Non-local stabilization of a spherical inverted pendulum

We design a non-linear stabilizing control law for a four degree of freedom spherical inverted pendulum. The pendulum is a slim cylindrical beam attached to a horizontal plane via a universal joint; the joint is free to move in the plane under the influence of a planar force. The upright position is an unstable equilibrium of the uncontrolled system because of gravity. The objective is to design a controller so that it stabilizes the upright position starting from any position in the upper hemisphere with arbitrary velocity. We achieve this by first transforming the original system to an appropriate upper triangular form and then designing a controller which incorporates a high gain design with the method of non-linear forwarding. The control law is evaluated through computer simulations.