Stabilization of Nonholonomic Euler–poincaré Mechanical Systems With Broken Symmetry by Controlled Lagrangians

We extend the method of Controlled Lagrangians to nonholonomic Euler–Poincaré mechanical systems with broken symmetry by considering the problem of stabilizing what we call a pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. By exploiting the symmetry of the system as well as taking care of the part of the symmetry broken by the gravity, the equations of motion are given as nonholonomic Euler–Poincaré equation with advected parameters. After that, we discovered the general form of the equilibrium points and presented the classification of two special ones, designated as sliding and spinning. Of our main interest is the stability of the sliding and spinning equilibria of the system. We show that the former is unstable and the latter is stable only under certain conditions. We use the method of Controlled Lagrangians to find a control to stabilize the sliding equilibrium and also show how to achieve the stabilization for the general equilibrium point.