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... In Fig. 6b, with a larger initial state deviation of δx 0 −3000; 1000; 1000; −0.1; −0.1; 0.1 m; m∕s, the Kalman filter converges on an ambiguous relative orbit for which the shape and size of the estimated orbit are different from those of the true relative orbit. More detailed analysis of these ambiguous relative orbits using range-only measurements have recently been presented for both circular [21,22] and elliptic [23] chief orbits. In the case of circular chief orbits and using HCW dynamics, it was shown that there exist seven ambiguous orbits (three are similar to Fig. 5a and the other four are similar to Fig. 6b) generating the same range history as the true relative orbit. ...
... performance of the third-order and the full nonlinear model are almost indistinguishable. Specifically, in Fig. B2a, it is noted that the EKF with linear model converges on the "persistent deformed ambiguous orbit" proposed by Wang et al. [23]. These results suggest that the conclusions obtained for relative motion models for circular chief orbits also apply for relative motion models for elliptic chief orbits. ...
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. Copyright © 2017 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Range-ambiguous spacecraft relative orbits based on linear dynamics for elliptic chief orbits were studied for the purpose of sequential relative orbit estimation with range-only measurements. Using a geometric solution of the Tschauner–Hempel equation (“Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel,” Acta Astronautica, Vol. 11, No. 2, 1965, pp. 104–109), the mirror range-ambiguous relative orbits, which were shown to have the same size as the true relative orbit for a circular chief reference orbit, are also shown to exist for elliptic chief orbits. However, it is found that deformed range-ambiguous relative orbits, which do not have the same size as the true relative orbit for the circular case, do not analytically exist for nonvanishing chief orbit eccentricity, although they persist in Kalman filtering simulations. This persistence is explained analytically through a perturbation analysis using the variation of the geometric quantities associated with range-ambiguous relative orbits. Furthermore, by altering the relative orbit configuration and model used in the filter, it is shown that the inclusion of chief orbit eccentricity and dynamic model nonlinearities in the filter model can help resist the tendency of an extended Kalman filter to converge to the range-ambiguous relative orbits.
