This paper addresses uncertainty propagation on unimodular matrix Lie groups that have a surjective exponential map. We derive the exact formula for the propagation of mean and covariance in a continuous-time setting from the governing Fokker-Planck equation. Two approximate propagation methods are discussed based on the exact formula. One uses numerical quadrature and another utilizes the expansion of moments. A closed-form second-order propagation formula is derived. We apply the general theory to the joint attitude and angular momentum uncertainty propagation problem and numerical experiments demonstrate two approximation methods. These results show that our new methods have high accuracy while being computationally efficient.

翻译：本文研究具有满射指数映射的单模矩阵李群上的不确定性传播问题。我们基于主导的福克-普朗克方程，推导出连续时间框架下均值和协方差传播的精确公式。在此基础上讨论了两种基于精确公式的近似传播方法：一种采用数值求积，另一种利用矩展开。我们推导出封闭形式的二阶传播公式，并将该通用理论应用于姿态与角动量联合不确定性传播问题，通过数值实验验证了两种近似方法。结果表明，本文提出的新方法在保持高精度的同时具有计算高效性。