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

TL;DR

This work addresses safe and efficient reconfiguration near quasi-periodic invariant tori (QPITs) in multi-body celestial mechanics by formulating a bi-level optimal control framework. The first level computes torus-space reference trajectories that minimize a fictitious phase-space input, using a Fourier-based torus function to map to the phase space. The second level transitions these references to realizable phase-space trajectories via a minimum-tracking-error homotopy or minimum-time patches, enabling practical low-thrust maneuvers with constrained acceleration. Modifications extend the approach to quasi-periodically forced systems and demonstrate minimum-time recovery strategies, showing favorable torus conformity at the cost of increased propellant use relative to torus-agnostic transfers. The methodology, validated on 2D tori in the CR3BP and ER3BP, provides a structured framework for mission design near QPITs with potential for generalization to higher-dimensional tori and ephemeris integration.

Abstract

A bi-level optimal control framework is introduced to solve the low-thrust re-phasing problem on quasi-periodic invariant tori in multi-body environments where deviations away from the torus during 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 non-zero 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 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.

Figures (15)

• Figure 1: Approximation order analysis for 2-dimensional $L_2$ quasi-halo torus in the Earth-Moon circular restricted problem.
• Figure 2: Torus function by component for 2-dimensional $L_2$ quasi-halo torus in the Earth-Moon circular restricted problem.
• Figure 3: Torus space solutions for rephasing with mixed boundary conditions on a 2-dimensional QPIT in the CR3BP. 1Legends from Figs. \ref{['fig:leg1']} and \ref{['fig:leg2']} apply to all subplots. Time histories are normalized to their respective maximum absolute values.
• Figure 4: $t_f$ free rendezous rephasing cost.
• Figure 5: Torus agnostic solutions for rephasing with fixed boundary conditions on a 2-dimensional QPIT in the CR3BP.