Representations of higher-dimensional Poincaré maps with applications to spacecraft trajectory design

Author-Highlights
• We seek to facilitate trajectory design in the spatial CR3BP.
• We represent Poincaré maps using strategies from data visualization.
• Higher-dimensional Poincaré maps are visualized using two-dimensional images.
• Applications to spacecraft trajectory design are explored.

The Circular Restricted Three-Body Problem (CR3BP) offers a useful framework for preliminary trajectory design in multi-body regimes. While the CR3BP offers a simplified model with useful symmetries, trajectory design in this dynamical regime is often nontrivial. It is essential to gain insight into the available solution space to generate trajectories that meet a variety of constraints. The Poincaré map is a powerful tool that, in combination with a constraint on the energy level, allows a reduction in dimension such that, for the planar problem, the system is reduced to two-dimensions and the phase space is fully represented by the projection onto a plane. In the spatial problem, however, Poincaré maps must represent at least four states and are therefore challenging to visualize. In this investigation, a method to represent the information contained in higher-dimensional Poincaré maps using a two-dimensional image for visualization is explored and is applied for trajectory design. Four-dimensional map representations are demonstrated to compute transfers, including heteroclinic and homoclinic connections, between periodic and quasi-periodic orbits in the spatial problem. Alternative maps, such as the periapse Poincaré map, require that at least five Cartesian states be represented. Map representations are generated to visualize the full state associated with crossings of the periapse map, and are employed to locate families of periodic orbits about the Moon, as well as transfers to these orbits via transit trajectories.