We address the problem of uncertainty propagation and Bayesian fusion on unimodular Lie groups. Starting from a stochastic differential equation (SDE) defined on Lie groups via Mckean-Gangolli injection, we first convert it to a parametric SDE in exponential coordinates. The coefficient transform method for the conversion is stated for both Ito's and Stratonovich's interpretation of the SDE. Then we derive a mean and covariance fitting formula for probability distributions on Lie groups defined by a concentrated distribution on the exponential coordinate. It is used to derive the mean and covariance propagation equations for the SDE defined by injection, which coincides with the result derived from a Fokker-Planck equation in previous work. We also propose a simple modification to the update step of Kalman filters using the fitting formula, which improves the fusion accuracy with moderate computation time.

翻译：本文研究单模李群上的不确定性传播与贝叶斯融合问题。首先从基于McKean-Gangolli注入定义在李群上的随机微分方程出发，将其转化为指数坐标下的参数化随机微分方程。针对该转换过程，分别讨论了在Ito和Stratonovich两种随机微分方程解释下的系数变换方法。随后，我们推导了由指数坐标上的集中分布所定义的李群概率分布的均值与协方差拟合公式。该公式可推导出通过注入定义的随机微分方程的均值与协方差传播方程，其结果与先前工作中基于福克-普朗克方程推导的结果一致。此外，我们利用拟合公式对卡尔曼滤波的更新步骤进行了简单改进，在适度计算时间内提升了融合精度。