rigid body dynamics ii solving the dynamics problems l.
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Rigid body dynamics II Solving the dynamics problems. Outline. Algorithm overview Computing constrained accelerations Computing a frictional impulse Extensions - Discussion. Outline. Algorithm overview Computing constrained accelerations Computing a frictional impulse

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

S. Redon - M. C. Lin

outline3
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

S. Redon - M. C. Lin

algorithm overview
Algorithm Overview

S. Redon - M. C. Lin

algorithm overview5
Algorithm Overview
  • Two modules
    • Collision detection
    • Dynamics Calculator
  • Two sub-modules for the dynamics calculator
    • Constrained motion computation (accelerations/forces)
    • Collision response computation (velocities/impulses)

S. Redon - M. C. Lin

algorithm overview6
Algorithm Overview
  • Two modules
    • Collision detection
    • Dynamics Calculator
  • Two sub-modules for the dynamics calculator
    • Constrained motion computation (accelerations/forces)
    • Collision response computation (velocities/impulses)
  • Two kinds of constraints
    • Unilateral constraints (non-penetration constraints…)
    • Bilateral constraints (hinges, joints…)

S. Redon - M. C. Lin

outline7
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

S. Redon - M. C. Lin

constrained accelerations
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

S. Redon - M. C. Lin

constrained accelerations9
Constrained accelerations

S. Redon - M. C. Lin

constrained accelerations10
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

S. Redon - M. C. Lin

constrained accelerations11
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

S. Redon - M. C. Lin

constrained accelerations12
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”

S. Redon - M. C. Lin

constrained accelerations13
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

S. Redon - M. C. Lin

constrained accelerations14
Constrained accelerations
  • The set of possible accelerations is obtained from the non-penetration constraints
  • This is a Projection problem

S. Redon - M. C. Lin

constrained accelerations15
Constrained accelerations
  • Example with a particle

The particle’s unconstrained acceleration is projected on the set of possible accelerations (above the ground)

S. Redon - M. C. Lin

constrained accelerations16
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

S. Redon - M. C. Lin

outline17
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

S. Redon - M. C. Lin

computing a frictional impulse

-j

+j

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

S. Redon - M. C. Lin

computing a frictional impulse19
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.

S. Redon - M. C. Lin

computing a frictional impulse20
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.

S. Redon - M. C. Lin

computing a frictional impulse21
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

S. Redon - M. C. Lin

computing a frictional impulse22
Computing a frictional impulse
  • jz is the impulse magnitude in the normal direction.
  • Wz is the work done in the normal direction.

jz

S. Redon - M. C. Lin

computing a frictional impulse23
Computing a frictional impulse
  • v-=v(0), v0=v(tmc), v+=v(tf), vrel=vz
  • Newton’s Empirical Impact Law:
  • Poisson’s Hypothesis:
  • Stronge’s Hypothesis:
    • Energy of the bodies does not increase when friction present

S. Redon - M. C. Lin

computing a frictional impulse24
Computing a frictional impulse
  • Sliding (dynamic) friction
  • Dry (static) friction
  • Assume no rolling friction

S. Redon - M. C. Lin

computing a frictional impulse25
Computing a frictional impulse

where:

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

S. Redon - M. C. Lin

the k matrix
The K Matrix
  • K is constant over the course of the collision, symmetric, and positive definite

S. Redon - M. C. Lin

collision functions
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 .

S. Redon - M. C. Lin

collision functions28
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)

S. Redon - M. C. Lin

sliding or sticking
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?

S. Redon - M. C. Lin

sliding formulation
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 Stronge’s hypothesis, we know that

S. Redon - M. C. Lin

sliding formulation31
Sliding Formulation
  • Compression phase equations are:

S. Redon - M. C. Lin

sliding formulation32
Sliding Formulation
  • Restitution phase equations are:

S. Redon - M. C. Lin

sliding formulation33
Sliding Formulation

where the sliding vector is:

S. Redon - M. C. Lin

sliding formulation34
Sliding Formulation
  • These equations are based on the sliding mode
  • Sometimes, sticking can occur during the integration

S. Redon - M. C. Lin

sticking formulation
Sticking Formulation

S. Redon - M. C. Lin

sticking formulation36
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.

S. Redon - M. C. Lin

impulse based experiment
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.

S. Redon - M. C. Lin

outline38
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

extensions discussion40
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

S. Redon - M. C. Lin

extensions discussion41
Extensions - Discussion
  • Problems with micro-collisions
    • creeping : a block on a ramp can’t be stabilized

S. Redon - M. C. Lin

extensions discussion42
Extensions - Discussion
  • Problems with micro-collisions
    • creeping : a block on a ramp can’t be stabilized

S. Redon - M. C. Lin

extensions discussion43
Extensions - Discussion
  • Problems with micro-collisions
    • creeping : a block on a ramp can’t be stabilized
    • A hybrid system is required to handle bilateral constraints (non-trivial)

S. Redon - M. C. Lin

extensions discussion44
Extensions - Discussion
  • Problems with micro-collisions
    • creeping : a block on a ramp can’t be stabilized
    • A hybrid system is required to handle bilateral constraints (non-trivial)
    • Objects stacks can’t be handled for more than three objects (in 1996), because numerous micro-collisions cause the simulation to grind to a halt

S. Redon - M. C. Lin

extensions discussion45
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 Stronge’s hypothesis in this formulation ?

S. Redon - M. C. Lin

extensions discussion46
Extensions - Discussion
  • Extending the projection problem
    • accelerations are replaced by velocities
    • constraints are expressed on velocities and forces

S. Redon - M. C. Lin

extensions discussion47
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

S. Redon - M. C. Lin