When the uncertainty of an orbit state estimate is large, the choice of coordinates is important in properly characterizing the probability distribution. Coordinates that naturally incorporate orbital motion in their definition, like the equinoctial elements, are better able to model the statistics of large deviations from the mean estimate. The shortcoming with equinoctial elements is their difficulty in characterizing position uncertainty apart from velocity uncertainty, making them inconvenient for visualizing position uncertainty. On the other hand, Hill's curvilinear coordinates (denoted herein as downrange coordinates) depend only on position and are defined relative to the Keplerian ellipse defined by the mean estimate. In these coordinates, an equal probability density surface for position uncertainty is shaped like an ellipsoid bent along the curved path of the ellipse. These coordinates are suitable for displaying large position covariance. Explicit equations are developed for the downrange coordinates, their derivatives, and the matrix of partials used for covariance transformations. The amount of the expected bending is assessed as the covariance size increases.
Using Bent Ellipsoids to Represent Large Position Covariance in Orbit Propagation
2015
Article (Journal)
English
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