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

Yan-Bin Jia, Matthew T. Mason, and Michael Erdmann
Eighth International Workshop on the Algorithmic Foundations of Robotics, December, 2008.


Abstract
This paper models a multibody collision in the impulse space as a state transition diagram, where each state represents a phase during which impacts are active at only a subset of the contact points. A state transition happens whenever an active impact finishes restitution, or an inactive impact gets reactivated, depending on whether the two involved bodies are instantaneously penetrating into each other or not. The elastic energy due to an impact is not only affected by the impulse at the corresponding contact point, but also by other impulses exerted on the two involved bodies during the impact. Consequently, Poisson’s impulse-based law of restitution could result in negative energy. A new law governing the loss of elastic energy during restitution is introduced. Convergence of the impulse sequence generated by the state transition diagram is established. The collision outcome depends on the ratios of the contact stiffnesses rather than on their individual values. The collision model is then applied in an analysis of billiard shooting in which the cue stick impacts the cue ball, which in turn impacts the pool table.

Notes

Text Reference
Yan-Bin Jia, Matthew T. Mason, and Michael Erdmann, "A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting," Eighth International Workshop on the Algorithmic Foundations of Robotics, December, 2008.

BibTeX Reference
@inproceedings{Jia_2008_6821,
   author = "Yan-Bin Jia and Matthew T. Mason and Michael Erdmann",
   title = "A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting",
   booktitle = "Eighth International Workshop on the Algorithmic Foundations of Robotics",
   month = "December",
   year = "2008",
}