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A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting

This paper introduces a new model for modeling simultaneous collisions in billiard shooting using a state transition diagram. The model includes a new law of restitution and allows for the analysis of frictional impact. The paper also includes experiments and simulations to validate the model.

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A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting

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  1. 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 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213, USA December 7, 2008

  2. 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) Han & Gilmore (1989); Wang & Mason (1991); Ahmed et al. (1999) Hardly applicable in 3D, where impulse builds along a curve. Keller (1986) – differential equation often with no closed form solution.

  3. 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).

  4. 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)

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

  6. 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.

  7. Compression An impact starts with compression (of the virtual spring). The virtual spring stores energy. The phase ends when the spring length stops decreasing. 2 Maximum elastic energy. 1 2

  8. Restitution The virtual spring releases energy. Compression Restitution Impulse Poisson’s law of impact: coefficient of restitution

  9. State Transition Diagram  The two impacts almost never start or end restitution at the same time.  An impact may be reactivated after restitution.

  10. 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. Our model: limit the amount of energy released during restitution to be a fixed ratio of that accumuluated during compression.

  11. 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!

  12. Example kg m/s ball-ball compression lines -3 0.61 0.97 ball-table 0.36 0 2.44 2.44 0.74 -0.12

  13. Energe Curve energy loss of lower ball ending restitution total loss of energy: 1.2494.

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

  15. Ping Pong Experiment

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

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

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

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

  20. Tangential Impulses & Contact Modes Coloumb’s law of friction. Involved analysis based on State Sliding or sticking Compression or restitution

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

  22. Mechanical Cue Stick

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

  24. Acknowledgment Iowa State University Carnegie Mellon University DARPA (HR0011-07-1-0002) Amir Degani & Ben Brown (CMU)

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