A low-order automatic domain splitting approach for nonlinear uncertainty mapping

TL;DR

The paper tackles nonlinear uncertainty propagation in transformations by introducing a da-based nonlinear index $\nu$ that quantifies nonlinearity through polynomial bounding of the Jacobian variation. This index drives a low-order automatic domain splitting (LOADS) algorithm that partitions the uncertainty into locally quasi-linear domains, enabling accurate mapping through nonlinear transforms without Monte Carlo sampling. The LOADS framework is extended with merging to control domain growth and further combined with Gaussian Mixture Models (LOADS-GMM) to provide an analytical approximation of the propagated pdf, using Gaussian kernels attached to each domain and UT sigma points for propagation. Numerical experiments in orbital propagation show that LOADS achieves comparable accuracy to Monte Carlo with substantially reduced computation, while LOADS-GMM delivers efficient, scalable pdf representations and favorable lam overlaps with MC results. Overall, the approach offers a flexible, parameter-tunable toolkit for reliable nonlinear uncertainty propagation in astrodynamics and related fields. $\nu$ and $\varepsilon_{\nu}$ are central to controlling accuracy and computational load across the LOADS and LOADS-GMM components.

Abstract

This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second order Taylor expansions and polynomial bounding techniques is first introduced to rigorously estimate the Jacobian variation of a nonlinear transformation. This index is then embedded into a low-order automatic domain splitting algorithm that accurately describes the mapping of an initial uncertainty set through a generic nonlinear transformation by splitting the domain whenever some imposed linearity constraints are non met. The algorithm is illustrated in the critical case of orbital uncertainty propagation, and it is coupled with a tailored merging algorithm that limits the growth of the domains in time by recombining them when nonlinearities decrease. The low-order automatic domain splitting algorithm is then combined with Gaussian mixtures models to accurately describe the propagation of a probability density function. A detailed analysis of the proposed method is presented, and the impact of the different available degrees of freedom on the accuracy and performance of the method is studied.

Figures (8)

• Figure 1: Merging phase description: \ref{['subfig:Merging_1']} recombination of three sets at splitting level 3; \ref{['subfig:Merging_2']} merging failure at level 3, and $\tilde{M}_h$ update; \ref{['subfig:Merging_3']} merging failure at level 2 and $\tilde{M}_h$ update; \ref{['subfig:Merging_4']} final $\tilde{M}_h$ composition.
• Figure 2: Splitting routines for loads and loads-gmm. On the left, the standard loads subsets generation with 3 domains without overlapping. On the right, the splitting library used for gmm generation ($L=3$, $\lambda = 1e-3$).
• Figure 3: Comparison between $\nu$ and $\nu_{JS}$ over three revolutions. The plot reports 50$\nu_{JS}$ samples out of 150 for visualization purposes.
• Figure 4: Orbital uncertainty propagation results: \ref{['subfig:LOADS_t_33']} 1.88 orbital periods, pericenter approach, 1183 sets; \ref{['subfig:LOADS_t_42']} 2.4 orbital periods, apocenter approach, 497 sets ($\varepsilon_{\nu}=0.02$, $N_{max,j}=10$, $\alpha_{\lambda_j}=3$, $n_{b/r}=20$).
• Figure 5: Orbital uncertainty propagation results (detail): \ref{['subfig:LOADS_t_33_focus']} 1.88 orbital periods, pericenter approach, 1183 sets; \ref{['subfig:LOADS_t_42_focus']} 2.4 orbital periods, apocenter approach, 497 sets ($\varepsilon_{\nu}=0.02$, $N_{max,j}=10$, $\alpha_{\lambda_j}=3$, $n_{b/r}=20$).