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: Coulombs law of friction. Poissons hypothesis of restitution. Law of energy conservation. Impulse accumulation – Rouths 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 (Poissons law).

Two-Ball Collision Problem: One rigid ball impacts another resting on the table. Q: Ball velocities after the impact? contact points virtual springs (kinematics) (dynamics)

Impulse Impact happens in infinitesimal time. Use impulse: Velocities in terms of impulses

Elastic Energies Stored by the two virtual springs at contacts. Dependent on the impulses: Relationship between the two impulses: relative stiffness Governing differential equation of the impact.

Compression An impact starts with compression (of the virtual spring) The phase ends when the spring length stops decreasing. The virtual spring stores energy. Maximum elastic energy.

Restitution The virtual spring releases energy. Poissons law of impact: Compression Restitution Impulse 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 Poissons 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. Our model : limit the amount of energy released during restitution to be a fixed ratio of that accumuluated during compression.

A Couple of Theorems Theorem 1 (Stiffness Ratio) : Outcome of collision depends on but not on their individual values. Theorem 2 (Bounding Ellipse) : The impulses satisfy Inside an ellipse!

Example kgm/s ball-ball ball-table compression lines

Energe Curve energy loss of lower ball ending restitution total loss of energy:

Convergence : impulses at the end of the ith state. monotone nondecreasing. bounded within the ellipse. Sequence : Theorem 3 (Convergence) : The state transition will either terminate or the sequence will converge with either or.

Ping Pong Experiment

Experiment vs. Simulation upper ball velocity lower ball velocity kg estimated cofficients of restitution: (ball-ball) (ball-table)

Billiard Shooting Simultaneous impacts: cue-ball and ball-table!

Change in Velocities cue stick: cue ball: contact velocities: (cue-ball) (ball-table) M m

Normal Impulses Three states based on active impacts: Apply the state transition diagram based on the normal impulses. 1.Cue-ball and ball-table impacts. 2.Ball-table impact only. 3.Cue-ball impact only.

Tangential Impulses & Contact Modes Sliding or sticking Coloumbs law of friction. Compression or restitution Involved analysis based on State

A Simulated Masse Shot kg m rolling sliding ball cue After the shot:

Mechanical Cue Stick

Extensions of Collision Model Rigid bodies of arbitrary shapes Linear dependence of velocities on impulses carries over. 3 impact points on each body Within a state, a subset of impacts are active. angular inertia matrix

Acknowledgment Iowa State University Carnegie Mellon University DARPA (HR ) Amir Degani & Ben Brown (CMU)