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.

翻译：本文提出了一种在通用变换中自动检测与处理非线性的新方法。首先引入了一种利用二阶泰勒展开和多项式边界技术的非线性指标，用于严格估计非线性变换的雅可比矩阵变化。随后将该指标嵌入低阶自动域分割算法中，通过每当预设线性约束条件不满足时对域进行分割，精确描述初始不确定性集通过通用非线性变换的映射过程。该算法以轨道不确定性传播这一关键案例进行说明，并耦合了专门设计的合并算法——当非线性减弱时通过重新组合域来限制域随时间增长。随后将低阶自动域分割算法与高斯混合模型相结合，精确描述概率密度函数的传播过程。本文对所提方法进行了详细分析，并研究了不同自由度对方法精度与性能的影响。