Two-Phase Directional Low Thrust Optimal Control for Spacecraft Proximity Operations

The growing use of small satellites, such as CubeSats, in near-Earth space missions poses significant challenges to autonomous guidance and control for spacecraft proximity operations, particularly under constrained actuation capabilities. This paper addresses the orbital control problem of an underactuated spacecraft using directional low thrust. The proposed method employs a two-phase optimal control strategy, where low thrust is applied in a fixed direction during each phase: one phase utilizes tangential or radial thrust for in-plane motion control, while the other applies normal thrust for out-of-plane motion control. Additionally, a user-defined coast phase can be introduced between these two control phases, allowing for thrust redirection adjustments. First, three optimal control problems considering tangential, radial, and normal thrust separately are formulated as Two-Point Boundary Value Problems (TPBVPs) in the Earth-centered inertial dynamic frame for a given control duration. These TPBVPs are then transformed into initial value problems and solved analytically using the state transition matrix method. Based on these solutions, four independent control strategies are proposed: Tangential-Coast-Normal, Normal-Coast-Tangential, Radial-Coast-Normal, and Normal-Coast-Radial. The effectiveness of these strategies is demonstrated through applications to spacecraft rendezvous and relative hovering control missions. In addition, performance analyses, considering fuel cost and operational safety, are conducted to identify the most suitable strategy for different mission scenarios. Numerical simulations with varying parameters validate the efficiency and feasibility of the proposed approach.