Enhanced Indirect and Convex Optimization Methods for Generating Minimum-Fuel Low-Thrust Trajectories

Abstract

The high costs of space missions necessitate optimization across all mission aspects, particularly fuel consumption, which impacts both economic feasibility and payload capacity. Electric propulsion systems, with higher fuel efficiency but lower thrust output (compared to their chemical counterparts), have been used for several decades as a promising alternative propulsion system. To optimize spacecraft equipped with electric propulsion systems, researchers employ two optimization paradigms: indirect and direct methods.

Indirect methods, based on the calculus of variations and Pontryagin’s minimum principle, transform the task of solving an optimization problem into finding the roots of nonlinear boundary-value problems (BVPs). This is achieved by introducing time-varying Lagrange multipliers (a.k.a. costates) and constant Lagrange multipliers. While indirect methods offer high-precision (i.e., with respect to time and resolution) and the solutions are guaranteed to be extremal, the resulting BVPs exhibit high sensitivity to the unknown decision variables (typically costates) and can become challenging to form and solve, when they are applied to constrained nonlinear optimization problems.

Direct methods, on the other hand, discretize variables early, converting the problem into a nonlinear programming (NLP) problem. A convex optimization problem is a specific type of NLP problem where the objective function and constraint set are all ``convex,'' meaning that any local minimum is also a global minimum, making it significantly easier to solve with guaranteed globally optimal solutions, while a general NLP problem can have multiple local minima, making it potentially much harder to find the true optimal solution; essentially, convex optimization is a subset of nonlinear programming with the added benefit of guaranteed global optimality due to its convex structure. By incorporating convexification techniques and leveraging convex optimization tools, direct methods offer a computationally feasible approach. This research proposes methods that alleviate some of the difficulties associated with solving minimum-fuel low-thrust trajectory optimization problems using both indirect and direct methods.