Revisiting the orbital tracking problem
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Consider a space object in an orbit about the earth. An uncertain initial state can be represented as a point cloud which can be propagated to later times by the laws of Newtonian motion. If the state of the object is represented in Cartesian earth centered inertial (Cartesian-ECI) coordinates, then even if initial uncertainty is Gaussian in this coordinate system, the distribution quickly becomes non-Gaussian as the propagation time increases. Similar problems arise in other standard fixed coordinate systems in astrodynamics, e.g. Keplerian and to some extent equinoctial. To address these problems, a local "Adapted STructural (AST)" coordinate system has been developed in which uncertainty is represented in terms of deviations from a "central state". In this coordinate system initial Gaussian uncertainty remains Gaussian for all propagation times under Keplerian dynamics. Further, it can be shown that the transformation between Cartesian-ECI and AST coordinates is approximately linear. Statistical tests are carried out to confirm the quality of the Gaussian approximation. Given a sequence of angles-only measurements, it is straightforward to implement any of the standard variants of the Kalman filter. It turns out that iterated nonlinear extended (IEKF) and unscented (IUKF) Kalman filters are often the most appropriate variants to use. In particular, they can be much more accurate than the more commonly used non-iterated versions, the extended (EKF) and unscented (UKF) Kalman filters, especially under high ellipticity. In addition, iterated Kalman filters can often be well-approximated by two new closed form filters, the observation-centered extended (OCEKF) and unscented (OCUKF) Kalman filters.
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