On-Manifold Low-Thrust Rephasing of Quasi-Periodic Orbits

A bilevel optimal control framework is introduced to solve the low-thrust rephasing problem on quasi-periodic invariant tori in multibody environments where deviations away from the torus during a maneuver are considered unsafe or irresponsible. It is shown for a large class of mechanical systems that conformity to the torus manifold during periods of nonzero control input is infeasible. The most feasible trajectories on the torus surface are generated through the minimization of fictitious control input in the torus space using phase space control variables mapped via the torus function. These reference trajectories are then transitioned to the phase space both through a minimum-tracking-error homotopy and minimum-time patched solutions. Results are compared to torus-agnostic low-thrust transfers using measures of fuel consumption, cumulative torus error, and coast time spent on the torus during a maneuver. Modifications to the framework are made for the inclusion of quasi-periodically forced dynamical systems. Lastly, minimum-time recovery trajectories with free final torus conditions expose the disparity between the proposed framework and torus-agnostic approaches. Examples are drawn from the circular and elliptical restricted three-body problems.