On the exact tracking of the spherical inverted pendulum via an homotopy method

A spherical pendulum consists of a mass (the body) linked to a moving mass (the base), to which an external force is applied. In recent literature, considerable attention has been given to the spherical pendulum with particular emphasis on the stabilization problem, but there are fewer results related to tracking.This paper considers an “homotopy method” for solving the exact tracking problem for the inverted pendulum. Recently, this method has allowed us to find precise bounds for the internal dynamics of some well-known nonminimum phase systems with two-dimensional internal dynamics, and to characterize precisely the set of curves that can be tracked in the case of the VTOL aircraft. This paper is a first step for finding similar results for a nonminimum phase system with internal dynamics of dimension greater than 2, such as the spherical pendulum. We also discuss the problems related to the numerical implementation of the proposed method.