Modelling and control of a spherical pendulum via a non–minimal state representation

Abstract

A spherical pendulum is a 2 degree-of-freedom mechanism consisting on a rod whose tip moves on the surface of a sphere. It is common to use two angular coordinates to describe such a system. This paper proposes the use of a non-minimal set of coordinates for modelling and controlling a fully-actuated torque-driven spherical pendulum. These coordinates is merely for the purpose of showing the application of unit quaternions as a useful tool for dealing with the orientation of rigid bodies. First, we recall the properties of unit quaternions, and explain how they can be employed for the definition of such non-minimal pendulum coordinates. Later, the control objective for orientation regulation is established and an inverse-dynamics controller, which uses joint displacement and velocity measurements but also some non-minimal states for the orientation error, is proposed. The stability analysis shows the fulfilment of the control objective and is validated through simulations.