Game Physics
References:
PBM 2001
Advanced character dynamics
Content
Newton 2nd law
Euler’s method
Collision detection & response
Particle system data structure
Verlet integration
Particles with constraints
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Constraint dynamics
Point based dynamics
“character animation paper” (ragdoll, hitman)
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Basics
Newton 2nd Law
f  ma
Equation of Motion of a Newtonian Particle
x  m1 f
Second order ODE requires two conditions
Initial value problem: give x(0) and v(0)
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Euler’s Method
x  v
v  m1 f
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 x   v 
 v    1 f 
  m 
Euler’s Method
dx
 f x 
dt
x  f x t
xt  t   xt   f x t
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Point-Plane Collision Detection
Collision might occur if
n
x  p nˆ  
x
p
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
v
Close enough
and
v  nˆ  0
Head toward
plane
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Point-Plane Collision Response
x
n
p
vN
vT
frictionless
v

x :
CR: Coefficient of Restitution
0  CR  1
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move x onto the plane
v  v T c R v N
 v  v  nˆ nˆ   cR v  nˆ nˆ
 v  1  cR v  nˆ nˆ
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Case Study: rocket
Gravity + boosters
Time-based
animation
Euler’s method
Collision detection
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Case: Sprite animation
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Case Study: fly
Fake physics for aircraft motion control
Particle based dynamics
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Particle System
Data Structure
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Force Objects
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Case Study
Particle system
Damped spring
“Mouse spring”
manipulation
Gravity + wall
collisions
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Spring with Geometry Shader
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Verlet Integration (1, 2, 3)
Numerical method especially for
Loup Verlet
molecular dynamics simulations (loo vuhr-LEH)
Offer greater stability than Euler
integration
Developed by French physicist Loup
Verlet in 1967
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Derivation
Taylor’s Expansion
(1)
(2)
(1) + (2) :
(1) – (2) :
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Formula
Euler Integration
Verlet Integration
Velocity is implicit
Verlet Integration (damped version: e.g., air resistance in games)
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f: Percentage loss of
velocity due to friction
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Implementation
Starting… xt 0t   xt0   vt0 t  at 2
Iterating … xt 0t   2xt0   xt 0t   at 2
x new  2 x  xold  at 2
xold  x
x  xnew
st ate variable: x
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st atic variable: xold
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Does not really work …
Collision Response in Verlet Integration
xold
n
xnew
x
p
x’old v
N
vT
v
Find the mirror image of xold
w.r.t. the obstacle
Continue Verlet integration
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Reflection Matrix
Q  I  2uuT


Qu  I  2uuT u
 
u
 u  2u u T u
 u  2u  u

  I  2nnT xold  p  p
xold
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Regarding (basic) Verlet Collision
From:
here
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Verlet & Velocity Verlet1
2

x(t  h)  x(t )  hx (t )  h
x(t  h)  x(t )  hx(t )  h
x ( t )
2
2 x ( t )
2
3 x ( t )
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 O(h )
3 x ( t )
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 O(h 4 )
h
h
4
x(t  h)  x(t  h)  2 x(t )  h 2 x(t )  O(h 4 )
x(t  h)  2 x(t )  x(t  h)  h 2 x(t )  O(h 4 )
v(t )  x(t  h)  x(t  h)  / 2h  O(h 2 )
Basic Verlet
Position error O(h4)
Velocity error O(h2)
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Verlet & Velocity Verlet2
x(t  h)  x(t )  hx(t )  h 2
x(t  h)  x(t )  hv(t )  h 2
x ( t )
2
x ( t )
2
 O(h 3 )
 O(h 3 )
Good for velocity limiting
and collision response
v(t  h)  v(t )  hf (t )  O(h 2 )
(from t to t+h)
v(t )  v(t  h)  hf (t  h)  O(h 2 )
(from t+h to t)
 v(t  h)  v(t )  hf (t  h)  O(h 2 )
f (t )  f (t  h)
v(t  h)  v(t )  h
 O(h 2 )
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Velocity Verlet
Position error O(h3)
Velocity error O(h2)
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Implementation (Wiki)
System Implementation
State_x(3n), state_v (3n), state_a(3n)
First, update state_x from (1)
Derive state_a with (2)
Update state_v with (3)
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Comparison (free fall and rebound)
Exact
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Verlet
Euler Velocity Verlet
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Ragdoll Physics
Ref: advanced character dynamics (url)
Youtube videos
Physics for Games
Accuracy not really the primary concern
Goals:
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believability (stability), and
speed of execution
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Techniques used in Hitman
Particle/stick configuration to represent human anatomy
Verlet integration
Simple constraint solver
Square root approximation (next page)
Collision engine for penetration depth
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Square Root Approximation
Newton’s method with initial guess r.
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Handling Constraints (ref)
Containment (boundary)
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If out of bound, move the point to its
projection on the boundary
Concave boundary: more complicated
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Particle-Distance Constraint
From wikipedia
Remark:
x1new  x1old  delta* 0.5 * diff
x2 new  x2 old  delta* 0.5 * diff
x2 new  x1new  x2 old  x1old  delta* diff
 delta  delta* diff
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 deltalen restlen 
 1 
delta
deltalen


restlen
x2old  x1old   restlen x2old  x1old 

deltalen
x2old  x1old
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Moving on …
Connect sticks to form rigid bodies
Connect rigid bodies to form articulated
objects
Joint limit: one-sided stick constraint
x2  x1  l
Or dot product
constraints
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Extension (ref)
cloth, hair, rigid body, …
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Physics Engines
Box2D
Bullet
ODE
Havok
PhyX
Fall 2012
In-depth look at
Box2D Lite
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