Higher-Order Tensor-Based Deferral of Gaussian Splitting for Orbit Uncertainty Propagation

Accurate propagation of orbital uncertainty is essential for a range of applications within space domain awareness. Adaptive Gaussian mixture-based approaches offer tractable nonlinear uncertainty propagation through splitting mixands to increase resolution in areas of stronger nonlinearities, as well as by reducing mixands to prevent unnecessary computational effort. Recent work introduced principled heuristics that incorporate information from the system dynamics and initial uncertainty to determine optimal directions for splitting. This paper develops adaptive uncertainty propagation methods based on these robust splitting techniques. A deferred splitting algorithm tightly integrated with higher-order splitting techniques is proposed and shown to offer substantial gains in computational efficiency without sacrificing accuracy. Second-order propagation of mixand moments is also seen to improve accuracy while retaining significant computational savings from deferred splitting. Different immediate and deferred splitting methods are compared in three representative test cases, including a geostationary orbit, a Molniya orbit, and a periodic three-body orbit.

Nonlinearity and Uncertainty Informed Moment-Matching Gaussian Mixture Splitting

Many problems in navigation and tracking require increasingly accurate characterizations of the evolution of uncertainty in nonlinear systems. Nonlinear uncertainty propagation approaches based on Gaussian mixture density approximations offer distinct advantages over sampling based methods in their computational cost and continuous representation. State-of-the-art Gaussian mixture approaches are adaptive in that individual Gaussian mixands are selectively split into mixtures to yield better approximations of the true propagated distribution. Despite the importance of the splitting process to accuracy and computational efficiency, relatively little work has been devoted to mixand selection and splitting direction optimization. The first part of this work presents splitting methods that preserve the mean and covariance of the original distribution. Then, we present and compare a number of novel heuristics for selecting the splitting direction. The choice of splitting direction is informed by the initial uncertainty distribution, properties of the nonlinear function through which the original distribution is propagated, and a whitening based natural scaling method to avoid dependence of the splitting direction on the scaling of coordinates. We compare these novel heuristics to existing techniques in three distinct examples involving Cartesian to polar coordinate transformation, Keplerian orbital element propagation, and uncertainty propagation in the circular restricted three-body problem.