Gaussian Mixture Approximation of Angles-Only Initial Orbit Determination Likelihood Function

A method is developed to approximate the angles-only orbit determination likelihood function using a Gaussian mixture to incorporate information about an admissible region. The resulting probability density function can provide the a priori information for a Gaussian mixture orbit determination filter. This admissible region approach can compensate for the effects of measurement uncertainty over a short track of angles-only data better than deterministic approaches can. The new technique starts with a nonlinear batch least-squares solution. The solution enforces soft constraints on an admissible region defined in terms of minimum periapsis and maximum apoapsis. Although this soft-constrained solution lies in or near the admissible region, it does not characterize that region well. It provides a starting point to develop a Gaussian mixture approximation of the batch least-squares likelihood function as modified through multiplication by a finite-support function that is zero outside the admissible region and 1 inside that region. This Gaussian mixture is optimized to fit the resulting probability density in the two-dimensional subspace of position/velocity space that has the most uncertainty. The procedure starts by approximating the finite-support distribution using a single optimized Gaussian mixand. It automatically increases the number of mixands and reoptimizes until the resulting mixture fits the finite-support distribution to a prespecified precision. This optimal fitting procedure allows the Gaussian mixture to use a low number of mixands while fitting the finite-support probability density function well. By approximating the product of a finite-support function and the original likelihood function, the new method gains the capability to transition smoothly between regimes where the admissibility constraints dominate, i.e., high-altitude/short-measurement-arc cases, and those where they are less important, i.e., low-altitude/long-measurement-arc cases.