Trajectory design in the spatial circular restricted three-body problem exploiting higher-dimensional Poincare maps

With recent motivation for manned missions to asteroids or Mars, interest in demonstrating human exploration technologies within the Earth-Moon system has increased. The Circular Restricted Three-Body Problem (CR3BP) offers a unique solution space and serves as a useful framework for preliminary trajectory design in a multi-body force environment, such as the Earth-Moon system. While the CR3BP provides a simplified model with useful symmetries, trajectory design in this dynamical regime is often nontrivial. To generate trajectories that meet a variety of mission constraints, tools that provide insight into the available solution space are essential. 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, strategies from the field of data visualization are employed to represent the information contained in higher-dimensional Poincaré maps using a two-dimensional image. Four-dimensional map representations are demonstrated to compute free and low-cost transfers between periodic libration point orbits in the Earth-Moon system. Alternative Poincaré maps, such as the periapse map, require visualization of the full six-dimensional Cartesian state space. Sample six-dimensional periapse map representations are demonstrated to locate families of periodic orbits about the Moon, as well as transfers to these orbits from Earth. Figure 1. Equilibrium, or libration, points in the Earth-Moon CR3BP Figure 2. Example Lyapunov (black), halo (blue), and vertical (magenta) orbits in the Earth-Moon CR3BP Figure 5. Representing four state variables on a Poincaré map Figure 6. Representing all six state variables on a Poincaré map Figure 11. (a) Grid of intitial conditions used to initiate lunar periapse map; (b) projection of returns to the periapse map corresponding to long-term captures; (c) close view of map crossings, representing all six states, reveals a nearby periodic orbit; (d) a periodic orbit is computed from initial guess located on the map Figure 10. A low-cost transfer requiring a maneuver of only 7.8 m/s is located from the map References 1Paskowitz, M. E. , Scheeres, D. J.: Robust capture and transfer trajectories for planetary satellite orbiters. J. of Guid. , Control, and Dyn. 29, 342-353 (2006) 2Pickett, R. , Grinstein, G. : Iconographic displays for visualizing multidimensional data. Proceedings IEEE Conference on Systems, Man, and Cybernetics 1, 514519 (1988) 3 Villac, B. F. , Scheeres, D. J.: Escaping trajectories in the Hill three-body problem and applications. J. of Guid. , Control, and Dyn. 26, 224-232 (2003) Conclusions The Circular Restricted Three-Body Problem (CR3BP) serves as a useful framework for preliminary trajectory design in a multi-body force environment, however mission design in this dynamical regime is nontrivial. To facilitate trajectory design in such an environment, it is desirable to develop tools that provide insight into the available solution space. Methods to display the information contained in higher-dimensional Poincaré maps are explored and prove useful for mission design applications. Maps for which four or more states are represented visually yield much insight into the available solution space. Free and low-cost transfers are quickly located that may otherwise remain difficult to detect. Structures on the map are identified that reveal the existence of nearby periodic orbits. Such map representations present opportunities to increase the ease of mission design in multi-body regimes. (b) (a) (b)