Approximate output regulation for a spherical inverted pendulum

The present study applies an approximate output regulator for controlling a MIMO nonlinear non-minimum phase system, considering the example of a four degree of freedom spherical inverted pendulum. The spherical pendulum consists of a slim cylinder attached to a universal joint upon which the planar control force acts. The aim in output tracking is to control the pendulum such that the base follows a desired reference trajectory as closely as possible while maintaining the upright position. The standard output regulator requires the solution to a mixed algebraic partial differential equation, which entails finding the manifold on which the system state trajectories result in exact tracking of the reference signal. This can be very difficult to solve in practice for the non-minimum phase case thus motivating the use of approximation methods. A local approximation method, based on a Taylor series expansion of the system dynamics is used, such that an output regulator may be applied to the spherical inverted pendulum. This gives the first application of output regulation for the output tracking of this system.