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

Jul 16, 2026

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Nonlinear uncertainty propagation in perturbed astrodynamics is addressed through rapid characterization of non-Gaussian distributions and the construction of three-dimensional “banana-shaped” confidence boundaries. The work bridges computationally intensive high-fidelity methods and inaccurate linear approximations by providing a fully analytical, sample-free framework for higher-order moment extraction. Using Differential Algebra to avoid repeated numerical integration, statistical moments are obtained analytically via Isserlis’ theorem and a monomial-to-Hermite basis transformation. A pair-product projection reduces fourth-order tensor bottlenecks, enabling accurate skewness and kurtosis parameterization for non-elliptical confidence geometries validated in cislunar NRHO and Apophis flyby scenarios.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

Introduction

Uncertainty quantification in nonlinear astrodynamics

Limits of Monte Carlo and linear covariance propagation

Existing nonlinear UQ methods

Differential Algebra and tensor-based moment extraction

Proposed analytical framework and computational optimization

arXiv :2607 . 10095v1 [ ee ss . SY] 11 Jul 2026 (Preprint) AAS AAS-26-909 ANALYTICAL CONFIDENCE BOUNDARIES FOR NON-GAUSSIAN UNCERTAINTY IN PERTURBED SPACECRAFT DYNAMICS Niccol Michelotti*, Ethan R. Burnett†, and Francesco Topputo‡ 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. 1 INTRODUCTION Reliable and efficient uncertainty quantification (UQ) is a persistent challenge in modern astrodynamics. As mission architectures increasingly target highly nonlinear and weakly stable regimes, capturing the true evolution of state uncertainty becomes critical for the robustness and efficiency of the mission. In cislunar environments, such as Near-Rectilinear Halo Orbits (NRHOs), spacecraft are subjected to sensitive multi-body gravitational fields and repeated low-altitude perilune passages.1 Similarly, close-proximity operations around small bodies, such as the asteroid Apophis, are dominated by irregular gravity fields and rapid dynamical timescales. In these environments, initial Gaussian state uncertainties quickly deform into non-Gaussian distributions driven by strong localized gravitational gradients and resonances.2 Consequently, standard geometric descriptions based uniquely on the mean and covariance fail to capture relevant features of the distribution. Accurately bounding the spacecraft’s state is a fundamental requirement for robust trajectory optimization and space domain awareness. In the presence of non-Gaussian distributions, this requires the *PhD Student, Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa, 34, Milan, 20156, [Italy. niccolo.michelotti@polimi.it](Italy. niccolo.michelotti@polimi.it) †Assistant Professor, Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, 2617 Wichita St., Austin TX, 78712, [USA. ethan.burnett@utexas.edu](USA. ethan.burnett@utexas.edu). ‡Full Professor, Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa, 34, Milan, 20156, [Italy. francesco.topputo@polimi.it. AIAA Senior Member](Italy. francesco.topputo@polimi.it. AIAA Senior Member). evaluation of higher-order statistical moments, such as skewness and kurtosis, to properly represent the asymmetry and tail-stretching of the probability density function. Historically, Monte Carlo (MC) methods have served as the highest-fidelity benchmark for capturing these non-Gaussian features. However, their reliance on thousands of numerical integrations makes them computationally expensive for rapid and preliminary analyses, large-scale trajectory design campaigns, or onboard applications for a

FAQs

What problem does the paper address in spacecraft uncertainty quantification?

The paper targets reliable and efficient uncertainty quantification when non-Gaussian state uncertainty emerges under nonlinear, perturbed astrodynamics, where mean and covariance alone fail to represent key distribution features.

How does the proposed method avoid repeated numerical integration?

It introduces a fully analytical, sample-free framework using Differential Algebra, extracting higher-order moments directly from expansion coefficients rather than repeatedly sampling the full dynamics.

Why is the pair-product projection strategy important?

It mitigates the computational bottleneck from dense fourth-order tensor contractions by computing only relevant terms through efficient polynomial algebra, improving scalability for confidence boundary reconstruction.

Jul 16, 2026

Upload by Liam