Ambiguous relative orbits in sequential relative orbit estimation with range-only measurements

This paper describes a manifold of ambiguous spacecraft relative orbits that arise in sequential relative orbit estimation. The development herein assumes linear relative dynamics, a circular reference orbit, and range-only measurements. Using a formulation based on relative orbit elements, the ambiguous orbits are categorized into two cases: mirror orbits, which conserve the size and shape but transform the orientation of the true relative orbit, and deformed orbits, which both distort the shape and change the orientation. A special case, that of central ambiguous relative orbits, which are geometrically symmetric relative to the chief's local-vertical-local-horizontal frame is also discussed. The multiplicity of mirror ambiguous orbits, deformed ambiguous orbits and central ambiguous orbits are shown to be three, four and infinity, respectively. Numerical results using an extended Kalman filter are provided to confirm the existence of these ambiguous orbits. Furthermore, the observability is studied analytically with a nonlinear observability criterion using Lie derivatives. It is also shown by numerical results that the inclusion of nonlinearities in the filter model can help resist the tendency of an extended Kalman filter to converge to the ambiguous relative orbits. Finally, the persistence of these ambiguous orbits under unmodeled chief eccentricity error and J2 perturbation is studied.