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Low-thrust trajectory design in the context of the three-body problem, particularly in the vicinity of liberation points, is addressed in this paper with a focus on efficiency and accuracy. The Fourier series, previously successful in approximating near-coplanar low-thrust trajectories within two-body dynamics, is applied to the challenging landscape of the three-body problem, marked by instability and nonlinearity. This paper introduces a novel approach for constructing an initial guess trajectory, that combines coast arcs from periodic orbits and intermediate trajectory arcs from other natural dynamical structures. The resulting initial guess trajectory is instrumental in generating efficient initial estimates for the coefficients of the Fourier series. The study encompasses the analysis of homoclinic and heteroclinic transfers in both two-dimensional and three-dimensional scenarios. The approximated trajectories derived from this methodology serve as invaluable starting points for high-fidelity optimization solvers, offering promise for enhancing the precision and efficiency of low-thrust transfers in cislunar space. • The shape method based on Fourier series is applied to the initial design of the transfer trajectories between periodic orbits in cislunar space. • An innovative approach to construct a continuous initial guess trajectory is introduced to derive initial guess values for the coefficients of Fourier terms. • An accurate approximation of the optimal transfer trajectory can be rapidly obtained. [ABSTRACT FROM AUTHOR]

Pseudo-arclength continuation is a scheme used to generate a set of solutions for nonlinear equations and has been applied in numerous fields. In continuation schemes, solutions are obtained by tracking a parameter associated with the system as it varies, thereby creating a solution path. Pseudo-arclength continuation employs the nullspace vector, which is tangent to the path, as an approximation of the arclength in order to determine the direction of variation at each point on the path. However, the nullspace vector has both positive and negative solutions that are equally valid, leading to uncertainty in the direction of travel along the path at each location. Selecting the wrong nullspace vector can result in returning to a previously located solution instead of continuing the mapping of the solution path. Techniques to ensure consistent path direction have been developed; however, their efficacy does not extend to all scenarios. This paper addresses this gap by introducing a "flipping condition" to track previously determined path directions and ensure continuous following of the same path direction. A clear description of the process used to incorporate the flipping condition into the pseudo-arclength continuation scheme, practical implementation of the scheme, and examples demonstrating its application to the nonlinear circular restricted three-body problem (CR3BP), is provided. The inclusion of the flipping condition ensures a consistent direction of travel along the solution path and allows for families of solutions to be developed. • Pseudo-arclength continuation develops solutions to nonlinear equations along a path. • Uncertainty in path direction comes from equally valid signs of the nullspace. • Previously-located solutions are repeatedly found without a consistent path direction. • Implementation process for pseudo-arclength continuation to track path direction. • Application to periodic orbits families in the circular restricted three-body problem. [ABSTRACT FROM AUTHOR]