CS 231 Animation Using Dynamic Simulationvbz/cs231f15/lecture07-15.pdf · What is dynamics for...
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CS 231 Animation Using Dynamic Simulation
Transcript of CS 231 Animation Using Dynamic Simulationvbz/cs231f15/lecture07-15.pdf · What is dynamics for...
CS 231
Animation Using Dynamic Simulation
What is dynamics for animation?
forward
dynamics
integrator
graphics
state
Loosely, using physical rules
or principles to animate, a
procedural approach
Usually, the use of a forward
dynamics model and numeric
routine to compute motion
Got its start in engineering
simulation and robotics
Forward Dynamics vs. Inverse Dynamics
force and
torque
acceleration,
velocity
forward
inverse
Forward: given forces and torques, compute
acceleration, velocity, and finally position
Inverse: given motion (positions), compute the
forces and torques
Why use forward dynamics?
Captures realistic movements because it is
based on physical constraints
Can provide high-level control over
animation
Can allow principled methods for animated
elements to communicate – leading to
rich simulated worlds
AKA: initial value problem
Types of simulations
Passive
Particle systems
Spring/Mass systems
Rigid bodies
Active
Articulated rigid bodies - linkages
Passive vs Active dynamics
model of
dynamics
integrator
graphics
control
user
inputs
model of
dynamics
integrator
graphicsuser
inputs
Passive
Active
Maya, etc support passive dynamics
Particles, deformations (cloth), rigid bodies
Gravity, collisions, friction
Spring-mass systems
2D meshes = cloth
3D meshes = jello,blobs
Particle (Mass) System
F(t) = ma
x’’ = F(t)/m - One second order equation
mtF
tv
tv
tx
dt
dY
dt
d
/)(
)(
)(
)(
or:
v’ = F(t)/m - Two first order equations
x’ = v
Particle (Mass) System
Use Euler Integration to update:
1) Find New Velocity:
2) Find New Position: tttvtxttx )()()(
tm
tFtvttv
)()()(
Gravity
(mg)
Particle (mass) system
Gravity
External Force
Particle (mass) system
xtktxf )(),(
Example:
Vector field
The differential equation
defines a vector field
over x
),( txfx
Differential equation – visual review
Initial value problem
Given the starting point,
follow the integral curve
Euler integration
),()()( txtftxttx
Simplest numerical solution method
Discrete time steps
Bigger steps, bigger errors
Inaccuracy
Error turns this x(t),
a circle into different
spirals defined by
step size
Inaccuracy
Inaccuracy can lead to
instability
Midpoint method
a) Compute a “full” Euler step
b) Evaluate f at midpoint, fmid
c) Take a step from the original point
using the midpoint f value:
midtftxttx )()(
Approximation of Taylor Series
Many kinds of integrators
...!3
)()(
!2
)()(
!1
)()()()(
32
ax
afax
afax
afafxf
Further rolling out of the Taylor Series leads to:
Fourth Order – Runga-Kutta Integration
Many kinds of integrators
))(,(
)2
1)(,
2(
)2
1)(,
2(
))(,(
)22(6
)()(
34
23
12
1
4321
tktxttfk
tktxt
tfk
tktxt
tfk
txtfk
kkkkt
txttx
Leapfrog integrator
Instead of
Leapfrog splits as
Many kinds of integrators
mtF
tv
tv
tx
dt
dY
dt
d
/)(
)(
)(
)(
tvxx iii 2/11
tavv iii )2/1()2/1(
)2/1()2/1( ii vvAt i+=1
Many kinds of integrators
Adaptive Steps
Take a full step and two half steps from starting
point
Compare results
If difference is greater than threshold, then reduce
time step
Else, possibly increase time step
Also, implicit integrator - to be covered
x0(t)
v0(t)
Particle Physics
x1(t)
v1(t)
mtF
tv
tv
tx
dt
dY
dt
d
/)(
)(
)(
)(
What is different about rigid body motion?
position,
velocity
rotation
angular velocity
Also, in addition to mass, m,
and rotational inertia, I
iFtF )(
Net force
x(t)
v(t)
F1F2
F3
?
)(
?
)(
tF
tv
Yx(t)
v(t)
Body State
?
)(
?
)(
?
)(
?
)(
tF
tv
tMv
tx
dt
dY
dt
d
Rigid Body Equation of Motion
Lets represent angular velocity as a vector w(t)
which is both spin direction and speed
Orientation change
How are R(t) and w(t) related?
w(t)
Angular Velocity
Consider w(t) affect on simple velocity:
xx w
x
wx
)(
0)()(
)(0)(
)()(0
)( tx
tt
tt
tt
txdt
d
xy
xz
yz
ww
ww
ww
Angular Velocity
Change in x(t) due to w(t) can be specified as:
)(
0)()(
)(0)(
)()(0
)( tR
tt
tt
tt
tRdt
d
xy
xz
yz
ww
ww
ww
Angular Velocity
w*(t) will be shorthand for this matrix
Change in R(t) due to w(t) is specified as:
?
)(
)()(
)(
?
)(
)(
)(
*
tF
tRt
tv
tMv
tR
tx
dt
dY
dt
d w
Rigid Body Equation of Motion
Need to relate w(t) and mass to F(t)
?
)(
)()(
)(
)(
)(
)(
)(
*
tF
tRt
tv
t
tMv
tR
tx
dt
dY
dt
d w
w
Rigid Body Equation of Motion
Inertia Tensor
IzzIzyIzx
IyzIyyIyx
IxzIxyIxx
tI )(
Relate angular velocity derivative and mass
distribution
Rigid Body Equation of Motion
?
)(
)()(
)(
)(
)(
)(
)(
*
tF
tRt
tv
t
tMv
tR
tx
dt
dY
dt
d w
w
?
)(
)()(
)(
)()(
)(
)(
)(
*
tF
tRt
tv
ttI
tMv
tR
tx
dt
dY
dt
d w
w
Rigid Body Equation of Motion
Rigid Body Equation of Motion
)(
)(
)()(
)(
)()(
)(
)(
)(
*
t
tF
tRt
tv
ttI
tMv
tR
tx
dt
dY
dt
d
w
w
iFtF )(
Net force
x(t)
v(t)
F1F2
F3
Net torque
x(t)
F1F2
F3
ii Ftxpt ))(()(
p1
p2
p3
A few notes on the Inertia Tensor…
IzzIzyIzx
IyzIyyIyx
IxzIxyIxx
tI )(
dm
r = moment arm
dmrI2
Computing Inertia
Parallel-Axis
Ix = Iy = 1/12 m (3r2 + l2)
Iz = mr2/2
I = Io + md2
d
Computing Inertia
Approximating Ibody – Point Sampling
Pros: Simple, fairly accurate
Cons: Expensive, requires volume test
Inertia tensor varies in world space,
but not in body space for a rigid body
Rigid Body Equations of Motion
)(
)(
)()(
)(
)()(
)(
)(
)(
*
t
tF
tRt
tv
ttI
tMv
tR
tx
dt
dY
dt
d
w
w
)(
)(
)()(
)(
)()(
)(
)(
)(
*
t
tF
tRt
tv
ttI
tMv
tR
tx
dt
dY
dt
d
w
w
Rigid Body Equations of Motion
L(t) – linear momentum
H(t) – angular momentum
Additional forces
Gravitational forces
Wind/air resistance
Collisions/impacts
Friction
Sliding – box on plane
Rolling - wheel
Forces for collision response
Articulated rigid bodies
Linked chain of bodies,
Possibly branching
Joint types:
Slider, telescoping
Hinge
Universal Joint
Gimbal
Ball and socket
Equations of motion often computed
automatically (commercial solvers)
No control = rag doll
Must provide control
Articulated rigid bodies
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