Numerical simulations of the frictional collisions of solid balls on a rough surface

Three-dimensional simulations of the frictional collision between solid balls moving on a rough surface are analyzed in this paper. The analysis is performed in the context of pool and snooker, two popular games in the pocket billiards family. Accurate simulations of ball motion in billiard games are useful for television broadcasts, training systems and any robotic game playing systems. Studying solid ball collisions in a three-dimensional space requires careful consideration of the different phenomena involved in ball motion such as rolling, sliding and ball spin about a general axis. A set of differential equations are derived describing ball dynamics during collisions. In the absence of explicit analytical solutions to the differential equations, a numerical procedure is performed to determine post-collision ball velocities and spins after collision. In addition, the paper also presents a methodology to analyze the curved, slip trajectories of balls immediately after impact. The results presented here, when compared with some experimental shots, show that the percentile errors in post-collision velocities are reduced by the proposed method. The prediction accuracies for ball travel direction are increased twofold by the proposed impact simulation algorithm.