Spacetime Constraints

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Spacetime Constraints

• David Coyne
• Joe Ishikura

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The challenge of kinematics

• Successful animation requires control, but looks
  real
• Traditional principles of animation look right

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Problem
Control
Accuracy

• Keyframe animation
• Artist controls each pose
• Time consuming
• Takes an expert to make it look good

• Physics simulations
• Looks realistic
• Almost no kinematic control
• Forward simulation using time-dependent force
  functions looks bad

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If only

• We could produce motion to achieve a goal, rather
  than just simulate starting conditions
• The solution would show how a real model would
  move

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The Spacetime Solution

• Represent motion as a set of equations
• Set constraints to represent physical forces and
  goals (e.g. start at Pi, end at Pf)
• Optimize solution with respect to some objective
  (e.g. minimize force)

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Propelled Particle Example

• We want to animate a particle that is affected by
  gravity and has jet propulsion force that can
  propel it
• We want to be able to specify a starting position
  and ending position and want a program to figure
  out how to propel itself so that it uses the
  least amount of energy

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Basic Terminology

• Governing Equation
• Particle affected by gravity and a jet force
• Boundary Conditions
• Given start and end position
• Objective Function
• Minimize consumed energy

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Translate to Continuous Functions

• Governing Equation (affected by gravity and
  propulsion force function)
• Boundary Conditions (given start and end
  position)
• Objective Function (minimize consumed energy)

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Translate to Discrete Functions

• We want to represent x(t) and f(t) as a set of
  independent variables that we can solve for
• Do this by discretizing x(t) and f(t) into n 1
  samples with h step size
• Then translate all other equations

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Discretizing the Governing Equation

• Translate
• New Governing Equation

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Discretizing the Rest

• Boundary Conditions
• Objective Function

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Goal

• Find values for f0, f1, fn that minimizes R
  while adhering to constraints
• Do this by finding f values where

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Sequential Quadratic Programming

• Essentially, the method computes a second-order
  Newton-Raphson step in R, and a first-order
  Newton-Raphson step in the (constraint
  functions), and combines the two steps by
  projecting the first onto the null space of the
  second (that is, onto the hyperplane for which
  all the constraint functions are constant to
  first order)

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Newton-Raphson Method

• An iterative process used to attempt to converge
  on a root of an function given the function and
  its derivative
• Start with a guess, call it x0
• We converge on the answer by finding

source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Example

• So lets say were trying to find one root of
• We then guess at a value
• Then begin iterating

source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Example contd
xn f(xn) f(xn) dx xn-dx
x0 6 32 12 2.67 3.33
x1 3.33 7.09 6.66 1.06 2.27
x2 2.27 1.15 4.54 0.26 2.01
x3 2.01 0.04 4.02 .01 2.00
x4 2.00 0 4.00 0 2.00
source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Graphical Representation
source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Graphical Representation
source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Graphical Representation
source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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NR Graphical Representation
source http//www.shodor.org/UNChem/math/newton/i
ndex.html
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SQP and NR

• With SQP we are performing the Newton-Raphson
  method on our constraint functions and our
  objective function
• Assuming our system is not over-constrained we
  should be able to get close

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The SQP Notation

• Represent each guess as Si, a vector of all
  independent parameters at each iteration
• Turn all of the boundary conditions into
  constraint functions, call the set of them C
• C(S) and R(S) must equal 0 so we can use NR

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Step 1 Second Order NR on R

• For now, ignore constraints
• Start with an initial guess S0
• Find Hessian of objective function (do once)
• In our example

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Step 1 Continued

• Recall Taylor expansion
• With our equation
• We know that

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Step 1 Continued

• Our new equation becomes
• We can calculate and solve for (S -
  S0)
• S - S0 is the difference between the actual
  solution (root) and S0
• Because our Taylor series expansion is not
  complete (i.e. infinite), the value that we
  actually get is only an approximation of (S - S0)

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Step 1 Continued

• Adding our approximation of (S - S0) (call it ?S)
  to our current guess S0 should bring us closer to
  the actual solution
• True to the Newton Raphson method, our new guess
  at the end of this iteration is
• This new S1 ignores our constraint conditions

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Step 2 First Order NR on C

• If Ci(S1) 0 for all constraint functions we are
  done
• Otherwise, we must similarly converge onto an S
  value that will make C(S)0
• Find the Jacobian of each Ci
• Use Taylor Expansion on each Ci

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Step 2 Continued

• We rewrite it in terms of the Jacobian and set
  C(S)0
• Once again, we solve for (S - S1) which is again
  an approximation that we can call ?S
• After this iteration, our new guess becomes

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Iterating

• This new S2 value is fed back into Step 1
• The process repeats until C(Sx)0 and any further
  decrease in R requires violating the constraints

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Graphical Explanation of SQP
C(S)
S1
S2
S2
S
S0
S1
Slide taken from http//www.cs.virginia.edu/gfx/c
ourses/2005/Animation.spring.05/
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Using constraints to animate Luxo

• Define the model and its laws of motion
• Set constraints for desired result
• Choose a criteria and optimize solution

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Define model

• Define model
• Four rigid massive links
• Derive laws of motion
• Muscles Three springs produce arbitrary
  time-dependent joint forces

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Constraints

• Initial and final positions and poses
• No motion in contact with floor (simulates
  inelastic collision)

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Solving

• Set optimization criteria
• Minimize applied muscle power (muscle force times
  angular velocity)

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Adding different constraints

• Landing force

• Height of jump

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• Increase mass

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Ski Jump

• Removed Constraints
• Base free to slide
• Initial velocity

• Added Constraints
• Base tangent to surface
• Height of base in air at one time step
• Optimization includes style

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More about Spacetime

• Original paper by Witkins and Kass written in
  1988
• A number of applications and further
  optimizations studied since

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Spacetime Constraints Revisited

• J. Thomas Ngo, Harvard, Joe Marks, Cambridge 1993
• Instead of using perturbational analysis, use
  global search to find optimal solution
• Generate possible solutions and use a genetic
  search algorithm to find the best

source http//citeseer.ist.psu.edu/cache/papers/c
s/1776/httpzSzzSzwww.merl.comzSzpeoplezSzmarkszSz
spacetime.pdf/ngo93spacetime.pdf
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Human Motion with Spacetime Constraints

• Charles Rose, Brian Guenter, Bobby Bodenheimer,
  Michael Cohen, Microsoft Research 1996
• Using Inverse Kinematics and Spacetime
  Constraints, the Microsoft team was able to
  simulate realistic human motion with 44 degrees
  of freedom
• Biggest problem with so many degrees of freedom
  and so many constraints, difficult to do quickly

source http//research.microsoft.com/cohen/Effic
ientSIG96.pdf
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Human Motion with Spacetime Constraints
source http//research.microsoft.com/cohen/Effic
ientSIG96.pdf
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Motion Editing with Spacetime Constraints

• Michael Gleicher, Apple Research Laboratories
  1997
• Summary
• Given an animation, allow the animator to use
  direct manipulation to edit any joint in any time
  step and, using spacetime constraints, a program
  figures out, in real time, what the new animation
  will be, attempting to mimic the style of the
  original as best as possible

source http//www.cs.wisc.edu/graphics/Papers/Gle
icher/California/SpacetimeEditing.pdf
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Motion Editing with Spacetime Constraints

• Best or optimal motion is one where as much of
  the style is preserved as possible
• Animator can specify what parts of the animation
  she wants to preserve
• Uses spacetime techniques to propagate changes
  across entire animation

source http//www.cs.wisc.edu/graphics/Papers/Gle
icher/California/SpacetimeEditing.pdf
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Spacetime Constraints for Biomechanical Movements

• David Brogan, Kevin Granata, Pradip Sheth,
  University of Virginia, 2002
• Use the Spacetime method to see how pathological
  constraints can affect movement

source http//www.cs.virginia.edu/dbrogan/Public
ations/Papers/iasted02.pdf
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Arm Motion with Spacetime Constraints

• Dengming Zhu, Zhaoqi Wang, He Huang, Min Shi,
  Chinese Academy of Sciences 2003
• Simulate natural arm movement using Spacetime
  Constraints

source http//vh.ict.ac.cn/news/upload/2003Arm.do
c
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Questions?