Local Constant-Direction Low-Thrust Optimal Control for Spacecraft Close-Range Operations

Near-Earth space missions are placing growing emphasis on efficient control methods for executing proximity maneuvers, while the extensive deployment of small satellites introduces further limitations that must be considered in the design of these control laws. This paper investigates a constrained low-thrust optimal control problem in which the thrust direction is fixed in the local radial–transverse–normal frame, with a focus on its use for proximity operations of an underactuated CubeSat. The aim is to rapidly find a near-optimal constant thrust direction and design a corresponding fuel-optimal control solution to conduct a full-state orbital control mission. The energy-optimal problem is first studied assuming a constant thrust direction and is solved analytically using the state transition matrix approach. By adopting the least-squares solution, constant directions with favorable operational performance can be rapidly identified to provide initial guesses for solving the fuel-optimal problem, which is formulated as a nonlinear programming problem and then efficiently solved with analytical or semi-analytical state propagation. Numerical simulations are conducted for close-range rendezvous and relative hovering missions based on near-circular orbits to verify the effectiveness of the proposed methods.