Transcript Dynamics 101 - Essential Math for Games Programmers

Dynamics 101
Marq Singer
Red Storm Entertainment
[email protected]
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Talk Summary
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Going to talk about:
A brief history of motion theory
 Newtonian motion for linear and rotational
dynamics
 Handling this in the computer
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Physically Based-Motion
Want game objects to move consistent with
world
 Match our real-world experience
 But this is a game, so…
 Can’t be too expensive
 (no atomic-level interactions)
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History I: Aristotle
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Observed:
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From this, deduced:
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Push an object, stop, it stops
Rock falls faster than feather
Objects want to stop
Motion is in a line
Motion only occurs with action
Heavier object falls faster
Note: was not actually beggar for a bottle
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History I: Aristotle
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Motion as changing position
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History I: Aristotle
Called kinematics
 Games: move controller, stop on a dime,
move again
 Not realistic
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History II: Galileo
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Observed:
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Object in motion slows down
Cannonballs fall equally
Theorized:
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Slows due to unseen force: friction
Object in motion stays in motion
Object at rest stays at rest
Called inertia
Also: force changes velocity, not position
Oh, and mass has no effect on velocity
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History II: Galileo
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Force as changing velocity
Velocity changes position
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Called dynamics
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History III: Newton
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Observed:
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Planet orbit like continuous falling
Theorized:
Planet moves via gravity
 Planets and small objects linked
 Force related to velocity by mass
 Calculus helps formulate it all
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History III: Newton
Sum of forces sets acceleration
 Acceleration changes velocity
 Velocity changes position
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g
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History III: Newton
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Games: Move controller, add force, then
drift
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History III: Newton
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As mentioned, devised calculus
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Differential calculus:
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rates of change
Integral calculus:
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(concurrent with Leibniz)
areas and volumes
antiderivatives
Did not invent the Fig Newton™
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Differential Calculus Review
Have position function x(t)
 Derivative x'(t) describes how x changes
as t changes (also written dx/dt, or x )
 x'(t) gives tangent vector at time t
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x(ti)
y
y(t)
x'(ti)
t
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Differential Calculus Review
Our function is position:x(t )
 Derivative is velocity:
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dx
v(t )  x(t ) 
 x
dt
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Derivative of velocity is acceleration
dv
 v
dt
d 2x
 x(t )  2  x
dt
a(t )  v(t ) 
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Newtonian Dynamics
Summary
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All objects affected by forces
Gravity
 Ground (pushing up)
 Other objects pushing against it
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Force determines acceleration (F = ma)
dv
 a)
 Acceleration changes velocity (
dx dt
 Velocity changes position (
v )

dt
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Dynamics on Computer
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Break into two parts
Linear dynamics (position)
 Rotational dynamics (orientation)
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Simpler to start with position
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Linear Dynamics
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Simulating a single object with:
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Last frame position xi
Last frame velocity vi
Mass m
Sum of forces F
Want to know
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Current frame position xi+1
Current frame velocity vi+1
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Linear Dynamics
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Could use Newton’s equations
a  F /m
x i1  x i  v i t  1/2at 2
v i1  v i  at
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Problem: assumes F constant across frame
Not always true:
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E.g. spring force: Fspring = –kx
E.g. drag force: Fdrag = –mv
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Linear Dynamics
Need numeric solution
 Take stepwise approximation of function
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Linear Dynamics
Basic idea: derivative (velocity) is going
in the right direction
 Step a little way in that direction (scaled
by frame time h)
 Do same with velocity/acceleration
 Called Euler’s method
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Linear Dynamics
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Euler’s method
x i1  x i  hv i
a  F(x i ,v i ) /m
v i1  v i  ha
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Linear Dynamics
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Another way: use linear momentum
P  mv
P  F
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Then
x i1  x i  hvi
Pi1  Pi  hF(x i ,v i )
vi1  Pi1 /m
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Linear: Final Formulas
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Using Euler’s method with time step h
F   Fk (x i ,v i )
k
F
ai 
m
x i1  x i  hv i
v i1  v i  ha i
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Rotational Dynamics
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Simulating a single object with:
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Last frame orientation Ri or qi
Last frame angular velocity i
Inertial tensor I
Sum of torques 
Want to know
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Current frame orientation Ri+1 or qi+1
Current frame ang. velocity i+1
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Rotational Dynamics
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Orientation
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Represented by
Rotation matrix R
 Quaternion q
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Which depends on your needs
 Hint: quaternions are cheaper
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Rotational Dynamics
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Angular velocity
Represents change in rotation
 How fast object spinning
 3-vector
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Direction is axis of rotation
 Length is amount of rotation (in radians)
 Ccw around axis (r.h. rule)
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Rotational Dynamics
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Angular velocity
Often need to know linear velocity at point
 Solution: cross product
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 r  v
v
r
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Moments of Inertia
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Inertial tensor
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I is rotational equivalent of mass
3 x 3 matrix, not single scalar factor (unlike m)
Many factors - rotation depends on shape
Describe how object rotates around various axes
Not always easy to compute
Change as object changes orientation
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Rotational Dynamics
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Computing I
Can use values for closest box or cylinder
 Alternatively, can compute based on
geometry
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Assume constant density, constant mass at each
vertex
 Solid integral across shape
 See Mirtich,Eberly for more details
 Blow and Melax do it with sums of tetrahedra
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Rotational Dynamics
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Torque
Force equivalent
 Apply to offset from center of mass – creates
rotation
 Add up torques just like forces
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Rotational Dynamics
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Computing torque
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Cross product of vector r (from CoM to point
where force is applied), and force vector F
r
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F
 r F
Applies torque ccw around vector (r.h. rule)
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Rotational Dynamics
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Center of Mass
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Point on body where applying a force acts just like
single particle
“Balance point” of object
Varies with density, shape of object
Pull/push anywhere but CoM, get torque
Generally falls out of inertial tensor calculation
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Rotational Dynamics
Have matrix R and vector 
 How to compute Ri1  Ri  hi ?
 Convert  to give change in R
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Convert to symmetric skew matrix
 Multiply by orientation matrix
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 Can use Euler's method after that
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Computing New Orientation
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If have matrix R, then
˜ iRi
Ri1  Ri  h
where

 0

˜   3


 2
 3
0
1
 2 

1 

0 
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Computing New Orientation
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If have quaternion q, then
h
q i 1  q i  w i q i
2
where
wi  (0,i )
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See Baraff or Eberly for derivation
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Computing Angular Velocity
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Can’t easily integrate angular velocity from
angular acceleration:
  I
 I    I
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Can no longer “divide” by I and do Euler step
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Computing Angular
Momentum
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Easier way: use angular momentum
L  I
 
L
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Then
L i1  L i  h (Ri , i )
 i1  I L i1
1
i1
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Using I in World Space
Remember,   I1L
 I computed in local space, must
transform to world space
 If using rotation matrix R, use formula
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1
i
1
i 0
I Li  R I R Li
T
i
If using quaternion, convert to matrix
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Rotational Formulas
   rk  Fk
k
~
R i 1  R n  hi R i
L i 1  L i  h
I
1
i 1
1
i 1 0
R I R
i 1  I i11L i 1
T
i 1
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Impulses
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Normally force acts over period of time
F
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E.g., pushing a chair
t
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Impulses
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Even if constant over frame
F
sim assumes application over entire time
t
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Impulses
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But if instantaneous change in velocity?
Discontinuity!
F
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t
Still force, just instantaneous
Called impulse - good for collisions/constraints
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Summary
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Basic Newtonian dynamics
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Linear simulation
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Position, velocity, force, momentum
Force -> acceleration -> velocity -> position
Rotational simulation
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Torque -> ang. mom. -> ang. vel. ->
orientation
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Questions?
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References
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Burden, Richard L. and J. Douglas Faires, Numerical
Analysis, PWS Publishing Company, Boston, MA,
1993.
Hecker, Chris, “Behind the Screen,” Game Developer,
Miller Freeman, San Francisco, Dec. 1996-Jun. 1997.
Witken, Andrew, David Baraff, Michael Kass,
SIGGRAPH Course Notes, Physically Based Modelling,
SIGGRAPH 2002.
Eberly, David, Game Physics, Morgan Kaufmann,
2003.