Analytical Confidence Boundaries for Non-Gaussian Uncertainty in Perturbed Spacecraft Dynamics

This work investigates nonlinear uncertainty propagation in perturbed astrodynamics, focusing on the rapid characterization of non-Gaussian distributions and the construction of three-dimensional "banana-shaped" confidence boundaries. To bridge the gap between computationally intensive high-fidelity methods and inaccurate linear approximations, this paper introduces a fully analytical, sample-free framework for higher-order moments extraction. Leveraging Differential Algebra to bypass repeated numerical integration, statistical moments are extracted analytically via Isserlis' theorem and a monomial-to-Hermite basis transformation. A pair-product projection strategy is exploited to overcome the severe computational bottleneck of full fourth-order tensor contractions and compute only relevant terms via efficient polynomial algebra. The extracted skewness and kurtosis components directly parameterize non-elliptical confidence geometries that capture spatial bending and out-of-plane coupling of typical non-Gaussian distributions in astrodynamics. The approach is validated in high-fidelity environments including a cislunar Near-Rectilinear Halo Orbit and close-proximity trajectories around Apophis during Earth's flyby, where the analytical approach achieves geometric accuracy comparable to expensive Monte Carlo simulations while reducing computational runtime by orders of magnitude.