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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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slide2

Enforcing Soft and Hard Constraints

Energy minimization

Space-time constraints

slide4

Energy minimization

Useful Functions

Parametric position function P(u,v)

Surface normal function, N(u,v)

Implicit function I(x)

slide6

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

slide7

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)

slide8

Spacetime constraint example

Need to determine f(t)

x(t0) = a

x(t1) = b

Subject to constraints

Objective: to minimize total force

Objective function:

slide9

Spacetime constraint example

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

Time derivatives approximated by finite differences

R – minimize discrete version subject to constraints

slide10

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

slide11

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

slide12

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
slide13

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

slide14

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:

slide15

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

slide16

Spacetime constraint example

Final update:

Reaches fixed point:

- When Ci’s = 0

- Any further decrease in R violates constraints

slide17

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)

slide18

Spacetime constraint example

Matrix evaluation

i=j

i=j-1, j+1

otherwise

i=j

otherwise

slide19

Spacetime constraint example

Matrix evaluation

i=j

otherwise

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