On choosing quaternion equilibrium point in attitude stabilization

Due to the parametrization of the attitude for closed loop rigid body systems we either encounter an inherent geometric singularity using Euler representation, or obtain dual equilibrium points using the unit quaternion. In order to save energy during attitude maneuvers the choice of equilibrium point and thus rotational direction is imperative for quaternion feedback systems. Normally the shortest rotation is preferred, but in this paper we present schemes where both initial attitude and angular velocity are considered for choosing the preferable rotational direction for a rigid body, thus taking advantage of the initial angular velocity. The solution is based on a set of simple rules where two initial parameters are analyzed and the sign of the solution decides which rotational direction is preferable. The check is not computationally consuming, and may therefore be implemented on i.e. a spacecraft where computational resources are limited. When the preferable equilibrium is chosen, it is kept throughout the maneuver. A tracking controller is derived, resulting in uniform asymptotic stability for both equilibrium points, and the performance of our results are shown through a large number of simulations using randomized initial values.