A New Higher-Order Moment-Matching Unscented Transform (MM-UT) for Uncertainty Analysis

Uncertainty quantification (UQ) for highly nonlinear dynamical systems poses a major challenge, requiring a balance between accuracy and computational effort. In recent years, several publications have introduced the Higher-Order Unscented Transform (HOUT). However, most are limited to estimating seventh- or eighth-order statistical moments, thereby neglecting very high-order moments (such as the thirteenth or fourteenth), and often cross-moments. This paper proposes a moment-matching UT method (MM-UT) that employs a 3^D + 2D sigma-point configuration. Compared to classical UQ methods (including various UT variants), the proposed method is distinguished by: a) a unique funnel-shaped architecture capable of handling high-order statistical moments, and b) minimizing the impact of very high-order moments and cross-moments. The primary advantage of MM-UT over existing deterministic UQ methods, however, lies in the ability to compute scaling factors and weights offline, which can then be reused for the online computation of various uncertainty analysis problems; the proposed method is thus independent of the specific problem being addressed. Several examples are presented, including one concerning the long-term propagation of orbital uncertainty, to demonstrate the enhanced accuracy and efficiency of MM-UT over other existing methods.