A Spherical Coordinate Parametrization for an In-Orbit Bearings-Only Navigation Filter

Abstract

In-orbit rendezvous is a key enabling technology for many space missions. Implementing it employing only bearing measurements would simplify the relative navigation hardware currently required, increasing robustness and reliability by reducing complexity, launch mass and cost. The problem of bearings-only navigation has been studied intensively by the Naval and Military communities. Several authors have proposed that a polar or spherical coordinate parametrization of the underlying dynamics produces a more robust navigation filter due to the inherent de-coupling of the observable and un-observable states. Nevertheless, the complexity of this problem increases significantly when the underlying dynamics follow those of relative orbital motion. This paper develops a spherical coordinate parametrization of the linearized relative orbital motion equations for elliptical orbits and uses an approximation of these equations for circular orbits to develop an Extended Kalman Filter (EKF) for bearings-only navigation. The resulting filter is compared to its equivalent based on the well known Hill Equations in cartesian coordinates via a Monte Carlo analysis for a given reference trajectory. Simulations show that a spherical coordinate based EKF can perform better than its cartesian coordinate counterpart in terms of long-term stability tracking of the reference trajectory, with little additional computational effort.