A Frequency-Domain Differential Corrector for Quasi-Periodic Trajectory Design and Analysis

TL;DR

The paper tackles the challenge of constructing high-dimensional quasi-periodic orbits in realistic dynamical environments by shifting from explicit multi-frequency invariance enforcement to a frequency-domain correction framework. The Frequency-Domain Differential Corrector (FDDC) leverages L-NAFF and GMS-C refinements to identify and adjust dominant spectral components, enabling efficient single- and multi-shooting formulations across Earth–Moon dynamics. Key contributions include model-agnostic formulation, derivation of spectral sensitivities via STM, and demonstrated applications to DROs, ELFOs, and NRHOs, including constellation design and high-fidelity model reconciliation. The approach offers robust trajectory design and mission-planning capabilities in complex, time-varying systems, with potential to improve scalability and controllability of oscillatory mission profiles in cislunar space.

Abstract

This paper introduces the Frequency-Domain Differential Corrector (FDDC), a model-agnostic approach for constructing quasi-periodic orbits (QPOs) across a range of dynamical regimes. In contrast to existing methods that explicitly enforce an invariance condition in all frequency dimensions, the FDDC targets dominant spectral components identified through frequency-domain analysis. Leveraging frequency refinement strategies such as Laskar-Numerical Analysis of Fundamental Frequency (L-NAFF) and Gómez-Mondelo-Simó-Collocation (GMS-C), the method enables efficient and scalable generation of high-dimensional QPOs. The FDDC is demonstrated in both single- and multiple-shooting formulations. While the study focuses on the Earth-Moon system, the framework is broadly applicable to other celestial environments. Sample applications include Distant Retrograde Orbits (DROs), Elliptical Lunar Frozen Orbits (ELFOs), and Near Rectilinear Halo Orbits (NRHOs), illustrating constellation design and the recovery of analog solutions in higher-fidelity models. With its model-independent formulation and spectral targeting capabilities, FDDC offers a versatile tool for robust trajectory design and mission planning in complex dynamical systems.

Figures (14)

• Figure 1: Transformations across various state representations and frames
• Figure 2: Sample cr3bp periodic-dro within the brf
• Figure 3: dft on $x$-position components for the periodic-dro in Fig. \ref{['fig:dro-compare-l-g-orbit']}
• Figure 4: Errors for refined frequencies at the multiples of $\nu_{C}$
• Figure 5: cr3bp quasi-dro initiated with the state in Table \ref{['table:dro-ig']}