The computation of optimal rendezvous trajectories using the sequential gradient-restoration algorithm
Weeks, Michael W.
Doctor of Philosophy
In recent years, there has been a growing demand for the autonomous rendezvous and docking capability of a spacecraft. Current guidance methods in existence are based on the human control of the chaser spacecraft and are not suitable nor sufficient for an autonomous vehicle. The optimal solution of the rendezvous problem investigated in this thesis consists of finding an allowable finite control distribution which minimizes some prescribed performance index (i.e. time, fuel, etc) and brings a chaser vehicle into coincidence with a target vehicle. This thesis first derives the well-known Clohessy-Wiltshire (CW) differential equations (Ref. 1) and focuses on the optimal solution of a linearized three-dimensional rendezvous with bounded thrust and limited fuel. To accomplish this, the sequential gradient-restoration algorithm is utilized to optimize several rendezvous trajectories for the case of a target spacecraft in a circular orbit at the International Space Station (ISS) altitude and a chaser spacecraft with typical initial conditions during the terminal phase of the rendezvous with bounded thrust and bounded DeltaV. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max thrust acceleration via the sequential gradient-restoration algorithm (SGRA). Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, two of which determine the thrust direction in space and one which determines the thrust magnitude. The main conclusion is that the optimal control distribution can result in two, three, or four subarcs depending on the performance index and the constraints. The time-optimal case results in a two-subarc solution with max thrust. The fuel-optimal case results in a four subarc solution consisting of an initial coasting period, followed by a maximum thrust phase, followed by another coasting period, followed by another maximum thrust phase. Regardless of the number of resulting subarcs, the optimal thrust distribution requires the thrust magnitude to be either at the maximum value or at zero. The coasting periods are finite in duration and their length increases as the time to rendezvous increases and/or as the max allowable thrust increases. Another finding is that, for the fuel-optimal rendezvous with the time unconstrained, the minimum fuel required is nearly constant and independent of the max available thrust. Based on the above observations, the final potion of this thesis applies the multiple-subarc version of SGRA to solve the guidance problem based on the implementation of constant-control finite-thrust functions during each subarc. (Abstract shortened by UMI.)