Gaussian Mixture Filter for Angles-Only Orbit Determination in Modified Equinoctial Elements

Abstract

A Gaussian mixture orbit determination filter is developed using a state vector that consists of equinoctial-like osculating elements. This new filter seeks to reduce the number of Gaussian mixture elements that are needed in order to accurately model the Bayesian posterior distributions that apply to the angles-only orbit determination problem. The modified version of the equinoctial elements replaces the elements $h$ and $k$ , whose sum squared is bounded by 1. The two replacement elements are unbounded, and $h$ and $k$ can be determined from them. This modification allows the orbit determination filter to operate in an unbounded state space. The new state requires the development of a corresponding mixand covariance upper bound due to the particular type of Gaussian mixture filter that is implemented. The upper bound is used in a mixture resampling algorithm in a way which ensures that extended Kalman filter calculations will be sufficiently accurate for each mixand’s computations. The resulting filter is able to reduce the required number of mixands from 5000 to 500 for angles-only orbit determination of a geosynchronous spacecraft.