Nonlinear representation of the confidence region of orbits determined on short arcs

The total number of active satellites, rocket bodies, and debris larger than 10 cm is currently about 20,000. If all resident space objects larger than 1 cm are considered, this number increases to an estimate of 700,000 objects. The next generation of sensors will be able to detect small-size objects, producing millions of observations per day. However, due to observability constraints, long gaps between observations will be likely to occur, especially for small objects. As a consequence, when acquiring observations on a single arc, an accurate determination of the space object orbit and the associated uncertainty is required. This work aims to revisit the classical least squares method by studying the effect of nonlinearities in the mapping between observations and state. For this purpose, high-order Taylor expansions enabled by differential algebra are exploited. In particular, an arbitrary-order least squares solver is implemented using the high-order expansion of the residuals with respect to the state. Typical approximations of differential correction methods are then avoided. Finally, the confidence region of the solution is accurately characterized with a nonlinear approach, taking advantage of these expansions. The properties and performance of the proposed methods are assessed using optical observations of objects in LEO, HEO, and GEO.