Abstract
Strategies for designing three-dimensional spacecraft trajectories in a multi-body dynamical environment are investigated using four-dimensional Poincaré maps. Unlike the planar circular restricted three-body problem, where a two-dimensional map provides a simplified view of a portion of the vast and often chaotic design space, the spatial problem requires a four-dimensional map to achieve an equivalent perspective. Such higher-dimensional maps present a visualization challenge. Furthermore, a spacecraft in the spatial problem can exhibit fundamentally more diverse and complex behavior than in the planar problem. A novel approach to four-dimensional-map-based design in the spatial circular restricted three-body problem is developed and applied to practical examples with real-world spaceflight applications involving three-dimensional trajectories in the Earth-Moon, Sun-Earth, and Uranus-Titania systems. Included in the approach is a method for representing, interpreting, and manipulating four-dimensional Poincaré maps in an interactive, three-dimensional visual environment in which the fourth dimension is displayed using color. This "space-plus-color" method expands on the "color and rotation" method of Patsis and Zachilas (used for the study of motion in a galaxy) by applying additional tools and techniques enabling design in the circular restricted three-body problem. Design is often based on maps generated by many trajectories. Image manipulation in both spatial and color dimensions is accomplished iteratively using MATLAB® and Avizo®. Four-dimensional-map-based design in the spatial circular restricted three-body problem is practical, and success is enabled by interactive tools and techniques in a visual environment. The design strategy is methodical and not restricted to any particular map formulation. Human insight is leveraged to determine reference solutions in a problem without a closed-form analytical solution. Estimates obtained through visual inspection of a map are fed into automated processes, leading to precise and/or locally-optimal solutions, including transfers to and between libration/Lagrange point orbits as well as capture, departure, and transit maneuvers near a planet or moon. Additionally, the long-term variations in instantaneous eccentricity of a high-altitude Earth orbit perturbed by lunar gravity are correlated with the shape and evolution of the surface of a deformed torus on a four-dimensional map.
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Aeronautics and Astronautics
Committee Chair
Kathleen C. Howell
Date of Award
Fall 2013
Recommended Citation
Geisel, Christopher D., "Spacecraft Orbit Design in the Circular Restricted Three-Body Problem Using Higher-Dimensional Poincaré Maps" (2013). Open Access Dissertations. 109.
https://docs.lib.purdue.edu/open_access_dissertations/109
First Advisor
Kathleen C. Howell
Committee Member 1
James M. Longuski
Committee Member 2
William A. Crossley
Committee Member 3
Martin J. Corless