Non-Gaussian Distribution Steering in Nonlinear Dynamics with Conjugate Unscented Transformation

TL;DR

The paper addresses steering non-Gaussian distributions arising in nonlinear astrodynamics by extending covariance steering to higher-order moments. It introduces Conjugate Unscented Transformation (CUT) to quantify distribution moments through discrete sigma points and embeds these into a convex optimization via sequential convex programming (SCvx*). The key contributions include a comprehensive convex formulation for mean, covariance, skewness, and higher moments; centralized sigma-point treatment; and validation on two-body transfers and CR3BP halo orbits, showing improved control over non-Gaussian uncertainties. Practically, this framework enables onboard autonomous guidance with explicit moment constraints, improving robustness to non-Gaussian disturbances in nonlinear spaceflight applications, though CUT remains an approximation requiring Monte Carlo validation.

Abstract

In highly nonlinear systems such as the ones commonly found in astrodynamics, Gaussian distributions generally evolve into non-Gaussian distributions. This paper introduces a method for effectively controlling non-Gaussian distributions in nonlinear environments using optimized linear feedback control. This paper utilizes Conjugate Unscented Transformation to quantify the higher-order statistical moments of non-Gaussian distributions. The formulation focuses on controlling and constraining the sigma points associated with the uncertainty quantification, which would thereby reflect the control of the entire distribution and constraints on the moments themselves. This paper develops an algorithm to solve this problem with sequential convex programming, and it is demonstrated through a two-body and three-body example. The examples show that individual moments can be directly controlled, and the moments are accurately approximated for non-Gaussian distributions throughout the controller's time horizon in nonlinear dynamics.

Figures (20)

• Figure 1: Flowchart of methodology for statistical moment steering.
• Figure 2: Monte Carlo ($n_{\text{samples}}=10,000$) for two-body example: Initial non-Gaussian, highly skewed, distribution. Axes are not equalized to better show skewness. Origin normalize to mean predicted by CUT.
• Figure 3: Convergence profile of SCvx* for two-body example. Y-axis in log scale.
• Figure 4: Left: initial orbit in blue, final orbit in magenta. Right: transfer trajectory with control nodes marked in red and initial/final points marked with a green/red $\star$.
• Figure 5: Left: transfer trajectory with nominal control vectors marked in red and initial/final points marked with a green/red $\star$. Right: CUT points ($\times$) along transfer trajectory, with initial/final CUT points marked in green/red. Deviations enlarged $2{\times}$ to better show the individual points.