Rigid body dynamics II
Solving the dynamics problems
UNC Chapel Hill
S. Redon - M. C. Lin
Outline
Algorithm overview
 Computing constrained accelerations
 Computing a frictional impulse
 Extensions - Discussion

UNC Chapel Hill
S. Redon - M. C. Lin
Outline
Algorithm overview
 Computing constrained accelerations
 Computing a frictional impulse
 Extensions - Discussion

UNC Chapel Hill
S. Redon - M. C. Lin
Algorithm Overview
UNC Chapel Hill
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)
UNC Chapel Hill
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…)
UNC Chapel Hill
S. Redon - M. C. Lin
Outline
Algorithm overview
 Computing constrained accelerations
 Computing a frictional impulse
 Extensions - Discussion

UNC Chapel Hill
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

UNC Chapel Hill
S. Redon - M. C. Lin
Constrained accelerations
UNC Chapel Hill
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
UNC Chapel Hill
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
UNC Chapel Hill
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”
UNC Chapel Hill
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
UNC Chapel Hill
S. Redon - M. C. Lin
Constrained accelerations

The set of possible accelerations is
obtained from the non-penetration
constraints

This is a Projection problem
UNC Chapel Hill
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)
UNC Chapel Hill
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
UNC Chapel Hill
S. Redon - M. C. Lin
Outline
Algorithm overview
 Computing constrained accelerations
 Computing a frictional impulse
 Extensions - Discussion

UNC Chapel Hill
S. Redon - M. C. Lin
Computing a frictional impulse

We consider -one- contact point only
 The problem is formulated in the collision
coordinate system
-j
+j
UNC Chapel Hill
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
– v̂ t : 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.
UNC Chapel Hill
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.
UNC Chapel Hill
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
UNC Chapel Hill
vz is the relative
normal velocity.
(We used vrel
before)
S. Redon - M. C. Lin
Computing a frictional impulse
jz


UNC Chapel Hill
jz is the impulse
magnitude in
the normal
direction.
Wz is the work
done in the
normal
direction.
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: v z  εv z
 Poisson’s Hypothesis: jz  j0z  εj0z
jz  (1 ε)j0z
 Stronge’s Hypothesis: Wz  Wz0  ε 2 Wz0
Wz  (1 ε 2 )Wz0

– Energy of the bodies does not increase
when friction present
UNC Chapel Hill
S. Redon - M. C. Lin
Computing a frictional impulse

Sliding (dynamic) friction
v t  0  ft  μ fn vˆ t

Dry (static) friction
v t  0  ft  μ fn

Assume no rolling friction
UNC Chapel Hill
S. Redon - M. C. Lin
Computing a frictional impulse
 1

1 
~
~
~
~

1

1
Δv(t)   
1  r1I1 r1  r2I2 r2  j(t)  Kj(t)
 m1 m2 

where:
r = (p-x) is the vector from the center of
mass to the contact point
0
~r   r
 z
- ry
UNC Chapel Hill
- rz
0
rx
ry 

- rx 
0 
S. Redon - M. C. Lin
The K Matrix

K is constant over the course of the
collision, symmetric, and positive
definite
UNC Chapel Hill
S. Redon - M. C. Lin
Collision Functions

Change variables from t to something else
that is monotonically increasing during the
collision: Δv(  )  Kj(  )

Let the duration of the collision  0.
The functions v, j, W, all evolve over the
compression and the restitution phases
with respect to .

UNC Chapel Hill
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)
UNC Chapel Hill
S. Redon - M. C. Lin
Sliding or Sticking?

Sliding occurs when the relative
tangential velocity v t  0
– Use the friction equation
to formulate v t  0  ft  μ fn vˆ t

Sticking occurs otherwise
– Is it stable or instable?
– Which direction does the instability get
resolved?
UNC Chapel Hill
S. Redon - M. C. Lin
Sliding Formulation

For the compression phase, use   v z
– v z is the relative normal velocity at the
start of the collision (we know this)
– At the end of the compression phase, v 0z  0

For the restitution phase, use   Wz
– Wz0 is the amount of work that has been
done in the compression phase
– From Stronge’s hypothesis, we know that
Wz  (1 ε 2 )Wz0
UNC Chapel Hill
S. Redon - M. C. Lin
Sliding Formulation

Compression phase equations are:
 vx 
k xξ(θ)
d  
1 
vy 
k yξ(θ)

dv z   k zξ(θ) 
 Wz 
 v z 
UNC Chapel Hill
S. Redon - M. C. Lin
Sliding Formulation

Restitution phase equations are:
v x 
k x ξ(θ)
d   1
1
v y  K ξ(θ) 
k y ξ(θ)

dWz   v z
vz 
 v z 
k zξ(θ)
UNC Chapel Hill
S. Redon - M. C. Lin
Sliding Formulation
where the sliding vector is:
 μv x

2  v2 
v

x
y
 μcosθ 


μv
y
ξ(θ)    μsinθ   

2
2

 
vx  vy 
 1  
1



UNC Chapel Hill
S. Redon - M. C. Lin
Sliding Formulation

These equations are based on the
sliding mode
v t  0  ft  μ fn vˆ t

Sometimes, sticking can occur during
the integration
UNC Chapel Hill
S. Redon - M. C. Lin
Sticking Formulation
UNC Chapel Hill
S. Redon - M. C. Lin
Sticking Formulation

Stable if K
  K   μ K 
1 2
13
1 2
23
2
1 2
33
– 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.
UNC Chapel Hill
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.

UNC Chapel Hill
S. Redon - M. C. Lin
Outline
Algorithm overview
 Computing constrained accelerations
 Computing a frictional impulse
 Extensions - Discussion

UNC Chapel Hill
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
UNC Chapel Hill
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
UNC Chapel Hill
S. Redon - M. C. Lin
Extensions - Discussion

Problems with micro-collisions
– creeping : a block on a ramp can’t be stabilized
UNC Chapel Hill
S. Redon - M. C. Lin
Extensions - Discussion

Problems with micro-collisions
– creeping : a block on a ramp can’t be stabilized
UNC Chapel Hill
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)
UNC Chapel Hill
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
UNC Chapel Hill
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


z
z
v  εv

Problem :
– constraints are expressed on velocities and forces (!)
This can add energy to the system
– Integrating Stronge’s hypothesis in this formulation ?
UNC Chapel Hill
S. Redon - M. C. Lin
Extensions - Discussion

Extending the projection problem
– accelerations are replaced by velocities
– constraints are expressed on velocities and forces


z
z
v  εv
UNC Chapel Hill
S. Redon - M. C. Lin
Extensions - Discussion

Extending the projection problem
– accelerations are replaced by velocities
– constraints are expressed on velocities


z
z
v  εv

Problem :
– constraints are expressed on velocities (!)
– This can add energy to the system
UNC Chapel Hill
S. Redon - M. C. Lin