Unscented and Higher-Order Linear Covariance Fidelity Checks and Measures of Non-Gaussianity

TL;DR

This work targets the fidelity of linear covariance (LinCov) uncertainty propagation in highly nonlinear systems by introducing a suite of fidelity measures grounded in higher-order statistics, constrained optimization, and the unscented transform. It develops both expectation-based and optimization-based metrics, including tensor-based moments and maximal nonlinearity directions, and contrasts them with Monte Carlo benchmarks. Through a cislunar three-body propagation example, the authors show that second-order and unscented approximations generally align with MC for moderate uncertainties, while non-Gaussian features become pronounced near perilune. The proposed measures offer computationally efficient verifications to decide when LinCov is trustworthy for spacecraft navigation and mission planning, and they provide insight into when higher-fidelity nonlinear uncertainty propagation is warranted.

Abstract

Linear covariance (LinCov) techniques have gained widespread traction in the modeling of uncertainty, including in the preliminary study of spacecraft navigation performance. While LinCov methods offer improved computational efficiency compared to Monte Carlo based uncertainty analysis, they inherently rely on linearization approximations. Understanding the fidelity of these approximations and identifying when they are deficient is critically important for spacecraft navigation and mission planning, especially when dealing with highly nonlinear systems and large state uncertainties. This work presents a number of computational techniques for assessing linear covariance performance. These new LinCov fidelity measures are formulated using higher-order statistics, constrained optimization, and the unscented transform.

Figures (17)

• Figure 1: Notional depiction of the contours of equal Mahalanobis distance with a difference in two means superimposed.
• Figure 2: Notional depiction of minimal and maximal ratios of two ellipses. Note that the two directions are not orthogonal.
• Figure 3: Notional depiction of two distributions with distinct means and covariance ellipsoids.
• Figure 4: The normalized skewness and kurtosis of three common distributions as reference to interpret skewness values in terms of the asymmetry of the distribution and kurtosis values in terms of the heavy-tailedness of the distribution.
• Figure 5: A schematic describing the quantity being maximized in computing wussos.