A theoretical analysis of billiard ball dynamics under cushion impacts

The last two decades have seen a growing interest in research related to billiards. There have been a number of projects aimed at developing training systems, robots, and computer simulations for billiards. Determination of billiard ball trajectories is important for all of these systems. The ball’s collision with a cushion is often encountered in billiards and it drastically changes the ball trajectory, especially when the ball has spin. This work predicts ball bounce angles and bounce speeds for the ball’s collision with a cushion, under the assumption of insignificant cushion deformation. Differential equations are derived for the ball dynamics during the impact and these equations are solved numerically. The numerical solutions together with previous experimental work by the authors predict that for the ball–cushion collision, the values of the coefficient of restitution and the sliding coefficient of friction are 0.98 and 0.14, respectively. A comparison of the numerical and experimental results indicates that the limiting normal velocity under which the rigid cushion assumption is valid is 2.5 m/s. A number of plots that show the rebound characteristics for given ball velocity–spin conditions are also provided. The plots quantify various phenomena that have hitherto only been described in the billiards literature

Ball positioning in robotic billiards: a nonprehensile manipulation-based solution

The development and testing of a robotic system to play billiards is described in this paper. The last two decades have seen a number of developments in creating robots to play billiards. Although the designed systems have uccessfully incorporated the kinematics required for gameplay, a system level approach needed for accurate shot- making has not been realized. The current work considers the different aspects, like machine vision, dynamics, robot design and computational intelligence, and proposes, for the first time, a method based on robotic non-prehensile manipulation. High-speed video tracking is employed to determine the parameters of balls dynamics. Furthermore, three-dimensional impact models, involving ball spin and friction, are developed for different collisions. A three degree of freedom manipulator is designed and fabricated to execute shots. The design enables the manipulator to position the cue on the ball accurately and strike with controlled speeds. The manipulator is controlled from a PC via a microcontroller board. For a given table scenario, optimization is used to search the inverse dynamics space to find best parameters for the robotic shot maker. Experimental results show that a 90% potting accuracy and a 100–200 mm post-shot cue ball positioning accuracy has been achieved by the autonomous system

Trajectory solutions for a game-playing robot using nonprehensile manipulation methods and machine vision

The need for autonomous systems designed to play games, both strategy-based and physical, comes from the quest to model human behaviour under tough and competitive environments that require human skill at its best. In the last two decades, and especially after the 1996 defeat of the world chess champion by a chess-playing computer, physical games have been receiving greater attention. Robocup TM, i.e. robotic football, is a well-known example, with the participation of thousands of researchers all over the world. The robots created to play snooker/pool/billiards are placed in this context. Snooker, as well as being a game of strategy, also requires accurate physical manipulation skills from the player, and these two aspects qualify snooker as a potential game for autonomous system development research. Although research into playing strategy in snooker has made considerable progress using various artificial intelligence methods, the physical manipulation part of the game is not fully addressed by the robots created so far. This thesis looks at the different ball manipulation options snooker players use, like the shots that impart spin to the ball in order to accurately position the balls on the table, by trying to predict the ball trajectories under the action of various dynamic phenomena, such as impacts. A 3-degree of freedom robot, which can manipulate the snooker cue on a par with humans, at high velocities, using a servomotor, and position the snooker cue on the ball accurately with the help of a stepper drive, is designed and fabricated. [Continues.