Continuous Low-Thrust Trajectory Optimization: Techniques and Applications

Trajectory optimization is a powerful technique to analyze mission feasibility during mission design. High-thrust trajectory optimization problems are typically formulated as discrete optimization problems and are numerically well-behaved. Low-thrust systems, on the other hand, operate for significant periods of the mission time. As a result, the solution approach requires continuous optimization; the associated optimal control problems are in general numerically ill-conditioned. In addition, case studies comparing the performance of low-thrust technologies for space travel have not received adequate attention in the literature and are in most instances incomplete. The objective of this dissertation is therefore to design an efficient optimal control algorithm and to apply it to the minimum-time transfer problem of low-thrust spacecraft. We devise a cascaded computational scheme based on numerical and analytical methods. Whereas other conventional optimization packages rely on numerical solution approaches, we employ analytical and semi-analytical techniques such as symmetry and homotopy methods to assist in the solution-finding process. The first objective is to obtain a single optimized trajectory that satisfies some given boundary conditions. The initialization phase for this first trajectory includes a global, stochastic search based on Adaptive Simulated Annealing; the fine tuning of optimization parameters — the local search — is accomplished by Quasi-Newton and Newton methods. Once an optimized trajectory has been obtained, we use system symmetry and homotopy techniques to generate additional optimal control solutions efficiently. We obtain optimal trajectories for several interrelated problem families that are described as Multi-Point Boundary Value Problems. We present and prove two theorems describing system symmetries for solar sail spacecraft and discuss symmetry properties and symmetry breaking for electric spacecraft systems models. We demonstrate how these symmetry properties can be used to significantly simplify the solution-finding process.