Impact controllability of an air hockey puck

Abstract

In this paper, we investigate the controllability of an air hockey puck sliding on a frictionless table and subject to impacts from a mallet. This problem is a special case of the more general problem of impulsive manipulation. We first characterize the set of velocities reachable with arbitrary impulsive inputs. In reality, one cannot specify impulsive forces as inputs but rather only the velocity of a mallet impacting the puck. Therefore, we refine the characterization to the case that the impulsive forces arise through impacts with a mallet whose velocity can be arbitrarily controlled. The characterization of the reachable puck velocities then depends on the material properties of the impacting objects. We use the two-dimensional Routh impact model to determine these reachable velocities in terms of the coefficients of friction and restitution. We show that the system satisfies an accessibility-like property as long as the coefficient of friction is nonzero.