Stabilization of an Inverted Pendulum on a Nonholonomic System

Abstract

In this paper we investigate the problem of stabilizing the roll dynamics of a nonholonomic system that is inspired by a Chaplygin sleigh whose center of mass is at some height above the ground. The sole actuation for the extruded Chaplygin sleigh, is via the motion of an internal reaction wheel which applies a torque in the yaw direction. This torque is used to propel the Chaplygin sleigh as well as stabilize its roll. This system is motivated by the problem of the stabilization of the roll of a fish-like underwater swimmer. The dynamics of a fish-like swimmer have been shown to be similar to that of a Chaplygin sleigh. We propose a feedback control, by considering the associated linear representation due to the action of a Koopman operator on the observables. Using the Koopman operator, a constrained optimal control problem is formulated in the lifted space which we solve using model predictive control. The approach has the advantage of being systematically generalized for increased model complexity for nonholonomic systems and actuator saturation.