Analytic Optimal Observability Maneuvers for In-Orbit Bearings-Only Rendezvous

B EARINGS-ONLY or angles-only navigation is the problem of fully determining the relative position between a chaser spacecraft and a target using only angular measurements. A few authors have already shown that the execution of known maneuvers in a suitable direction is necessary in order to render the problem observable: namely, the conditions under which there is enough information to fully determine the relative position [1,2]. Specifically, the work in [1] defines the conditions under which the problem is observable or not. In addition, our previous work in [2] presents an alternate derivation of these conditions that allows for a deeper understanding of the problem, and it expands the state of the art by developing a method to find the set of maneuvers that render the problem unobservable, thus defining the types of maneuvers that should be avoided when performing a rendezvous using angles-only measurements. Nonetheless, during a bearings-only rendezvous mission, it is not only desired to ensure observability but it is also of interest to obtain the best possible navigation estimate. The latter can only be obtained when the conditions are optimal for a navigation filter to make best use of the extra information introduced by a certain maneuver: that is, when the applied maneuvers yield maximum observability. To find these maneuvers, a “continuous”measure of observability is required, which has already been proposed in the literature [3]. However, due to the complexity of the resulting expressions, explicit results can only be found for simple trajectories that must be known a priori, and numerical optimization is required in order to provide results for more general trajectories. This Note presents an alternative analytic expression for an optimal observability objective that can be used to maximize observability for any given initial trajectory. Due to its simple form, the proposed objective function is more versatile than the previous state of the art, as it could easily be included within a higher-level trajectory optimization scheme. Moreover, its simple form also aids in the understanding of the conditions that maximize observability, thus broadening this understanding. In addition, using the proposed observability objective, an analytical solution for optimal observability maneuvers is developed. These closed-form solutions are developed in a generic manner, applicable to anymaneuver type and any a priori unknown trajectory. Thus, this constitutes a second further advancement from the previous state of the art [3]. Moreover, it is also shown how the framework proposed can be easily modified to provide closed-form solutions for a variety of different constraints, both on the maneuvers and/or on the trajectory. For these reasons, the proposed scheme is very amenable for inclusion in a global rendezvous guidance scheme in order to improve bearings-only observabilitywhile simultaneously considering other mission critical constraints, such as safety. Finally, numerical simulations are used to verify that these maneuvers yield the optimal conditions so that a bearings-only navigation filter can provide its best possible navigation estimate at a specified time. The simulations show that the resulting maneuvers not only provide optimal navigation performance at a desired future time but do so while simultaneously reducing the range to the target during a rendezvous.