Optimal Orbital Rendezvous Maneuvering for Angles-Only Navigation

A NGLES-ONLY navigation has great potential for orbital rendezvous, satellite formation flight, and other relative motion applications, but is often discarded because of its inherent and misunderstood limitation in determining range. A common consensus in the published literature, regardless of the application, is that maneuvers are generally required to obtain target observability when using only anglemeasurements [1–6]. For the bearings-only tracking problem, Nardone and Aidala [7] and Hepner and Geering [8] also showed that certain types of maneuvers, particularly those maneuvers that cause position changes that lie along the instantaneous bearing lines associated with a constant velocity trajectory, do not necessarily guarantee observability. If certain maneuvers produce observability and others do not, then the next natural step that has received considerable attention is to determinewhichmaneuvers and trajectories are optimal in maximizing observability [9–16]. Despite these significant advances in deriving optimal maneuvers for the bearings-only tracking problem, these results are typically established on the premise of a constant moving target, an assumption valid for many naval applications that motivated the earlier research efforts but not applicable to orbital rendezvous. Although the influence of maneuvering for angles-only navigation has been considered for orbital rendezvous [17–19], there does not exist in the published literature derivations of performing optimal maneuvers to maximize observability. This Note develops the mathematical framework to analytically derive optimal maneuvers for angles-only navigation using a previously derived observability criteria [20–22]. The concept of having levels or degrees of observability is formally defined as a function of the measurement error and then used to form the theoretical foundation to derive optimal maneuvers that maximize the observability of the relative state. A simple yet common orbital rendezvous example is provided to illustrate the possibility of designing optimal orbital rendezvous maneuvers for angles-only navigation. Although the topic is introduced in the context of orbital rendezvous, the fundamental concepts can be applied to any linear dynamic system (and extended to nonlinear systems) for which the relative position and velocity are estimated using only angular measurements.