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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
• 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