On Kalman filtering and observability in nonlinear sequential relative orbit estimation

To study the effects of incorporating higher-order nonlinearities with different measurement types on observability and filter performance in sequential relative orbit estimation, an extended Kalman filter is implemented with four dynamic models of different orders of nonlinearity and either angles- or range-only measurements. The Kalman filtering studies compare the filter performance for these estimation scenarios and illustrate the lack of observability when using the first-order (Hill-Clohessy-Wiltshire) dynamic model, as well as the benefits of using higher-order nonlinear dynamic models on increased observability and faster filter convergence. Observability properties are then studied analytically using Lie derivatives, including the new concept of an "observability angle" for the case of anglesonly measurements, as well as numerically with two observability measures obtained from the observability gramian for both angles- and range-only measurements and for a variety of different relative orbit scenarios, including for both circular and elliptic chief orbits. The analytical and numerical observability results are found to agree and to qualitatively confirm and verify the Kalman filtering studies.