On the Connection of Optimal Impulsive and Low-Thrust Trajectories

Abstract

Space missions are designed for a range of objectives, including scientific exploration, telecommunications, Earth observation, and establishing a sustained human presence around the Moon and Mars. Trajectory optimization plays an essential role in achieving these objectives, informing the spacecraft design and launch window selection. The spacecraft's propulsion system influences the resulting trajectory optimization problem. This study considers two primary propulsion systems: chemical and electric. The velocity change due to thrusting with a chemical thruster is modeled as an instantaneous velocity change, consistent with impulsive trajectories. Electric propulsion systems, however, must operate over a finite duration to achieve a velocity change, owing to their lower thrust compared to chemical thrusters. The resulting low-thrust trajectory optimization problems typically seek minimum-fuel or minimum-time solutions and are challenging to solve due to discontinuities in spacecraft states, nonlinear dynamics, and unknown thrust or impulse structures. For impulsive trajectories, the structure corresponds to the number, location, magnitude, and direction of impulses. In contrast, for low-thrust trajectories, the thrust structure comprises the number of thrust arcs, their durations, and the thrust vector direction. Numerical approaches for solving optimal control problems are categorized as direct or indirect, producing nonlinear programming or boundary-value problems.

This dissertation advances spacecraft trajectory optimization by proposing new methods and improving upon the existing numerical approaches. Since the initial guess is essential for achieving convergence in trajectory optimization problems, this dissertation investigates methods to bridge the gap between impulsive and low-thrust trajectories to improve initialization strategies. Therefore, the main goal is to investigate the relationships between impulsive and low-thrust trajectories and leverage them to improve numerical methods. Starting from low-thrust solutions, the initialization of impulsive trajectory optimization problems is investigated. The investigations led to the development of a novel analytical approach to impulsive trajectory generation, revealing infinitely many iso-impulse trajectories across a range of mission-specific parameters (e.g., maneuver time). The iso-impulse trajectories are then used to generate corresponding minimum-fuel low-thrust trajectories, completing the framework that connects the low-thrust and impulsive domains. Additionally, the dissertation explores methods for obtaining more realistic low-thrust trajectories, further improving practitioners' ability to generate these trajectories and broadening their applicability to mission design.