  Tracking a desired Pointing Direction and simultaneously obtaining a
reference Angular Velocity (PDAV) around the pointing direction constitutes a
very involved and complicated motion encountered in a variaty 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 variational 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/or have a rough estimate of the
evolution of the solutions.
