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：本文提出了一种新颖方法，用于在通用变换中自动检测和处理非线性特征。首先引入了一种基于二阶泰勒展开与多项式界限技术的非线性指数，以严格评估非线性变换的雅可比矩阵变化。随后将该指数嵌入到一种低阶自动域分割算法中，该算法通过当设定的线性度约束不满足时自动分割区域，准确描述初始不确定性集通过通用非线性变换的映射关系。在轨道不确定性传播这一关键案例中展示了算法的应用，并将其与定制化合并算法相结合——当非线性度降低时通过重新合并区域来限制区域数量随时间增长。进一步将低阶自动域分割算法与高斯混合模型相结合，以精确描述概率密度函数的传播过程。最后对提出的方法进行了详细分析，并研究了不同自由度的选取对方法精度与性能的影响。