Rigid Body Dynamics III Constraintbased Dynamics

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Rigid Body Dynamics (III)Constraint-based
Dynamics
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Bodies intersect ! classify contacts

• Bodies separating
• vrel gt ?
• No response required
• Colliding contact (Part I)
• vrel lt -?
• Resting contact (Part II)
• -? lt vrel lt ?
• Gradual contact forces avoid interpenetration
• All resting contact forces must be computed and
  applied together because they can influence one
  another

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Outline

• Algorithm overview
• Computing constrained accelerations
• Computing a frictional impulse
• Extensions - Discussion

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Outline

• Algorithm overview
• Computing constrained accelerations
• Computing a frictional impulse
• Extensions - Discussion

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Algorithm Overview
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Algorithm Overview

• Two modules
• Collision detection
• Dynamics Calculator
• Two sub-modules for the dynamics calculator
• Constrained motion computation (accelerations/forc
  es)
• Collision response computation (velocities/impulse
  s)

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Algorithm Overview

• Two modules
• Collision detection
• Dynamics Calculator
• Two sub-modules for the dynamics calculator
• Constrained motion computation (accelerations/forc
  es)
• Collision response computation (velocities/impulse
  s)
• Two kinds of constraints
• Unilateral constraints (non-penetration
  constraints)
• Bilateral constraints (hinges, joints)

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Outline

• Algorithm overview
• Computing constrained accelerations
• Computing a frictional impulse
• Extensions - Discussion

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Constrained accelerations

• Solving unilateral constraints is enough
• When vrel 0 -- have resting contact
• All resting contact forces must be computed and
  applied together because they can influence one
  another

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Constrained accelerations
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Constrained accelerations

• Here we only deal with frictionless problems
• Two different approaches
• Contact-space the unknowns are located at the
  contact points
• Motion-space the unknowns are the object
  motions

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Constrained accelerations

• Contact-space approach
• Inter-penetration must be prevented
• Forces can only be repulsive
• Forces should become zero when the bodies start
  to separate
• Normal accelerations depend linearly on normal
  forces
• This is a Linear Complementarity Problem

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Constrained accelerations

• Motion-space approach
• The unknowns are the objects accelerations
• Gauss principle of least contraints
• The objects constrained accelerations are the
  closest possible accelerations to their
  unconstrained ones

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Constrained accelerations

• Formally, the accelerations minimize the distance
  
• over the set of possible accelerations
• a is the acceleration of the system
• M is the mass matrix of the system

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Constrained accelerations

• The set of possible accelerations is obtained
  from the non-penetration constraints
• This is a Projection problem

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Constrained accelerations

• Example with a particle

The particles unconstrained acceleration is
projected on the set of possible accelerations
(above the ground)
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Constrained accelerations

• Both formulations are mathematically equivalent
• The motion space approach has several algorithmic
  advantages
• J is better conditionned than A
• J is always sparse, A can be dense
• less storage required
• no redundant computations

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Outline

• Algorithm overview
• Computing constrained accelerations
• Computing a frictional impulse
• Extensions - Discussion

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Computing a frictional impulse

• We consider -one- contact point only
• The problem is formulated in the collision
  coordinate system

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Computing a frictional impulse

• Notations
• v the contact point velocity of body 1 relative
  to the contact point velocity of body 2
• vz the normal component of v
• vt the tangential component of v
• a unit vector in the direction of vt
• fz and ft the normal and tangential
  (frictional) components of force exerted by body
  2 on body 1, respectively.

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Computing a frictional impulse

• When two real bodies collide there is a period of
  deformation during which elastic energy is stored
  in the bodies followed by a period of restitution
  during which some of this energy is returned as
  kinetic energy and the bodies rebound of each
  other.

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Computing a frictional impulse

• The collision occurs over a very small period of
  time 0 ? tmc ? tf.
• tmc is the time of maximum compression

vz is the relative normal velocity. (We used
vrel before)
vz
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Computing a frictional impulse

• jz is the impulse magnitude in the normal
  direction.
• Wz is the work done in the normal direction.

jz
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Computing a frictional impulse

• v-v(0), v0v(tmc), vv(tf), vrelvz
• Newtons Empirical Impact Law
• Poissons Hypothesis
• Stronges Hypothesis
• Energy of the bodies does not increase when
  friction present

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Computing a frictional impulse

• Sliding (dynamic) friction
• Dry (static) friction
• Assume no rolling friction

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Computing a frictional impulse

• where
• r (p-x) is the vector from the center of mass
  to the contact point

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The K Matrix

• K is constant over the course of the collision,
  symmetric, and positive definite

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Collision Functions

• Change variables from t to something else that is
  monotonically increasing during the collision
• Let the duration of the collision ? 0.
• The functions v, j, W, all evolve over the
  compression and the restitution phases with
  respect to ?.

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Collision Functions

• We only need to evolve vx, vy, vz, and Wz
  directly. The other variables can be computed
  from the results.
• (for example, j can be obtained by inverting
  K)

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Sliding or Sticking?

• Sliding occurs when the relative tangential
  velocity
• Use the friction equation
  to formulate
• Sticking occurs otherwise
• Is it stable or instable?
• Which direction does the instability get resolved?

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Sliding Formulation

• For the compression phase, use
• is the relative normal velocity at the start
  of the collision (we know this)
• At the end of the compression phase,
• For the restitution phase, use
• is the amount of work that has been done
  in the compression phase
• From Stronges hypothesis, we know that

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Sliding Formulation

• Compression phase equations are

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Sliding Formulation

• Restitution phase equations are

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Sliding Formulation

• where the sliding vector is

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Sliding Formulation

• These equations are based on the sliding mode
• Sometimes, sticking can occur during the
  integration

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Sticking Formulation
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Sticking Formulation

• Stable if
• This means that static friction takes over for
  the rest of the collision and vx and vy remain 0
• If instable, then in which direction do vx and vy
  leave the origin of the vx, vy plane?
• There is an equation in terms of the elements of
  K which yields 4 roots. Of the 4 only 1
  corresponds to a diverging ray a valid
  direction for leaving instable sticking.

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Impulse Based Experiment

• Platter rotating with high velocity with a ball
  sitting on it. Two classical models predict
  different behaviors for the ball. Experiment and
  impulse-based dynamics agree in that the ball
  rolls in circles of increasing radii until it
  rolls off the platter.
• Correct macroscopic behavior is demonstrated
  using the impulse-based contact model.

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Outline

• Algorithm overview
• Computing constrained accelerations
• Computing a frictional impulse
• Extensions - Discussion

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Extensions - Discussion

• Systems can be classified according to the
  frequency at which the dynamics calculator has to
  solve the dynamics sub-problems

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Extensions - Discussion

• Systems can be classified according to the
  frequency at which the dynamics calculator has to
  solve the dynamics sub-problems
• It is tempting to generalize the solutions (fame
  !)
• Lasting non-penetration constraints can be viewed
  as trains of micro-collisions, resolved by
  impulses
• The LCP / projection problems can be applied to
  velocities and impulses

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Extensions - Discussion

• Problems with micro-collisions
• creeping a block on a ramp cant be stabilized

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Extensions - Discussion

• Problems with micro-collisions
• creeping a block on a ramp cant be stabilized

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Extensions - Discussion

• Problems with micro-collisions
• creeping a block on a ramp cant be stabilized
• A hybrid system is required to handle bilateral
  constraints (non-trivial)

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Extensions - Discussion

• Problems with micro-collisions
• creeping a block on a ramp cant be stabilized
• A hybrid system is required to handle bilateral
  constraints (non-trivial)
• Objects stacks cant be handled for more than
  three objects (in 1996), because numerous
  micro-collisions cause the simulation to grind to
  a halt

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Extensions - Discussion

• Extending the LCP
• accelerations are replaced by velocities
• forces are replaced by impulses
• constraints are expressed on velocities and
  forces
• Problem
• constraints are expressed on velocities and
  forces (!) This can add energy to the system
• Integrating Stronges hypothesis in this
  formulation ?

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Extensions - Discussion

• Extending the projection problem
• accelerations are replaced by velocities
• constraints are expressed on velocities and
  forces

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Extensions - Discussion

• Extending the projection problem
• accelerations are replaced by velocities
• constraints are expressed on velocities
• Problem
• constraints are expressed on velocities (!)
• This can add energy to the system