Computer Animation
Algorithms and Techniques
Optimization & Constraints
Add mention of global techiques
Add mention of calculus of variations
Rick Parent
Computer Animation
Enforcing Soft and Hard
Constraints
Soft constraints - Minimizing energy terms
Hard constraints – constrained optimization
Example: Space-time constraints
Rick Parent
Computer Animation
Constrained optimization
Use whenever the best, shortest, least error, is needed:
Fit surface with least curvature to set of points
Reach for object with minumu torque
Find motion in database whose start pose is
closest to end of current motion
Rick Parent
Computer Animation
Minimum (or maximum)
of a function
f(x)
Ballistic motion
f’(x) = 0
1 2
f (t )  p0  v0t  gt
2
f (t )  v0  gt  0
v0
t
g
Analytic derivative & solution
Rick Parent
Computer Animation
Minimum (or maximum)
of a function
Multivariate case
 f 
x 
 1
f 


f ( x )   x2   0
 f 
 
 x3 
 ... 
Gradient / Jacobian
Rick Parent
Computer Animation
Energy function
Define energy terms
in terms of geometric features
=0 for desirable configuration
increases for less desirable configurations
Useful geometric features– easy to compute
Parametric position function P(u,v)
Surface normal function, N(u,v)
Implicit function I(x) – distance to surface
Search parameter space
modify parameters to reduce energy
Rick Parent
Computer Animation
Useful Constraints
E  P (u , v)  Q
2
E  P (ua , va )  P (ub , vb )
a
b
2
2
E  P (ua , va )  P (ub , vb )  N a (ua , va )  N b (ub , vb )  1.0
a
b
E  ( I b ( P a (ua , va ))) 2
Rick Parent
Computer Animation
Pipe
assembly
Rick Parent
Computer Animation
Finding the minimum

minimize f (x )
Analytic derivative/gradient

f (x )
Analytic solution

f ( x )  0
Rick Parent
Numeric approximation to
derivative/gradient


f ( xi )  f ( xi 1 )
f 
x
Numeric solution by iterative search
Newton’s method
Steepest Descent
Conjugate Gradient Method
Computer Animation
Newton’s method

f ( x )  0
Rick Parent
xi 1

f ' ( xi )
 xi 

f ( xi )
Computer Animation
Steepest descent – step in direction
of negative of gradient
Rick Parent
Computer Animation
Conjugate gradient method
Rick Parent
Computer Animation
Constrained Optimization
minimize

f (x )

gi ( x )  0
Equality constraints

hi ( x )  0
Inequality constraints
Rick Parent
Computer Animation
Lagrange multipliers
 ( x, y ,  )  f ( x, y )   g ( x, y )
( x, y,  )  f ( x, y )  g ( x, y )  0
 xy ( x, y,  )   xy f ( x, y )   xy g ( x, y )  0
 ( x, y,  )  g ( x, y)  0 .
Rick Parent
Computer Animation
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.
Rick Parent
Computer Animation
Spacetime constraint example
Particle position is function of time, x(t)
Time-varying force function, f(t)
Equation of motion
mx(t )  f (t )  mg  0
Given f(t), x(t0),
Rick Parent
x (t0 ) - integrate to get x(t).
Computer Animation
Spacetime constraint example
Need to determine f(t)
x(t0) = a
x(t1) = b
Subject to constraints
Objective: to minimize total force
Objective function:
2
t1
R   f dt
t0
Rick Parent
Computer Animation
Spacetime constraint example
Use discrete x(t), f(t), R, constraints
Time derivatives approximated by finite differences
xi  xi 1
xi 1  2 xi  xi 1
xi 
xi 
h
h2
Constraints:
xi 1  2 xi  xi 1
pi  m
 f i  mg  0
2
h
ca  x1  a  0
cb  xn  b  0
R – minimize discrete version subject to constraints.
2
R  h f
i
Rick Parent
Computer Animation
Spacetime constraint canonical form
Numerical Solution – canonical form
Sj – collection of scalar independent variables
x, y, z components of the xi’s and fi’s
R(Sj) – objective function to be minimized
sum of forces squared used at each time step
Ci (Sj) – collection of scalar constraint functions => 0.
components of pi’s, ca, and cb .
Rick Parent
Computer Animation
Spacetime constraint - canonical
Numerical problem statement
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 .
Rick Parent
Computer Animation
Spacetime constraint
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.
Rick Parent
Computer Animation
Spacetime constraint using SQP
Sequential Quadratic Programming (SQP)
Computes first-order Newton-Raphson step in the Ci’s
Ci
Jacobian:
J ij 
S j
Computes second-order Newton-Raphson step in R
2R
H ij 
Hessian
S i S j
Plus the first derivative vector:
Rick Parent
R
S j
Computer Animation
Spacetime constraint example
xi 1  2 xi  xi 1
pi  m
 f i  mg  0
2
h
Matrices
p i
 2m / h 2
x j
 m / h 2
0
pi
1
f j
0
Rick Parent
i=j
i=j-1, j+1
otherwise
i=j
Otherwise.
Computer Animation
Spacetime constraint example
Matrices
R
 2 fi
f i
2R
2
f i f j
0
Rick Parent
i=j
Otherwise.
Computer Animation
Spacetime constraint example
SQP step
Solve 2 linear systems in sequence
1
R

  H ij Sˆ j
S i
j
Yields a step that minimizes a 2nd order approx. to R
~ ˆ
2
 Ci   J ij (S j  S j )
j
Yields a step that drives linear approx. to Ci’s to zero
And projects optimization step Sj to null space of constraint
Jacobian.
Rick Parent
Computer Animation
Spacetime constraint example
Final update:
~ ˆ
Sj  Sj
Reaches fixed point:
- When Ci’s = 0
- Any further decrease in R violates constraints.
Rick Parent
Computer Animation
Constrained optimization
Rick Parent
Computer Animation
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).
Rick Parent
Computer Animation