Two-Filter Adaptive Gaussian Mixture Smoothing for Nonlinear Systems

Space object tracking poses challenging estimation problems due to the significantly non-Gaussian distributions that can arise, particularly under highly nonlinear dynamics or during periods of measurement unavailability. Adaptive Gaussian mixture filters can dynamically adjust their mixture resolution to systematically approximate these non-Gaussian distributions, but challenging estimation problems can still produce highly uncertain or inaccurate estimates, especially during prolonged observation gaps. Smoothing algorithms can significantly improve filtered estimates by incorporating future measurement information for applications where immediacy is not required, but smoothing in nonlinear, non-Gaussian settings poses additional theoretical and computational challenges. This work develops a new recursive Bayesian smoothing algorithm for nonlinear systems that refines Gaussian mixture posteriors produced by a forward adaptive Gaussian mixture filter. A two-filter smoothing approach approximates the future measurement information by an information-form Gaussian mixture in the state variable. Techniques from nonlinear Gaussian mixture filtering including splitting, merging, and recursive measurement updating are also incorporated to improve the accuracy and computational efficiency of this approximation. The proposed smoother's estimation capabilities are demonstrated on space object tracking problems for Molniya and Earth-Moon halo orbits and shown to significantly reduce estimation error and uncertainty compared to the forward filter.