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

Trajectory design in multi-body dynamical environments increasingly relies on understanding quasi-periodic structures that shape long-term motion. This paper develops a Fourier–collocation framework for constructing quasi-periodic orbit (QPO) families 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 an initial Fourier-based torus. The torus is refined to enforce dynamical consistency with the CR3BP dynamics, and continuation in a perturbation parameter generates full QPO families. To improve accuracy, missing higher-order frequencies are systematically identified and incorporated by harmonic addition. The framework produces semi-analytical approximations that remain dynamically consistent over extended timescales, as verified through phase-point drift analyses. Results for Lissajous, quasi-halo, and quasi-distant retrograde orbit families demonstrate the utility of the developed framework. Although computationally intensive, the method establishes a scalable foundation for analyzing quasi-periodic structures in multi-body systems and is amenable to potential transitions to higher-fidelity models for trajectory design in the cislunar environment.