Trajectory tracing in figure skating

TL;DR

This paper models figure skating movements using a nonholonomic system called the Chaplygin sleigh, enabling precise control of skater trajectories through optimization of control inputs for pattern reproduction.

Contribution

It introduces a novel control method for figure skating patterns based on the Chaplygin sleigh model, allowing exact tracing of prescribed trajectories with optimized controls.

Findings

Successfully reproduces a classical 'double flower' skating pattern

Demonstrates precise control of trajectories using circular arc approximation

Provides a framework for designing complex skating patterns with control algorithms

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.