Keyframe Control of Complex Particle Systems Using the Adjoint Method

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Keyframe Control of Complex Particle Systems
Using the Adjoint Method

• Chris Wojtan Peter J. Mucha Greg Turk
• ACM SIGGRAPH Symposium on Computer Animation 2006

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Motivation

• Simulated animation is hard to control
• High level control is necessary
• Propose a way to control interacting particles by
  keyframes
• Flock
• Cloth

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Related works

• Keyframe control simulation
• Interactive manipulation of rigid body simulation
  Popovic et. al. SIGGRAPH 00
• Directable animation of elastic objects Kondo
  R. et. al. SCA 05
• Keyframe control of smoke simulations Treuille
  A. et. al. SIGGRAPH03
• Fluid control using the adjoint method McNamara
  A. et. al. SIGGRAPH 04

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Related works (cont.)

• Flocks
• Flocks, herds, and schools A distributed
  behavioral model. Reynolds C. W. SIGGRAPH 87
• Cloth
• Large steps in cloth simulation. Baraff D.,
  Witkin A. SIGGRAPH 98
• Stable but responsive cloth. Choi K. J., Ko H.
  S. SIGGRAPH 02

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Interactive particle system

• Each Particle has its own properties
• position, velocity, mass, etc.
• Particles may exert forces between each other
• External forces may exist

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Keyframe matching
F0
F1
F2
Fn
Define the objective function
P0
P1
P2
Pn
n frame number qn state vector of keyframe Wn
Weights Cn Mapping from u to q a smoothness Q
aggregation of q0qN
State vector
Control vector
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Problem of finding minimum

• Gradient-based method
• Expensive computation
• Dimension explosion in normal way
• Solve by the adjoint method!

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Adjoint method

• Duality
• Given matrices A, C and vector g are known
• its adjoint vector, s
• It is exactly the same as the original one!

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Adjoint method (cont.)

• Generally speaking, qi1 fi(qi)
• fi updating operation
• Simply apply external forces in this system
• f(q) q Mtu
• Suppose all the forces are applied linearly
• Mt matrix converting u to an incremental state
• Rewrite in a aggregate form
• Q F(Q,u)

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Adjoint method (cont.)
gT
A
B
C
g
sT
A
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Adjoint method (cont.)

• Solving for the adjoint vector

recall that qi1 fi(qi)

• Simply use Gaussian elimination
• Solve backward in time

A
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Adjoint method (cont.)

• Workflow
• Huge memory consumption, but efficient.

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Control flocking

• Control force
• Collision avoidance
• Velocity matching force
• Flocking centering force
• Wander force
• Solving by forward Euler scheme

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Control flocking (cont.)

• Derivation
• F ?t V(Q) gt Qn1 ?t V(Qn)
• Resulting adjoint vector

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Control flocking (cont.)

• Extending some constraints for more intuitive
  control
• Center of mass constraints
• Shape constraints
• No need to modify objective function
• Compute difference between boundary shape
• Computer difference between centers of mass

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Control cloth

• A different story!
• Solving by backward Euler scheme

General form of its adjoint vector
?????(???)??
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Control cloth (cont.)

• Dimension explosion!
• Expensive both in space and in time
• Avoid exact calculation, use approximation!
• Calculate adjoints of backward Euler to
  approximate adjoints of linearized scheme

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Control cloth (cont.)

• Derivation
• F ?t V(Qn1)
• Result adjoint
• Introduce O(?t3) errors

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Results

• No benchmark, no spec of demo environmentjust
  video!

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Conclusion future work

• Efficient control of particle system
• Simple system like flocking
• Stiff system like cloth
• Main contribution!
• More robust cloth control
• Adjoint method is theoretically sensitive to
  discontinuities
• Large amount of collision detection cannot occur.