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Global closed-loop equilibrium properties of a geometric PDAV controller via a coordinate-free linearization

Tracking a desired Pointing Direction and simultaneously obtaining a reference Angular Velocity (PDAV) around the pointing direction constitutes a very involved and compli- cated motion encountered in a variety of robotic, industrial and military applications. In this paper through the utilization of global analysis and simulation techniques, the smooth closed- loop vector fields induced by the geometric PDAV controller from [1], are visualized to gain a deeper understanding of its global stabilization properties. First through the calculation of a coordinate-free form of the closed-loop linearized dynamics, the local stability of each equilibrium of the system is analyzed. The results acquired by means of eigenstructure analysis, are used in predicting the frequency of complex precession/nutation oscillations that arise during PDAV trajectory tracking; an important tool in actuator selection. Finally, by utilizing vari- ational integration schemes, the flow converging to the desired equilibrium and the flow “close” to the stable manifold of the saddle equilibrium of the closed-loop system is visualized and analyzed. Results offer intimate knowledge of the closed-loop vector fields bestowing to the control engineer the ability to anticipate and have an estimate of the evolution of the solutions.