Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos

This paper demonstrates the use of polynomial chaos expansions (PCEs) for the nonlinear, non-Gaussian propagation of orbit state uncertainty. Using linear expansions in tensor-products of univariate orthogonal polynomial bases, PCEs approximate the stochastic solution of the ordinary differential equation describing the propagated orbit, and include information on covariance, higher moments, and the spatial density of possible solutions. Results presented in this paper use non-intrusive, i.e., sampling-based, methods in combination with either least-squares regression or pseudo-spectral collocation to estimate the PCE coefficients at any future point in time. Such methods allow for the utilization of existing orbit propagators. Samples based on Sun-synchronous and Molniya orbit scenarios are propagated for up to ten days using two-body and higher-fidelity force models. Tests demonstrate that the presented methods require the propagation of orders of magnitude fewer samples than Monte Carlo techniques, and provide an approximation of the a posteriori probability density function that achieves the desired accuracy. Results also show that Poincaré-based PCEs require fewer samples to achieve a given accuracy than Cartesian-based solutions. In terms of pdf accuracy, the PCE-based solutions represent an improvement over the linear propagation and unscented transformation techniques.