Optimal Low-Thrust Spacecraft Rendezvous with Terminal Velocity Constraints

This paper considers the problem of spacecraft rendezvous by a low-thrust spacecraft while imposing constraints on the terminal velocity. It addresses the terminal phase of the rendezvous for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic optimal control problem with soft constraints on the terminal relative position and velocity. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law with a constrained terminal velocity is derived. Subsequently, a secondary optimization problem is formulated based on the optimal guidance law solution to constrain either the magnitude or the direction of the terminal velocity. The performance of the derived guidance law under each velocity constraint is evaluated in simulations and compared with a numerical direct-optimization solution. It is shown that the solution satisfies the constraints with a minimal cost and that the solution of the proposed guidance law agrees with the numerical solution.