Space situational awareness requires estimation of satellite tracks with sparse observations. The sparsity of novel information makes critical tasks such as maneuvering target tracking and track initialization challenging. Bayesian inference via Markov-Chain Monte Carlo (MCMC) can improve robustness when compared to recursive estimation, but high-fidelity orbit propagation yields objectionable runtimes in such solvers. This challenge may be mitigated via multi-fidelity methods, i.e., the use of varying fidelity propagators that trade some accuracy for reduced runtime. In this paper, multi-fidelity methods are combined with a delayed acceptance approach in Metropolis-Hastings-based MCMC to reduce runtime for satellites in low-Earth orbits. The delayed acceptance approach is a two-step process. The first considers a sample propagated via a rapid, multifidelity model. In the second step, any preliminarily accepted samples are propagated via the full-fidelity model. This reduces the number of high-fidelity propagations. The multi-fidelity approach also includes proposal density adaptation and produces mixing ratios consistent with optimal rates. This MCMC method is demonstrated for sparse orbit determination in a low-Earth orbit scenario. Accuracy of the multi-fidelity approach is consistent with full-fidelity inference while reducing the runtime by almost a factor of two in the cases considered.

The proliferation of large satellite constellations in low Earth orbit (LEO) is dramatically increasing demand on existing systems for space domain awareness. The rapidly growing number of objects in LEO will reduce the average rate of observations per object, necessitating the development of multi-target algorithms that can handle higher levels of data sparsity without sacrificing computational efficiency. In this paper, we demonstrate that a multi-target filter that combines a number of useful capabilities is able to track and maintain custody of a simulated population of over 16,000 LEO objects without requiring the use of high-performance computing facilities. The filter is based on the generalized labeled multi-Bernoulli filter, and includes three previously-presented features: label space partitioning based on sensor fields of view, the ensemble Gaussian mixture filter (EnGMF), and bi-fidelity orbit uncertainty propagation. We also introduce a new, algorithmically simple method for adjusting the number of EnGMF particles to balance accuracy and computational efficiency, which we refer to as progressive resampling.

This paper introduces the concept of the Gaussian integral filter (GIF), the limit of the Gaussian sum filter (GSF) for when the number of mixands tends to infinity. The GIF is obtained via a combination of GSF, quadrature, and interpolation. While it is a very general concept, in this paper the GIF is used to represent multiviariate Laplace (ML) distributions defining the process noise when tracking a maneuvering target. The filter is first applied to a linear three-dimensional toy problem, and then to a maneuvering target tracking problem in Earth orbit. For the more complex maneuvering target tracking problem, the filter requires only 1.4 times the computational resources of an unscented Kalman filter (UKF), while having errors up to 11 times smaller. For the same problem, the UKF slowly diverges.

Filters relying on the Gaussian approximation typically incorporate the measurement linearly, i.e., the value of the measurement is pre-multiplied by a matrix-valued gain in the state update. Nonlinear filters that relax the Gaussian assumption, on the other hand, typically approximate the distribution of the state with a finite sum of point masses or Gaussian distributions. In this work, the distribution of the state is approximated by a polynomial transformation of a Gaussian distribution, allowing for all moments, central and raw, to be rapidly computed in closed form. Knowledge of the higher-order moments is then employed to perform a polynomial measurement update, i.e., the value of the measurement enters the update function as a polynomial of arbitrary order. A filter employing a Gaussian approximation with linear update is, therefore, a special case of the proposed algorithm when the order of the update is set to one. At the cost of more computations, the new methodology guarantees performance better than the linear/Gaussian approach for nonlinear systems. This work employs monomial basis functions and Taylor series, but it is readily extendable to an orthogonal polynomial basis.