Transcript Handling of Resting Contact  Global vs. Local Methods – Constraint-based vs. Impulse-based Colliding Contacts  Resting Contacts  Force application  Friction  UNC Chapel Hill M.

Handling of Resting Contact

Global vs. Local Methods
– Constraint-based vs. Impulse-based
Colliding Contacts
 Resting Contacts
 Force application
 Friction

UNC Chapel Hill
M. C. Lin
Impulse vs. Constraint

Impulse-based dynamics (local)
– Faster
– Simpler
– No explicit contact constraints

Constraint-based dynamics (global)
– Must declare each contact to be a resting
contact or a colliding contact
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M. C. Lin
Impulse vs. Constraint

Impulse-based dynamics (local)

Constraint-based dynamics (global)
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M. C. Lin
Impulse vs. Constraint
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M. C. Lin
Resting Contact Response
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
M. C. Lin
Resting Contact Response
UNC Chapel Hill
M. C. Lin
Resting Contact Response

The forces at each contact must
satisfy three criteria
– Prevent inter-penetration: di (t 0 )  0
– Repulsive -- we do not want the objects
to be glued together: fi  0
– Should become zero when the bodies
start to separate: f d (t )  0
i i

0
To implement hinges and pin joints:
di (t 0 )  0
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M. C. Lin
Resting Contact Response

We can formulate using LCP:
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M. C. Lin
Resting Contact Response

Need to solve a quadratic program to
solve for the fi’s
– A is symmetric positive semi-definite
(PSD) making the solution practically
possible

There is an iterative method to solve
for without using a quadratic
program [3]
UNC Chapel Hill
M. C. Lin
Global / Constraint-Based:
Apply_BB_Forces()
0
1
2
3
4
5
6
7
8
for each pair of bodies (A,B) do
if Distance(A,B) <  then
if Relative_Normal_Velocity(A,B) < - then
Cs = Contacts(A,B);
Apply_Impulses(A,B,Cs);
else if Relative_Normal_Velocity(A,B) <  then
Flag_Pair(A,B);
Solve_For_Forces(flagged pairs);
Apply_Forces(flagged pairs);
UNC Chapel Hill
M. C. Lin
Local / Impluse-Based:
Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2
if Relative_Normal_Velocity(A,B) < - then
3
Cs = Contacts(A,B);
4
Apply_Impulses(A,B,Cs);
5
else if Relative_Normal_Velocity(A,B) <  then
6
Compute_Restitution(A,B);
7
Cs = Contacts(A,B);
8
Apply_Impulses(A,B,Cs);
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M. C. Lin
Algorithm Overview
0 Initialize();
1 for t = 0; t < tf; t += h do
2 Read_State_From_Bodies(S);
3 Compute_Time_Step(S,t,h);
4 Compute_New_Body_States(S,t,h);
5 Write_State_To_Bodies(S);
6 Zero_Forces();
7 Apply_Env_Forces();
8 Apply_BB_Forces();
UNC Chapel Hill
M. C. Lin
Global / Constraint-Based:
Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2
if Relative_Normal_Velocity(A,B) < - then
// Colliding Contact
3
Cs = Contacts(A,B);
4
Apply_Impulses(A,B,Cs);
5
else if Relative_Normal_Velocity(A,B) <  then
// Resting Contact
6
Flag_Pair(A,B);
7 Solve_For_Forces(flagged pairs);
8 Apply_Forces(flagged pairs);
UNC Chapel Hill
M. C. Lin
Local / Impulse-Based:
Apply_BB_Forces()
0 for each pair of bodies (A,B) do
1 if Distance(A,B) <  then
2
if Relative_Normal_Velocity(A,B) < - then
// Normal collision
3
Cs = Contacts(A,B);
4
Apply_Impulses(A,B,Cs);
5
else if Relative_Normal_Velocity(A,B) <  then
// Micro-collision
6
Compute_Restitution(A,B);
7
Cs = Contacts(A,B);
8
Apply_Impulses(A,B,Cs);
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M. C. Lin
Compute_Restitution(A,B)

є = f( Distance(A,B) )
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M. C. Lin
Frictional Forces Extension

Constraint-based dynamics
– Reformulate constraints and solve
– No more on this here

Impulse-based dynamics
– Must not add energy to the system in the
presence of friction so we have to
reformulate the impulse to be applied
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M. C. Lin
Collision Coordinate System

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p is the
applied
impulse.
We use j
because P
is for linear
momentum
M. C. Lin
Impulse Reformulation

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.
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M. C. Lin
Impulse Reformulation


The collision is instantaneous but we can
assume that it occurs over a very small
period of time: 0  tmc  tf.
tmc is the time of maximum compression
uz is the relative
normal velocity.
We used vrel
before. From
now on we will
use vz.
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M. C. Lin
Impulse Reformulation


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pz is the
impulse
magnitude in
the normal
direction. We
used j before.
From now on
we will use jz.
Wz is the work
done in the
normal
direction.
M. C. Lin
Impulse Reformulation
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
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M. C. Lin
Friction Formulae

Assume the Coulomb friction law:
– At some instant during a collision between
bodies 1 and 2, let v be the contact point
velocity of body 1 relative to the contact
point velocity of body 2. Let vt be the
tangential component of v and let vˆ t be a
unit vector in the direction of vt. Let fz and
ft be the normal and tangential (frictional)
components of force exerted by body 2 on
body 1, respectively.
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M. C. Lin
Friction Formulae

Sliding (dynamic) friction
v t  0  ft  μ fn vˆ t

Dry (static) friction
v t  0  ft  μ fn

Assume no rolling friction
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M. C. Lin
Impulse with Friction

Recall that the impulse looked like this
for frictionless collisions:

Recall also that vz = j/m
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Impulse with Friction
 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
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- rz
0
rx
ry 

- rx 
0 
M. C. Lin
The K Matrix

K is constant over the course of the
collision, nonsingular, symmetric, and
positive definite
k x 


K  ky
 
k z 
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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 .

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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.
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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?
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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
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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 
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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)
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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



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M. C. Lin
Sliding Formulation

Notice that there is a problem at the
point of maximum compression
because vz = 0:
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)
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M. C. Lin
Sliding Formulation
Let us integrate using vz a bit into the
restitution phase (extension integration)
so that we never divide by 0.
 But we don’t want to integrate too far!
Otherwise we exceed the amount of work
that is to be done in the restitution
phase.
 We are safe if we stop at:
2  K2 
v z  2Wz  Wz0 K 33  μ K 31
32

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Sliding Formulation
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M. C. Lin
Sliding Formulation
There is another problem if the
tangential velocity becomes 0 because
the equations that we have derived
were based on v t  0  ft  μ fn vˆ t
which no longer holds.
 This brings us to the sticking
formulation

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Sticking Formulation
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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.
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M. C. Lin
Sticking Formulation

If sticking occurs, then the remainder
of the collision may be integrated
analytically due to the existence of
closed form solutions to the resulting
simplified equations.
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M. C. Lin
Static Contacts
What about a block on a ramp when the
static friction force is greater than
gravity? Will it not creep down?
 The solution is to create another type
of micro-collision which reverses the
initial relative velocity. It is applied if it
lies within the friction “cone” otherwise
the standard collision integration is
performed.

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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.

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M. C. Lin