Optimal Glideslope Guidance for Spacecraft Rendezvous

Abstract

M ANY space applications involve rendezvous with a vehicle in circular orbit. A subset of these applications requires the visiting vehicle to approach with a constant direction as seen by the target. This is the case for vehicles approaching the International Space Station (ISS), for example. By approaching it in a straight line the crew onboard the station can easily monitor nonnominal situations. The space shuttle employs a straight-line guidance law called glideslope [1]. Vehicles visiting the ISS usually employ a fixed-direction terminal approach, including H-II Transfer Vehicle (HTV) [2], Automated Transfer Vehicle (ATV) [3], and Cygnus [4]. In this work a constant direction guidance law is developed to rendezvous a target in circular orbit. This type of trajectory is referred to as glideslope. Much work exist in the general area of optimal space trajectories, an illustrative early work is that by Lawden [5]. Carter [6] studied minimum delta-v maneuvers to rendezvous with a vehicle in circular orbit. The approach used by Carter and by many authors after him is to optimize the system subject to the linearized dynamics, the socalled Clohessy–Wiltshire equations [7]. The rendezvous strategy by Lembeck and Prussing [8] is to add to an initial impulsive phase a low-thrust phase. Since continuous thrust is necessary to guide on the glideslope, this work also assumes low-thrust propulsion. Various aspects of this problemwere generalized. Carter andHumi [9] study the impulsive rendezvous in proximity of a general Keplerian orbit, while Carter [10] studies the continuous-thrust case. Power limitations and thrust bounds are also studied [11]. Guelman and Aleshin [12] develop a two-stage solution for the fixed-direction terminal approach. The first stage consists of an unconstrained optimization that puts the vehicle on the glideslope. The second stage is along the glideslope. In this work only the terminal phase is considered, when the spacecraft is required to fly on the glideslope. The current work differs considerably from the work of Guelman andAleshin [12]. In their work the constraint is not enforced directly, but the squared distance to the glideslope is added to the performance index with a weighting parameter. The bigger the parameter the closer the constraint is to be satisfied. This work’s approach is to satisfy the constraint exactly. Another difference between the two works is that Guelman and Aleshin solve their optimization numerically, while a closed-form solution is presented in this Note. The optimal guidance solution applies when the vehicle is on the glideslope. In practice, an inner-loop controller is needed tomaintain the vehicle on the desired terminal direction. II. Optimal Guidance