A New Probability-one Homotopy Method for Solving Minimum-Time Low-Thrust Orbital Transfer Problems

TL;DR

This paper introduces a novel probability-one homotopy method that guarantees finding the optimal low-thrust orbital transfer solution with high probability, improving reliability over existing methods.

Contribution

A new homotopy formulation based on Sard's theorem is developed, ensuring the optimal solution is obtained with probability one in low-thrust trajectory optimization.

Findings

Successfully applied to a 43-revolution orbital transfer problem

Guarantees optimal solution with probability one

Demonstrates effectiveness through numerical examples

Abstract

Homotopy methods have been widely utilized to solve low-thrust orbital transfer problems, however, it is not guaranteed that the optimal solution can be obtained by the existing homotopy methods. In this paper, a new homotopy method is presented, by which the optimal solution can be found with probability one. Generalized sufficient conditions, which are derived from the parametrized Sard's theorem, are first developed. A new type of probability-one homotopy formulation, which is custom-designed for solving minimum-time low-thrust trajectory optimization problems and satisfies all these sufficient conditions, is then constructed. By tracking the continuous zero curve initiated by an initial problem with known solution, the optimal solution of the original problem is guaranteed to be solved with probability one. Numerical demonstrations in a three-dimensional time-optimal low-thrust orbital transfer problem with 43 revolutions is presented to illustrate the applications of the method.