Impact Controllability of an Air Hockey Puck

In this paper we investigate the controllability of an air hockey puck subject to impacts from a mallet, which is a special case of the more general problem of impulsive manipulation. We rst formulate the control problem on the tangent bundle of the three dimensional connguration manifold of the puck and 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 reene 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 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 coeecients of friction and restitution. We show that the system satisses an Accessibility-like property as long as the coeecient of friction is nonzero.