Trajectory tracing in figure skating

Abstract

In this work, we model the movement of a figure skater gliding on ice by the Chaplygin sleigh, a classic pedagogical example of a nonholonomic mechanical system. The Chaplygin sleigh is controlled by a movable added mass, modeling the movable center of mass of the figure skater. The position and velocity of the added mass act as controls that can be used to steer the skater in order to produce prescribed patterns. For any piecewise smooth prescribed curve, this model can be used to determine the controls needed to reproduce that curve by approximating the curve with circular arcs. Tracing of the circular arcs is exact in our control procedure, so the accuracy of the method depends solely on the accuracy of approximation of a trajectory by circular arcs. To reproduce the individual elements of a pattern, we employ an optimization algorithm. We conclude by reproducing a classical "double flower" figure skating pattern and compute the resulting controls.