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Computer Animation Algorithms and Techniques. Dynamics. Enforcing Soft and Hard Constraints. Energy minimization Space-time constraints. Energy minimization. Energy minimization. Useful Functions. Parametric position function P(u,v) Surface normal function, N(u,v) Implicit function I(x).

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Computer animation algorithms and techniques
Computer AnimationAlgorithms and Techniques

Dynamics


Enforcing Soft and Hard Constraints

Energy minimization

Space-time constraints



Energy minimization

Useful Functions

Parametric position function P(u,v)

Surface normal function, N(u,v)

Implicit function I(x)



Spacetime constraints

http://www.cs.cmu.edu/~aw/pdf/spacetime.pdf

Constrained optimization problem in time and space

Constraints – e.g.

locating certain points in time and space

Non penetration constraints

Objective function – e.g.

Minimize the amount of force used over time interval

Minimize maximum torque


Spacetime constraint example

Particle position is function of time, x(t)

Time-varying force function, f(t)

Equation of motion

Given f(t), x(t0),

- integrate to get x(t)


Spacetime constraint example

Need to determine f(t)

x(t0) = a

x(t1) = b

Subject to constraints

Objective: to minimize total force

Objective function:


Spacetime constraint example

Use discrete x(t), f(t), R, constraints

Time derivatives approximated by finite differences

R – minimize discrete version subject to constraints


Spacetime constraint example

Numerical Solution – canonical form

Sj – collection of scalar independent variables

R(Sj) – objective function to be minimized

Ci (Sj) – collection of scalar constraint functions => 0


Spacetime constraint example

Numerical Solution – example in canonical form

Sj – x, y, z components of the xi’s and fi’s

R(Sj) – objective function to be minimized

Ci (Sj) – components of pi’s, ca, and cb


Spacetime constraint example

Canonical numerical problem

Find Sj that minimizes R(Sj) subject to Ci(Sj) = 0

  • Numerical solution method:

  • Request values of R and Ci for given Sj

  • Access to derivatives of R and Ci with respect to Sj

  • Iteratively provides updated values for solution vector Sj


Spacetime constraint example

Sequential Quadratic Programming (SQP)

Computes second-order Newton-Raphson step in R

Computes first-order Newton-Raphson step in the Ci’s

Projects the first onto the null space of the second

to the hyperplane for which all the Ci’s are constant to the first order


Spacetime constraint example

Sequential Quadratic Programming (SQP)

Computes second-order Newton-Raphson step in R

Hessian

Computes first-order Newton-Raphson step in the Ci’s

Jacobian:

Plus the first derivative vector:


Spacetime constraint example

SQP step

Solve 2 linear systems in sequence

1

Yields a step that minimizes a 2nd order approx. to R

2

Yields a step that drives linear approx. to Ci’s to zero

And projects optimization step Sj to null space of constraint Jacobian


Spacetime constraint example

Final update:

Reaches fixed point:

- When Ci’s = 0

- Any further decrease in R violates constraints


Spacetime constraint example

Note about Linear system solving

Large matrices often with spacetime problems

Inverting is O(n3)

Spacetime problems almost always sparse

Over and under constrainted systems easily arise

Under constrained: pseudo-inverse

Pseudo-inverse for sparse matrix

sparse conjugate gradient algorithm: O(n2)


Spacetime constraint example

Matrix evaluation

i=j

i=j-1, j+1

otherwise

i=j

otherwise


Spacetime constraint example

Matrix evaluation

i=j

otherwise


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