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 inversedynamics 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.

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

“…In this connection, the approach by Altmann & Herzog [1] should also be applicable. Lately, unit quaternions have been used for the modeling of the 3D mathematical pendulum in [12], which therefore seems to be yet another interesting application case of unit quaternions for rigid body rotations. To this end, the dynamics on S 2 could be regarded as a special case of SO(3) allowing for unit quaternion representations with additional constraints.…”