Optimization of Transfers linking Ballistic Captures to Earth-Moon Periodic Orbit Families

TL;DR

The paper tackles the problem of designing low-energy transfers from ballistic capture states to Earth–Moon periodic orbit families within the circular restricted three-body problem ($CR3BP$). It introduces a unified, high-order framework that marries a differential-algebra-based polynomial expansion of the dynamics with a differentiable abacus of periodic orbits, enabling rapid targeting and sensitivity analysis across planar and spatial families. Bi-impulsive transfers are optimized with seeds from mono-impulse solutions, using Poincaré-section insights and a robust DA-based mapping to minimize total impulse $\Delta v = \Delta v_0 + \Delta v_f$, with convex refinement providing near-optimal, fixed-time trajectories. The approach yields low-cost transfer options to Lyapunov, halo, butterfly, and NRHO-like targets, generates informative Pareto fronts across energy levels, and demonstrates practical applicability to mission design, including integration into an existing transfer database and rapid refinement for mission-specific needs.

Abstract

The design of transfers to periodic orbits in the Earth-Moon system has regained prominence with NASA's Artemis and CNSA's Chang'e programs. This work addresses the problem of linking ballistic capture trajectories - exploiting multi-body dynamics for temporary lunar orbit insertion - with bounded periodic motion described in the circular restricted three-body problem (CR3BP). A unified framework is developed for optimizing bi-impulsive transfers to families of periodic orbits via a high-order polynomial expansion of the CR3BP dynamics. That same expansion underlies a continuous parameterization of periodic orbit families, enabling rapid targeting and analytic sensitivity. Transfers to planar periodic orbit families - such as Lyapunov L1/L2 and distant retrograde orbits (DROs) - are addressed first, followed by extension to spatial families - such as butterfly and halo L1/L2 orbits - with an emphasis towards near-rectilinear halo orbits (NRHOs). Numerical results demonstrate low-Δv solutions and validate the method's adaptability for designing lunar missions. The optimized trajectories can inform an established low-energy transfer database, enriching it with detailed cost profiles that reflect both transfer feasibility and underlying dynamical relationships to specific periodic orbit families. Finally, the proposed transfers provide reliable estimates for rapid refinement, making them readily adaptable for further optimization across mission-specific needs.

Figures (17)

• Figure 1: Capture set $\mathcal{C}(\Gamma = 0.84)$ from BC-ETDjournal.
• Figure 2: Sample trajectories $\#1$ and $\#2$, with driving depicted with colored dashed lines.
• Figure 3: Sample $\#1$ nodes discretization (see \ref{['fig: sample 1 trajectory']}).
• Figure 4: Sample $\#2$ from \ref{['fig: sample 2 trajectory']} crossings on the Poincaré section (black plus signs) numbered in order of occurrence.
• Figure 5: Sketch of a bi-impulsive transfer from to the family.