Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint

Abstract

The Chaplygin sleigh is a canonical problem of mechanical systems with nonholonomic constraints. Such constraints often arise due to the role of a no-slip requirement imposed by friction. In the case of the Chaplygin sleigh, it is well known that its asymptotic motion is that of pure translation along a straight line. Any perturbations in angular velocity decay and result in an increase in asymptotic speed of the sleigh. Such motion of the sleigh is under the assumption that the magnitude of friction is as high as necessary to prevent slipping. We relax this assumption by setting a maximum value to the friction. The Chaplygin sleigh is then under a piecewise-smooth nonholonomic constraint and transitions between “slip” and “stick” modes. We investigate these transitions and the resulting nonsmooth dynamics of the system. We show that the reduced state space of the system can be partitioned into sets of distinct dynamics and that the stick–slip transitions can be explained in terms of transitions of the state of the system between these sets.