Best Initial Conditions for the Rendezvous Maneuver

Abstract The majority of the papers dedicated to the rendezvous problem have employed the assumption of given initial position and given initial velocity of the chaser spacecraft vis-à-vis the target spacecraft. In this research, the initial separation velocity components are assumed free and the initial separation coordinates are subject to only the requirement that the chaser-to-target distance is given. Within this frame, two problems are studied: time-to-rendezvous free and time-to-rendezvous given. It is assumed that the target spacecraft moves along a circular orbit, that the chaser spacecraft has variable mass, and that its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that the fuel-optimal trajectory is zero-bang; namely, it includes a long coasting zero-thrust subarc followed by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance trajectory. The optimization of the initial separation coordinates and velocities as well as the time lengths of all the subarcs is performed for several values of the initial distance in the range 5 ≤ do ≤ 60 km with particular reference to the rendezvous between the Space Shuttle (SS) and the International Space Station (ISS). This study is of interest because, for a preselected initial distance SS-to-ISS, it supplies not only the best initial conditions for the rendezvous maneuver, but also the corresponding final conditions of the ascent trajectory.