Rigid body dynamics II Solving the dynamics problems

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# Rigid body dynamics II Solving the dynamics problems - PowerPoint PPT Presentation

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

Algorithm Overview

S. Redon - M. C. Lin

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 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

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

S. Redon - M. C. Lin

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 accelerations

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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 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

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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 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 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
• 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 Formulation
• Compression phase equations are:

S. Redon - M. C. Lin

Sliding Formulation
• Restitution phase equations are:

S. Redon - M. C. Lin

Sliding Formulation

where the sliding vector is:

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

Sticking Formulation

S. Redon - M. C. Lin

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
• 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

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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

S. Redon - M. C. Lin

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