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Computer Animation Algorithms and TechniquesPowerPoint Presentation

Computer Animation Algorithms and Techniques

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Computer AnimationAlgorithms and Techniques

Dynamics

Useful Functions

Parametric position function P(u,v)

Surface normal function, N(u,v)

Implicit function I(x)

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

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)

Need to determine f(t)

x(t0) = a

x(t1) = b

Subject to constraints

Objective: to minimize total force

Objective function:

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

Time derivatives approximated by finite differences

R – minimize discrete version subject to constraints

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

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

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

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

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:

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

Final update:

Reaches fixed point:

- When Ci’s = 0

- Any further decrease in R violates constraints

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)

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