Date of Award

Fall 2013

Degree Type


Degree Name

Doctor of Philosophy (PhD)


Aeronautics and Astronautics

First Advisor

Kathleen C. Howell

Committee Chair

Kathleen C. Howell

Committee Member 1

James M. Longuski

Committee Member 2

William A. Crossley

Committee Member 3

Martin J. Corless


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.