Two-phase optimal control strategy with predefined thrust directions for close-range operations

CubeSats have emerged as a promising solution for future near-Earth space missions, while their limited actuation capabilities pose significant challenges for autonomous proximity operations. Considering an underactuated scenario in which low-thrust control is constrained to a sequence of single predefined directions, this work proposes a two-phase optimal control strategy for performing full-state close-range maneuvers. In the proposed framework, thrust is applied along a constant direction within each control phase: one phase uses tangential or radial thrust to manage in-plane motion, while the other employs normal thrust for out-of-plane control. A coast phase is inserted between the two thrust phases to allow thrust-direction reorientation and accommodate mission-specific constraints. Three predefined-direction optimal control problems, including tangential, radial, and normal, are formulated in the Earth-centered inertial frame as two-point boundary value problems. Energy-optimal solutions are obtained analytically using an inertial State Transition Matrix method, whereas corresponding fuel-optimal solutions are determined through a nonlinear programming approach initialized by the energy-optimal results. In addition, a fast algorithm is developed to estimate near-optimal control durations for low-thrust proximity operations, providing the time allocation for the two thrust phases. Using a known time window, four candidate two-phase strategies are constructed from combinations of the predefined thrust directions. The effectiveness of the proposed strategies is analyzed through numerical simulations in close-range rendezvous and relative hovering, in terms of fuel cost, operational safety, and control accuracy, to identify the most effective strategy for different mission scenarios.