Global analysis of a geometric PDAV controller by means of coordinate-free linearization

TL;DR

This paper provides a comprehensive global analysis of a geometric PDAV controller, visualizing its stability and oscillatory behaviors to aid in control design for complex pointing direction tasks.

Contribution

It introduces a coordinate-free linearization approach and uses simulation to analyze the global stabilization properties of the PDAV controller.

Findings

Eigenstructure analysis predicts precession/nutation oscillations.

Flow visualization reveals stability and saddle points.

Results assist in actuator selection and control design.

Abstract

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.