Deployment and Retrieval Missions from Quasi-Periodic and Chaotic States under a Non-Linear Control Law

When the length of the tether remains constant, the relative planar motion of the tethered subsatellite with respect to the base satellite in a circular orbit around the Earth, is similar to a simple pendulum motion, i.e., there are two kinds of equilibrium points: local vertical and local horizontal positions, which are center and saddle points, respectively. However, when out-of-plane motion is initially excited, the relative motion of the subsatellite presents symmetric quasi-periodic and chaotic behavior. In the first part of this study, such trajectories are analyzed by means of Poincaré sections. In the second part, a non-linear tension force by using a Lyapunov approach is proposed for controlling the coupled pitch-roll motion during the deployment and retrieval phases. The goal of this paper is to guide the relative non-linear motion of the subsatellite to the local upward vertical position. The numerical results show that the non-linear tension control steered the subsatellite close to the local vertical direction very well, reducing the quasi-periodic and chaotic oscillations.