A Fourier–Collocation Framework for Constructing Quasi-Periodic Orbit Families in the CR3BP

Abstract

Long-term spacecraft motion in multi-body dynamical environments rarely remains strictly periodic; the phase space near periodic orbits is instead foliated by invariant tori supporting quasi-periodic motion, yet generic perturbations may equally destabilize trajectories along hyperbolic directions – making the systematic construction and characterization of these competing structures a central challenge in cislunar trajectory design. To meet this challenge, this paper develops a Fourier–collocation framework for computing families of quasi-periodic orbits (QPOs) in the Circular Restricted Three-Body Problem (CR3BP). Starting from a reference periodic orbit, fundamental frequencies are identified through frequency analysis and Floquet theory, enabling the construction of a full torus embedding expressed directly in angle coordinates. The torus is refined through a nonlinear optimization that enforces dynamical consistency with the CR3BP flow by minimizing Jacobi constant variation across the embedding, yielding a coherent quasi-periodic representation rather than a locally invariant curve. Arclength continuation in Fourier coefficient space then generates complete QPO families, including those surrounding unstable periodic orbits. To systematically improve accuracy, a novel harmonic addition algorithm identifies missing integer combinations of fundamental frequencies through spectral analysis of the Jacobi constant residual and incorporates them into the Fourier basis, reducing phase-point drift by over an order of magnitude for representative cases. Accuracy is assessed globally through phase-point drift analyses that compare the Fourier model against the true CR3BP flow across the full torus surface, providing a quantitative measure of dynamical consistency that is formulated over the entire manifold rather than localized to an invariant curve. Results for Lissajous, quasi-halo, and quasi-distant retrograde orbit families in the Earth–Moon system demonstrate the utility of the framework.