ))
 Iω
• Inertia tensor I
  mi (ri 2  x i 2 )

I    mi x i yi
  mi x i z i

 mi yi x i
2
2
m
(r

y
 i i i)
 m i z i x i
 mi z i x i 

 m i z i y i 
2
2 
m
(r

z
 i i i )
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Angular momentum
L  Iω
• Inertia tensor I in continuous case
I jk   (r) (r 2 jk  rjrk ) dV3
V
x,y,z coordinates
Kronecker-symbol
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Translation vs. Rotation
Mass
Inertia Moment (Trägheitsmoment)
I   (r)r 2dr
m
V
Velocity
v =dr/dt
Momentum
p = mv
Force
F = dp/dt
Angular Velocity
 = d/dt
Angular Momentum
L = I = r x p
Torque
T = r x F = dL/dt
Kinetc Energy
Kinetic Energy
E = ½ mv2
E = ½ I2
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
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)
Iri  Ii ri
(I  diag(Ii , Ii , Ii ))r  0  I  diag(Ii , Ii , Ii )  0  i  1, 2,3
• Eigenvectors are orthogonal: directions of inertia
(Hauptträgheitsachsen)
• Eigenvalues are real: moments of inertia
(Hauptträgheitsmomente)
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Inertia tensor
– In this (directions of inertia) coordinate system, I can be diagonalized by
RIRT, where R is a rotation matrix:
 I1

I 0
0

0
I2
0
0

0
I 3 
• Inertia moment (scalar) for rotation around axis n (normalized)
I n  n T  I  n   Iik ni n k
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Torque of single „body element“
Ti (t )  (ri (t )  rCM(t ))  Fi
• Total torque
T(t )  (ri (t )  rCM(t ))  Fi
• Important:
starting point of force
• Equations of motion for rotation
(Euler equations for fixed coordinate system)
L  T
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• State vector of a rigid body
 rCM (t ) 


 R(t ) 
X( t )  
pCM (t ) 


 L(t ) 


Position
Orientation (rotation matrix)
Impuls
Angular momentum
• Constants:
– Inertia tensor IKS
– Mass M
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Derived variables
p CM (t)
v CM (t) 
M
I(t)  R(t)I KSR(t) T
I 1 (t)  R(t)I KS1 R(t) T
ω(t)  I(t) 1 L(t)
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Equations of motion
 rCM (t )   v CM (t ) 

 

*
 (t )  d  R(t )    ω(t ) R(t ) 
X
dt  p CM (t )   Fext (t ) 

 

 L(t )   T(t ) 

 

Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• System of ordinary partial differential equations
• Initial boundary problem
• Structure x (t )  f (t , x(t ))
x(t 0 )  x 0
• In general, numeric solution (Integration)
– Explicit solve: Euler, Runge-Kutta
– Implicit solver
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Numerical Integration
• Initial value problem:
x (t )  f ( x, t )
• Simple approach: Euler
x(t  t )  x(t )  t f ( x, t )
• Derivation: Taylor expansion
x(t  t )  x(t )  t x (t )  O( t 2 )
• First order scheme
• Higher accuracy with smaller step size
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Numerical Integration
• Problems of Euler-Scheme
• Inaccurate
• Unstable
• Example: f (x, t)  kx
x(t)  e kt
Divergenz für t > 2/k
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Numerical Integration
• Midpoint method:
1. Euler-Step
x  t f ( x, t )
2. Evaluation of f at midpoint
x
t 

fmid  f  x 
,t  
2
2

3. Step with value at midpoint
x(t  t )  t fmid
• Second order scheme
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Numerical Integration
• Fourth order Runge-Kutta
k1  t f ( x, t )
k
t 

k 2  t f  x  1 , t  
2
2

k
t 

k 3  t f  x  2 , t  
2
2

k 4  t f x  k 3 , t  t 
x(t  t )  x 
 
k1 k 2 k 3 k 4



 O t 5
6 3
3
6
• Adaptive step size control
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Numerical Integration
• So far: Explicit techniques
• Stable integration by means of implicit integration
schemes
• Implicit Euler-Step
x(t  t )  x(t )  t f x(t  t ), t  t 
• „rewind“ the explicit Euler-Step
• Taylor-expansion around t + t instead of t
• Solving the non-linear system of equations for
x(t + t)
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Demo
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization
Rigid body simulation
• Summary
Translation
Rotation
Position rCM
Orientation R
Velocity vCM
Angular velocity 
Impuls pCM
Angular moment L
Forces Fext
Torque T
Mass M
Inertia tensor 
Simulation and Animation – SS07
Jens Krüger – Computer Graphics and Visualization Group
computer graphics & visualization