This work studies the autonomous rendezvous, proximity operations, and docking (ARPOD) problem wherein a "chaser" spacecraft must maneuver and dock with a "target" spacecraft in a safe manner. In this work, the chaser satellite is underactuated in that it can only thrust along a single axial direction (its x-axis) and rotate about its z-axis. The translational and attitudinal dynamics (modeled via the Clohessy-Wiltshire equations with inputs) of the chaser spacecraft are inherently coupled and nonlinear, and the linearized system is not controllable about the equilibrium point. To derive the inputs that steer the chaser to the docking state and provide a solution to the ARPOD problem, we leverage an approach based on geometric control theory. In particular, we employ a method based on a modified Lafferriere-Sussmann algorithm to account for uncontrolled drift dynamics. An in-depth numerical simulation is provided to verify the results.
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