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Rigid%20body%20dynamics%20II%20Solving%20the%20dynamics%20problems

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... vrel ~= 0 -- have resting contact. All resting contact forces must be ... that static friction takes over for the rest of the collision and vx and vy remain 0 ... – PowerPoint PPT presentation

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Title: Rigid%20body%20dynamics%20II%20Solving%20the%20dynamics%20problems


1
Rigid body dynamics II Solving the dynamics
problems
2
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

3
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

4
Algorithm Overview
5
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)

6
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)

7
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

8
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

9
Constrained accelerations
10
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

11
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

12
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

13
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

14
Constrained accelerations
  • The set of possible accelerations is obtained
    from the non-penetration constraints
  • This is a Projection problem

15
Constrained accelerations
  • Example with a particle

The particles unconstrained acceleration is
projected on the set of possible accelerations
(above the ground)
16
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

17
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

18
Computing a frictional impulse
  • We consider -one- contact point only
  • The problem is formulated in the collision
    coordinate system

19
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.

20
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.

21
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
22
Computing a frictional impulse
  • jz is the impulse magnitude in the normal
    direction.
  • Wz is the work done in the normal direction.

jz
23
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

24
Computing a frictional impulse
  • Sliding (dynamic) friction
  • Dry (static) friction
  • Assume no rolling friction

25
Computing a frictional impulse
  • where
  • r (p-x) is the vector from the center of mass
    to the contact point

26
The K Matrix
  • K is constant over the course of the collision,
    symmetric, and positive definite

27
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 ?.

28
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)

29
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?

30
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

31
Sliding Formulation
  • Compression phase equations are

32
Sliding Formulation
  • Restitution phase equations are

33
Sliding Formulation
  • where the sliding vector is

34
Sliding Formulation
  • These equations are based on the sliding mode
  • Sometimes, sticking can occur during the
    integration

35
Sticking Formulation
36
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.

37
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.

38
Outline
  • Algorithm overview
  • Computing constrained accelerations
  • Computing a frictional impulse
  • Extensions - Discussion

39
Extensions - Discussion
  • Systems can be classified according to the
    frequency at which the dynamics calculator has to
    solve the dynamics sub-problems

40
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

41
Extensions - Discussion
  • Problems with micro-collisions
  • creeping a block on a ramp cant be stabilized

42
Extensions - Discussion
  • Problems with micro-collisions
  • creeping a block on a ramp cant be stabilized

43
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)

44
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

45
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 ?

46
Extensions - Discussion
  • Extending the projection problem
  • accelerations are replaced by velocities
  • constraints are expressed on velocities and
    forces

47
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
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