标题： 矩阵李群上左不变和右不变扩展卡尔曼滤波器的等价性 显示英文标题

标题： Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups

摘要： 本文推导了连续时间系统在矩阵李群上的扩展卡尔曼滤波器（EKF），该系统通过离散时间测量值进行观测。 通过在李代数上建模系统噪声，并采用随机微分方程（SDE）的Stratonovich解释，我们确保解始终保持在流形上。 滤波器的推导遵循经典的EKF原理，自然地整合了必要的全阶协方差重置测量更新。 一个关键贡献是证明这种全阶协方差重置保证了李群值状态估计对EKF中使用的左不变或右不变误差定义具有不变性。 辅助惯性导航问题的蒙特卡洛模拟验证了这一不变性属性，并确认在使用降阶协方差重置时该属性不存在。 显示英文摘要

摘要： This paper derives the extended Kalman filter (EKF) for continuous-time systems on matrix Lie groups observed through discrete-time measurements. By modeling the system noise on the Lie algebra and adopting a Stratonovich interpretation for the stochastic differential equation (SDE), we ensure that solutions remain on the manifold. The derivation of the filter follows classical EKF principles, naturally integrating a necessary full-order covariance reset post-measurement update. A key contribution is proving that this full-order covariance reset guarantees that the Lie-group-valued state estimate is invariant to whether a left- or right-invariant error definition is used in the EKF. Monte Carlo simulations of the aided inertial navigation problem validate the invariance property and confirm its absence when employing reduced-order covariance resets.