Comparing Normal Form Representations for Station-Keeping near Cislunar Libration Points

TL;DR

This work compares the Birkhoff and resonant normal forms of the CR3BP near libration points, showing that the resonant form offers a larger region of validity, especially for out-of-plane motion and quasihalo trajectories. It presents a Floquet-like station-keeping strategy that minimizes the unstable coordinate $\tilde{x}$ via impulsive maneuvers, with three variations to handle Lyapunov, vertical, Lissajous, halo, and quasihalo trajectories. The resonant form uniquely enables parameterization of quasihalo tori and serves as accurate initial guesses for Poincaré-section methods, while the station-keeping results demonstrate significantly lower maneuver costs than the Birkhoff approach, particularly for non-halo trajectories. The study highlights practical implications for near-libration-point missions, offering a physically interpretable, computationally tractable framework for long-term trajectory maintenance in the cislunar regime.

Abstract

The normal forms provide useful approximations for many trajectories of interest within the circular restricted three-body problem. This paper aims to thoroughly compare two of these forms: the Birkhoff normal form and the resonant normal form, highlighting the strengths of each for the representation of center manifold trajectories. A method of station-keeping is introduced, analogous to Floquet modes, in which the unstable component is minimized at specific points along a trajectory through impulsive maneuvers. Three different formulations of the same station-keeping approach are posed, collectively spanning Lyapunov, vertical, and halo orbits, as well as Lissajous and quasihalo trajectories.

Figures (13)

• Figure 1: Comparison between the representation accuracy of the Birkhoff and resonant normal forms for numerical $L_1$ Lyapunov orbits, using the analytical transformation, $\mathcal{A}$.
• Figure 2: Comparison between the representation accuracy of the Birkhoff and resonant normal forms for numerical $L_1$ vertical orbits, using the analytical transformation, $\mathcal{A}$.
• Figure 3: Comparison between the representation accuracy of the Birkhoff and resonant normal forms for small numerical $L_1$ halo orbits, using the analytical transformation, $\mathcal{A}$.
• Figure 4: Comparison between two similar $L_2$ Lissajous trajectories obtained from the Birkhoff normal form (top) and the resonant normal form (bottom).
• Figure 5: Poincaré section comparison between three northern $L_1$ normal form quasihalo tori and their numerical counterparts.