Simultaneous Three-Dimensional Impacts
with Friction and Compliance
Yan-Bin Jia
Department of Computer Science
Iowa State University
Ames, IA 50010, USA
Sep 27, 2012
Department of Computer Science, Iowa State University
Why Impact?

Because impact is everywhere …
Accomplishing tasks otherwise very difficult.
Reduction of harmful impulsive forces
Efficiency over static and dynamic forces

Because it is relevant to robotics…
Collision between robots and environments, walking robots …
Impulsive manipulation being underdeveloped in robotics
Higuchi (1985); Izumi and Kitaka (1993); Hirai et al. (1999)
Huang & Mason (2000); Han & Park (2001), Tagawa et al. (2010)
Foundation of impact not fully laid out
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Describing Impact by Impulse
 “Infinitesimal” duration
before
after
 “Infinite” contact force
 Finite change in momentum
m(v1  v0 ) 

t
0
I
v0
v1
Fdt
impulse
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Impact Phase 1 – Compression
 Contact point  virtual spring.
 The spring changes its length by
storing elastic energy E .
x,
v
p
 Compression ends when the spring length
stops decreasing:
v0
E  Emax
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Transition at End of Phase 1
 Loss of energy at transition to restitution:
E  e Emax
2
 [0,1] : energy coefficient of restitution
v
p
 Increase in stiffness at transition to restitution:
k0  k0 / e 2
F  kx  F
x  e2 x
1 2
E  kx  e 2 E
2
change in spring length
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Impact Phase 2 – Restitution
 During restitution the normal spring (n-spring)
releases the remaining e 2 E .
 Restitution ends when
E 0
v
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Impulse-Energy Relationship
dE
I 

  x  v   v0  
dI
m

Energy is a piecewise quadratic function of impulse.
One-to-one correspondence between impulse and time.
Describe the process of impact in terms of impulse.
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Example of General 3D Impact – Billiard Shooting
 Two contacts:
- cue-ball
- ball-table
normal impulses
 At each contact:
- normal impulse
- tangential impulse
I1
I2
tangential
impulses
How are they related
at the same contact?
(compliance & friction)
 At different contacts:
How are the normal
impulses related?
(simultaneous impacts)
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Talk Outline
I. Impact with Compliance
Relationship of tangential impulse to normal impact at single contact
(also with friction)
II. Simultaneous Impacts
Relationships among normal impacts at different contacts
(no friction or compliance)
III. Model Integration (Billiard Shooting)
Simultaneous impacts with friction and compliance
Department of Computer Science, Iowa State University
Related Work on Impact
 Newton’s law (kinematic coefficient of restitution)
Maclaurin (1742); Bernoulli (1969); Ivanov (1995)
 Energy increase
Wang et al. (1992); Wang & Mason (1992)
 No post-impact motion of a still object
Brogliato (1999); Liu, Zhao & Brogliato (2009)
 Poisson’s hypothesis (kinetic coefficient of restitution)
Routh (1905); Han & Gilmore (1989); Ahmed et al. (1999); Lankarani (2000)
Darboux (1880); Keller (1986); Bhatt & Koechling (1994);
Glocker & Pfeiffer (1995); Stewart & Trinkle (1996); Anitescu & Portta (1997)
 Energy increase
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Related Work (cont’d)
 Energy-Based Restitution (energetic coefficient of restitution)
Stronge (1990); Wang et al. (1992);
 Energy conservation
 Simultaneous Collisions
Chatterjee & Ruina (1998); Ceanga & Hurmulu (2001); Seghete & Murphey (2010)
Liu, Zhao & Brogliato (2008, 2009); Jia, Mason & Erdmann (2008)
 Tangential Impulse & Compliance
Mindlin (1949); Maw et al. (1976)
Smith (1991); Bilbao et al. (1989); Brach (1989)
Stronge (1994; 2000); Zhao et al. (2009); Hien (2010) ; Jia (2010)
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Impact with Compliance
t
I   fdt I n  I 
0
Normal impulse:
1. accumulates during impact
(compression + restitution)
2. energy-based restitution
3. variable for impact analysis
Tangential impulse:
1. due to friction & compliance
2. dependent on contact modes
3. driven by normal impulse
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Compliance Model
 Gravity ignored in comparison with impulsive force.
 Extension of Stronge’s contact structure to 3D.
 Analyze impulse in contact frame:
I  (Iu , I w , I n )
F
tangential
impulse
opposing initial
tangential contact
velocity
massless
particle
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Normal vs Tangential Stiffnesses
k : stiffness of normal spring (value varying with impact phase)
k  : stiffness of tangential u- and v-springs (value invariant)
Stiffness ratio:
  k0 / k
2
0
Depends on Young’s moduli and Poisson’s ratios of materials.
  k / k
2
  0
  0 / e
(compression)
(restitution)
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Normal Impulse as Sole Variable
Idea: describe the impact system in terms of normal impulse.
Key fact:
In  dI n / dt  Fn  2kEn
 Derivative well-defined at the impact phase transition.
En '  dEn / dI n  vn

Iu '  

   1
  1
Eu
En

Iw '  

Ew
En
(1 if extension of tangential u- and w-springs
–1 if compression)
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Tangential Springs
 Cannot determine changes
u and v in length without knowing stiffness.
 Can keep track of integrals
of
u
,
En
v
En
Gu , Gv
over
In
 Elastic strain energies:
2
u
2
0
G
Eu 
4
2
w
2
0
G
Eu 
4
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System Overview
v
Impact
Dynamics
& Contact
Kinematics
In
integrate
Contact
Mode
Analysis
En
I
Iu , Iv
vn
integrate
Iu ' , Iv '
Eu , Ev
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Sliding Velocity
v :
tangential contact velocity
from contact kinematics
vs :
sliding velocity represented by
velocity of particle p
v
vs
vs  v  (u, w ,0)
Sticking contact if vs
0
.
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Stick or Slip? Energy-Based Criteria
By Coulomb’s law, the contact sticks , i.e., v
s
F  F  Fn
2
u
0
if
Iu2  Iw2  In
2
w
Eu  Ew    En
2
Slips if
2
Eu  Ew    En
2
2
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Contact Mode Transitions
 Stick
to slip when
Eu  Ew    En
2
 Slip
2
to stick when
vs  0
i.e,
 ,0)
v  (u, w
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Sticking Contact
 Rates of change in spring length.
0  vs  v  (u, w ,0)
u  v  (1,0,0)
w  v  (0,1,0)
 Particle p in simple harmonic motion like a spring-mass system.
(No energy dissipation tangentially)
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Sliding Contact

u, w
can also be solved (via involved steps).
 Energy dissipation rate (tangentially):

E'  I n vs
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System of Impact with Compliance
Differential equations with five functions and one variable
In
:
I 'u  f1 ( En , Gu )
Tangential impulses
Length (scaled) of
tangential springs
Energy stored by
the normal spring
I 'w  f 2 ( En , Gu )
G'u  f 3 ( I n , En , Gu , Gw )
G'w  f 4 ( I n , En , Gu , Gw )
E 'n  f 5 ( I n , I u , I w )
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Bouncing Ball
z
Physical parameters:
m 1
r 1
  0.4
e  0.5
 02  (2  0.3) /( 2  2  0.3)  1.2
Before 1st impact:
V0  (1,0,5)
0  (0,2,0)
After 1st impact:
V  (0.570982,0,2.5)
x
v0  0
  (0,1.92746,0)
All bounces are in one vertical plane
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Impulse Curve (1st Bounce)
contact mode
switch
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Non-collinear Bouncing Points
V0  (1,0,5)
0  (6,6,0)
v0  0
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Trajectory Projection onto Table
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Bounce of a Pencil
Pre-impact:
V0  5(cos

6
,0, sin

6
)
0  (1,0.5,0.5)
Post-impact:
V0  (0.3681,0.3908,3.0302)
0  (0.2362,1.8021,0.5)
z
x
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Impulse Curve
(3.962,0.3908,5.5302)
end of
compression
z
stick
y
slip
slip
Slipping direction varies.
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Where Are We?
I. Impact with Compliance
Relationship of tangential impulse to normal impact at single contact
(also with friction)
II. Simultaneous Impacts
Relationships among normal impacts at different contacts
(no friction or compliance)
III. Model Integration (Billiard Shooting)
Simultaneous impacts with friction and compliance
Department of Computer Science, Iowa State University
Simultaneous Collisions in 3D
 High-speed photographs shows >2 objects simultaneously
in contact during collision.
 Lack of a continuous impact law.
Our model
 Collision as a state sequence.
 Within each state, a subset of impacts are “active”.
 Energy-based restitution law.
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Two-Ball Collision
Problem: One rigid ball impacts another
resting on the table.
Question: Ball velocities after the impact?
virtual springs
x1  v1  v2
x2  v2
m1v1  F1  k1x1
m2 v2  F2  F1  k1 x1  k 2 x2
(kinematics)
(dynamics)
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Impulses, Velocities & Stain Energies
Ball-ball impulse:
I 1  F 1dt
Ball-table impulse:
I 2  F 2 dt
Velocities:
1
v1  v0  I 1
m1
v2 
1
( I 2 I 1)
m2
Rates of change in contact strain energies:

 1
dE1
1 
1 

 I1 
  v0   
I2 
dI1
m2 
 m1 m2 

dE2
1
 I1  I 2 

dI 2 m2
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State Transition Diagram
 The two impacts almost never start or end restitution at the same time.
 An impact may be reactivated after restitution.
spring 2 ends
restitution first
spring 1 ends
restitution first
e2  0 and
v2  0 before
spring 1 ends
restitution
e1  0 and
v1  v2 before
spring 2 ends
restitution
both springs end
restitution together
s4
otherwise, spring 2
ends restitution
otherwise, spring 1
ends restitution
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Some Facts
 A change of state happens when a contact disappears or a
disappeared contact reappears.
 Impulse & strain energy for one impact also depend on those
for the other (correlation).
 Compression may restart from restitution within an impact
state due to the coupling of impulses at different contacts.
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Assumptions on Simultaneous Impacts
Every end of compression of single impact:
 increase in stiffness
 invariance of contact force
No change in stiffness
 when restitution switches back to compression, or
 when a contact is reactivated.
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Stiffness, Mass, and Velocity Ratios
Theorem 2
 Collision outcome depends on the contact stiffness
ratio but not on individual stiffness.
 The outcome does not change if the ball masses
scale by the same factor.
 Output/input velocity ratio is constant (linearity).
Consider upper ball with unit mass and unit downward velocity.
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Impulse Curve
I2
Theorem 3 During the collision,
the impulses I 1,I 2 
accumulate along a curve
that is first order continuous
and bounded within an ellipse.
1 2
1
I1 
( I1  I 2 ) 2  v0 I1  0
2m1
2m2
lines of
compression
S4
v1  0.94
v2  0.30
S1
S2
S1
2
m1  1 m2 
v0  1
3
k2
 1 e1  e2  1
k1
I1
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Example with Energy Loss
I2
E2
energy
loss
I1
2
m

m1  1
2
3
k2
1
k1
e1  0.9 e2  0.7
energy
loss
E1
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Convergence
I
(i )
1

, I 2(i ) :
impulses at the end of the ith state in the sequence.


Sequence {( II(1(ii)),, II 2((ii)) )}:
1
2
 monotone non-decreasing
 bounded within an ellipse
Theorem 4 : The state transition will either terminate with
v1 v 2  0 or the sequence will converge with
either v1 v 2  0 or v1 v 2  0 .
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Ping Pong Experiment
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Experiment
m  0.00023 kg
(m / s)
v1
Measured values:
e1  0.950043
(ball-ball)
e2  0.846529
(ball-table)
v2
Guessed value:
k2
3
k1
same trial
 v0
(m / s)
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Where Are We?
I. Impact with Compliance
Relationship of tangential impulse to normal impact at single contact
(also with friction)
II. Simultaneous Impacts
Relationships among normal impacts at different contacts
(no friction or compliance)
III. Model Integration (Billiard Shooting)
Simultaneous impacts with friction and compliance
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Billiard Shooting
c
I1
n
z
I2
Simultaneous impacts: cue-ball and ball-table!
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Contact Structures
Cue-ball contact
Ball-table contact
 Normal impulses at the two contacts are described by the simultaneous
impact model.
 At each contact, normal impulse drives tangential impulse as described
by the compliance model.
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Combing the Two Impact Models
normal CB
impulse
vc ,
normal BT
impulse
The two normal impulses take turns to drive the system.
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Mechanical Cue Stick
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A Masse Shot
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reconstructed trajectory
from video
Shot video
if not ended by cushion
Trajectory fitting
Post-shot ball velocities
Impact model
predicted trajectory
by model
Predicted post-shot ball velocities
increasing cue-ball
compliance
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Conclusion
• 3D impact modeling with compliance and friction
•
•
•
•
elastic spring energies
impulse-based not time-based
contact mode analysis (stick / slip)
sliding velocity computation
• Multiple impacts
•
•
•
•
•
state transition diagram
impulse curve
stiffness ratio
scalability
convergence
• Physical experiment.
• Integration of two impact models
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Extensions of Collision Model
 Rigid
bodies with arbitrary geometry
 General simultaneous multibody collision
 ≥3
contact points
 State
transition templates
 Measurement
 Robot
of relative contact stiffness
pool player!
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Acknowledgement
HR0011-07-1-0002
Matt Mason, Michael Erdmann, Ben Brown (CMU)
Amir Degani (Israel Institute of Technology)
Rex Fernando, Feng Guo (ISU students)
Department of Computer Science, Iowa State University
Online Papers
International Journal of Robotics Research, 2012:
1. Yan-Bin Jia. Three-dimensional impact: energy-based modeling of tangential
compliance. DOI: 10.1177/0278364912457832.
http://www.cs.iastate.edu/~jia/papers/IJRR11a-submit.pdf
2. Yan-Bin Jia, Matthew T. Mason, and Michael A. Erdmann. Multiple impacts:
a state transition diagram approach. DOI: 10.1177/0278364912461539.
http://www.cs.iastate.edu/~jia/papers/IJRR11b-submit.pdf
Department of Computer Science, Iowa State University
Appendix 1: Start of Impact
Initial contact velocity
v0  (v0u ,0, v0 n )

I w ' (0)  0
 Under Coulomb’s law, we can show that

sticks if
v v
slips if
v v
2
0u
2
0u
2
0w
2
0w
  v
I u ' (0) 
  v
I u ' (0)  
2
2
4 2
0 0n
4 2
0 0n
1
02

v0u
v0 n
…
…
En ' (0)  v0 n
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Bouncing Ball – Integration with Dynamics
Velocity equations:
V  V0  I / m
5
  0 
zI
2mr
(Dynamics)
Contact kinematics
z  (0,0,1)
Iz
7
v  v0  z 
I
m
2m
Theorem 1 During collision, I  is collinear with v0  .
Impulse curve lies in a vertical plane.
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Impulse Curve (1st Bounce)
Tangential contact velocity
vs. spring velocity
v
contact mode
switch
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Bouncing Pencil

3
m 1 r  1
h1  3 h2  0.5
  0 .8
e  0.5
 02  1.2
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