Observability Criteria and Unobservable Maneuvers for In-Orbit Bearings-Only Navigation

This paper provides an analytical derivation of the observability conditions for in-orbit bearings-only navigation that can also be used to solve for the conditions under which unobservable maneuvers occur. The derivation is performed in a rigorous yet relatively simple manner by attempting to solve for the initial conditions in a discrete formulation of the problem. This newly proposed criterion is then used to derive the necessary and sufficient conditions for observability, previously established in the literature through a geometric interpretation of the problem. Moreover, the proposed scheme is used to mathematically derive this geometrical interpretation, thus providing an alternate derivation that reinforces these already established concepts. In addition, a new framework is proposed that allows solving for the unobservable maneuvers sets of a given type. This result, which can be directly applied to many maneuver types and easily extended to different maneuvering schemes, is demonstrated for the case of impulsive as well as constant thrust maneuvers. Like so, it is shown that unobservable impulsive maneuvers can only occur on a departure trajectory and that all impulsive maneuvers performed on an approach trajectory are observable. In addition, it is shown that unobservable constant thrust maneuvers are in fact plausible, even during an approach trajectory, thus establishing a set of guidelines of which maneuvers to avoid, applicable to any current or future bearings-only rendezvous mission.