a state transition diagram for simultaneous collisions with application in billiard shooting yan-bin...
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A State Transition Diagram for Simultaneous Collisions with Application
in Billiard Shooting
Yan-Bin Jia Matthew Mason Michael Erdmann
Department of Computer Science Iowa State University Ames, IA 50011, USA
December 7, 2008
School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213, USA
Frictional Impact
Controversy over analysis of frictional impact:
Coulomb’s law of friction.
Poisson’s hypothesis of restitution.
Law of energy conservation.
Impulse accumulation – Routh’s method (1913)
Hardly applicable in 3D, where impulse builds along a curve.
Han & Gilmore (1989); Wang & Mason (1991); Ahmed et al. (1999)
Keller (1986) – differential equation often with no closed form solution.
Simultaneous Collisions in 3D
No existing impact laws are known to model well.
Stewart & Trinkle (1996); Anitescu & Porta (1997) Chatterjee & Ruina (1998);
High-speed photographs shows >2 objects simultaneously in contact during collision.
Lack of a continuous impact law.
We introduce a new model:
Collision as a state sequence.
Within each state, a subset of impacts are “active”.
A new law of restitution overseeing the loss of elastic energy rather than growth of impulse (Poisson’s law).
Two-Ball Collision
Problem: One rigid ball impacts another resting on the table.
Q: Ball velocities after the impact?
contact points
virtual springs
211 vvx 22 vx
11111 xkFvm
22111222 xkxkFFvm
(kinematics)
(dynamics)
Impulse
Impact happens in infinitesimal time.Use impulse:
FdtI
Velocities in terms of impulses
11
)0(11
1I
mvv
)(1
122
)0(22 II
mvv
Elastic Energies
Stored by the two virtual springs at contacts.
Dependent on the impulses:
122
21
211
)0(1
)0(21
1)
11(2
1)( dII
mI
mmIvvE
212
22
22
)0(22
1
2
1dII
mI
mIvE
Relationship between the two impulses:
),( 211
)0(1
2)0(
2
1
2
1
2 IIfEE
EE
k
k
dI
dI
relative stiffness
Governing differential equation of the impact.
Compression
An impact starts with compression (of the virtual spring).
2
1
201 x 0
1)
11( 2
21
210 I
mI
mmv
02 x 21 II
The phase ends when the spring length stops decreasing.
The virtual spring stores energy.
Maximum elastic energy.
Restitution
The virtual spring releases energy.
Poisson’s law of impact:
Compression Restitution
Impulse cI rI
cr IeI
coefficient of restitution
State Transition Diagram The two impacts almost never start or end restitution at the same time.
An impact may be reactivated after restitution.
Energe-Based Restitution Law
Poisson’s law based on impulse is inadequate because
Impulse & elastic energy for one impact also depend on the impulse for the other.
Not enough elastic energy left to provide the impulse increment during restitution.
122
21
211
)0(1
)0(21
1)
11(2
1)( dII
mI
mmIvvE
Our model: limit the amount of energy released during restitution to bea fixed ratio of that accumuluated during compression.
cr EeE 2
A Couple of Theorems
Theorem 1 (Stiffness Ratio) : Outcome of collision dependson but not on their individual values. 12 / kk
Theorem 2 (Bounding Ellipse) : The impulses satisfy
0)(2
1
2
110
221
2
21
1
IvIIm
Im
Inside an ellipse!
Example
1I
2I121 mm
10/ 12 kk9.01 e 4.02 ekg 30 v m/s
-3 0.61
0.36
0.97
0
2.44
-0.12
2.44
0.74
1S 3S
1S2S
ball-ball
ball-table
compressionlines
Energe Curve
energy loss of lower ball ending restitution
total loss of energy: 1.2494.
Convergence
: impulses at the end of the ith state. ),( )(2
)(1
ii II
monotone nondecreasing.
bounded within the ellipse.
0)(2
1
2
110
221
2
21
1
IvIIm
Im
)},{( )(2
)(1
ii IISequence :
Theorem 3 (Convergence) : The state transition will either terminate or the sequence will converge with either or . 021 vv 021 vv
Ping Pong Experiment
Experiment vs. Simulation
upper ball velocity
lower ball velocity
00023.0m kg
estimated cofficientsof restitution:
8078.01 e
8465.02 e
(ball-ball)
(ball-table)
Billiard Shooting
1I
2In
z
c
Simultaneous impacts: cue-ball and ball-table!
Change in Velocities
ccIM
vc )(1
1 cue stick:
cue ball:
contact velocities:
)(1
12 IIm
v
))()((2
5212IrzIrn
mr
)(rnvvv ccb
)(rzvvbt
(cue-ball)
(ball-table)
M
m
Normal Impulses
zI2
nI1
Three states based on active impacts:
Apply the state transition diagram based on the normal impulses.
nI1 zI2
1. Cue-ball and ball-table impacts.
),(
),(
211
212
1
2
zncb
znbt
n
z
IIEk
IIEk
dI
dI
2. Ball-table impact only.3. Cue-ball impact only.
Tangential Impulses & Contact Modes
zI2
nI1
Sliding or sticking
Coloumb’s law of friction.
tI1
tI2
Compression or restitution
Involved analysis based on
State
A Simulated Masse Shot
1673.0m5018.0M
0286.0r 6565.0cbe5163.0bte
6565.0cb1525.0bt
kg
kg
m
rolling
sliding
ballcue
149
)12,1,2( n
)16,0,4()0( cv
)232.6,2938.0,3253.0(v
)178.3,8.179,48.88(
After the shot:
Mechanical Cue Stick
Extensions of Collision Model
Rigid bodies of arbitrary shapes
Linear dependence of velocities on impulses carries over.
QIr ii QQfr ii 0t
≥3 impact points on each bodyWithin a state, a subset of impacts are active.
),,(
),,(
1
1
jjj
jii
j
i
IIEk
IIEk
dI
dI
angular inertia matrix
Acknowledgment
Iowa State University
Carnegie Mellon University
DARPA (HR0011-07-1-0002)
Amir Degani & Ben Brown (CMU)