Characterization of Singular Arcs in Spacecraft Trajectory Optimization

TL;DR

This paper provides a theoretical analysis of singular arcs in spacecraft trajectory optimization, offering necessary conditions and explicit expressions for singular controls, which enhances understanding of their rarity and occurrence in low-thrust missions.

Contribution

It introduces analytical necessary conditions and explicit formulas for singular arcs that depend only on physical variables, not on costates, advancing the theoretical understanding of trajectory optimality.

Findings

Singular arcs are shown to be rare in practical low-thrust missions.

Necessary conditions for singular arcs depend solely on physical variables.

Explicit expressions for singular controls are derived, facilitating analysis.

Abstract

Low-thrust engines for interplanetary spacecraft transfers allow cost-effective space missions with flexible launch and arrival dates. To find fuel-optimal trajectories, an optimal control problem is to be solved. Pontryagin's Maximum Principle shows that the structure of the optimal control is bang-bang with the possibility of singular arcs. Even though the latter have been heuristically shown to rarely appear in practical applications, a full theoretical characterization does not exist in the literature. As a growing number of missions are expected to adopt low-thrust engines in the near future, such study is required to have a comprehensive understanding of the problem. This work presents analytical necessary conditions for the existence of singular arcs that only depend on three physical variables and not on the costates. Moreover, it provides an analytical expression of the singular control that depends on a limited set of physical variables. This is a fundamental feature, as simple evaluation of the necessary condition and of the singular control can be performed. Finally, it provides insightful information on the reasons why singular arcs are rare and it quantifies the possibility of their occurrence.