Ordinary Differential Equations

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Ordinary Differential Equations
Slides borrowed from Ronen Barzel http//www.ense
ignement.polytechnique.fr/profs/informatique/Ronen
.Barzel/
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What is an ODE?

• Ordinary Differential Equation
• Relates value of a function to its derivatives
• Ordinary function of one variable
• Partial Differential Equation (PDE) more
  variables

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Standard ODE

• Generic form for first-order ODE
• Note
• typically t is time
• sometimes use Y instead of X, sometimes x instead
  of t
• names sometimes confusing often

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Why do we care?

• Differential equations describe (almost)
  everything in the world
• physics
• chemistry
• engineering
• ecology
• economy
• weather
• Also useful for animation!
• ODEs are fundamental. PDEs build on ODEs

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Solving differential equations

• Analytic solutions to differential equations
• Many standard forms, e.g.
• But most cant be solved analytically
• 3-body problem

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Numerical solutions to ODEs

• Given a function f(X,t) compute X(t)
• Typically, initial value problems
• Given values X(t0)X0
• Find values X(t) for t gt t0
• Also, boundary value problems, constrained
  problems,

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Solving ODEs for animation

• For animation, want a series of values
• samples of the continuous function X(t)
• i.e., frames of an animation

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Path through a field

• f(X,t) is a vector field defined everywhere
• it may change based on t

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Path through a field

• f(X,t) is a vector field defined everywhere
• X(t) is a path through the field

X0
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Higher order ODEs

• E.g., Mechanics has 2nd order ODE (FmA)
• Express as 1st order ODE by defining v(t)

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E.g., for a 3D particle

• We have a 6 dimension ODE problem

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For a collection of 3D particles
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Still, a path through a field

• X(t) maybe path in multi-body phase space
• For ODE, its an array of numbers.

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Intuitive solution take steps

• Current state X
• Examine f(X,t) at (or near) current state
• Take a step to new value of X
• Most solvers do some form of this

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Aside Integral equation

• Note, differential equation
• is equivalent to integral equation

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Eulers method

• Simplest and most intuitive.
• Define step size h
• Given X0X(t0), take step
• Piecewise-linear approximation to the curve

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Step size controls accuracy

• Smaller steps more closely follow curve
• For animation, may need to take many small steps
  per frame

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Eulers method inaccurate

• Moves along tangent can leave curve, e.g.
• Exact solution is circle
• Eulers spirals outwardno matter how small h
  is

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Eulers method unstable

• Exact solution is decaying exponential
• Limited step size
• If k is big, h must be small

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Analysis Taylor series

• Expand exact solution X(t)
• Eulers method approximates
• First-order method Accuracy varies with h
• To get 100x better accuracy need 100x more steps

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2nd-order methods

• Midpoint
• ½ Euler step
• evaluate fm
• full step using fm
• Trapezoid
• Euler step
• evaluate f1
• full step using f1
• average (a) and (c)

Midpoint method
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Runge-Kutta method

• Follow same derivation for higher order
• 4th order
• This is what most people use most often

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Adaptive step size

• How do you pick the step size?
• Want large as possible given acceptable error.
• Best step size might change along path
• harder and easier parts of the path
• decrease/increase step size as needed
• Automatic adaptive step size control

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Adaptive step size

• Given h, take step and estimate error
• if error is too large, try again with smaller h
• if error is small, accept step and maybe increase
  h
• Estimate error compute X(t0h) two ways
• compute Xa by taking one step using h
• compute Xb by taking two steps using h/2
• estimated error err Xa Xb

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Step size for animation

• Want values at regular frame times
• Can make sure not to step past next frame
• OK if h ltlt df
• Can step past frame, then interpolate back
• OK if h df
• Need interpolation anyway when doing collisions

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Is there a solution?

• The function f(X,t) must be smooth
• None of these solvers handle discontinuity
• Adaptive step size may drop to smallest h
• Sometimes desired solution has discontinuity
• Rigid-body collisions have discontinuous v
• ODE solver cant step across discontinuity.

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Which solver to choose?

• Many more kinds of ODE solvers
• predictor-corrector, boundary-value problems,
  semi-implicit, differential-algebraic equations,
  
• Multi-step methods keep history of previous
  values
• Adaptive-order methods
• For hard problems, need to understand theory
• lots of theory!
• Runge-Kutta (4th order) is a standard workhorse
• Try to avoid stiff equations!
• Often need to experiment and see what works

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For animation

• Dont need to be accurate, just look good.
• Eulers method w/fixed step size
• Always a bad idea.
• Unless it works for you!

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Particle motion

• mass m, position x, velocity v
• equations of motion
• ODE

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Mass-spring systems

• Basically same as a particle system
• Structured, rather than free form
• Mass points part of model
• no generators, no lifetime, no aging
• Use spring forces to connect masses
• all forces no longer apply to all points
• Each force object knows which points it acts on

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Mass spring systems

• Connect points in 1D line
• used for hair, springs, etc
• Connect points in 2D mesh
• used for cloth
• Connect points in 3D lattice
• semi-rigid structures
• used for flesh/muscle motion
• e.g. Jurassic park dinosaurs

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Spring to fixed point

• Basic Hookes-law spring force
• Attracts mass to point x0
• Causes oscillation

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Damped spring

• Spring and dashpot
• like a car suspension
• only damps motion in direction of spring

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Damped spring

• Relationship between ks and kd determines
• underdamping, critical damping, overdamping
• (in isolation)
• Always want some damping.

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Constructions with springs

• connect mass points to make objects
• shapes can kink, shear, or collapse
• add extra supports
• can still be tricky to get desired behavior

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Multiple springs

• Each spring tries to achieve E0
• Competition between multiple forces
• Result is compromise when stable E?0
• To guarantee result, need constraints

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Rigid body state

• Same as points
• m is the total mass
• x is the position of the center of mass
• v is the velocity of the center of mass
• Plus new terms
• I is the rotational inertia tensor (scalar in
  2Dmatrix in 3D)
• ? is the orientation (angle in 2D messy in 3D)
• ? is the angular velocity (scalar in 2D vector
  in 3D)
• Body coordinates
• origin at x, orientation ? relative to World
• for any shape, m and I completely describe
  response

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Rigid body equation of motion

• Equation of motion has extra terms
• T is torque

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Torque

• Each force f acts at a point on body
• r is vector from center of mass to point of
  action
• Torque causes body to spin
• if force acts at center of mass, no torque
  (gravity)
• if force points towards center of mass, torque0
• f at r and f at r give net torque but no net
  force

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Collisions

• Collisions between two bodies
• Need point of collision, and normal vector
• many algorithms
• For body-particle collision
• transform to body coordinates
• see if particle is inside body shape
• compute collision normal in body coordinates
• compute relative velocity in body coordinates
• compute impulse in body coordinates
• transform back to world and apply

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Whats a constraint?

• A restriction on the position of a body
• A relationship between one or more bodies
• Permanent constraints
• holonomic expressed by equation C()0
• Joints between bodies
• Transient constraints
• nonholonomic expressed by equation C()?0
• contact
• joint limits

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Constraint forces

• First compute non-constraint forces
• gravity, springs, etc
• Then compute force due to constraint
• takes into account effect of non-constraint
  forces.
• Can have multiple constraints
• each constraint must take into account the others
• simultaneous equation
• Add constraint forces to non-constraint forces
• Simulate normally.