The error representation using the straight difference of two vectors in the inertial navigation system may not be reasonable as it does not take the direction difference into consideration. Therefore, we proposed to use the $SE_2(3)$ matrix Lie group to represent the state of the inertial-integrated navigation system which consequently leads to the common frame error representation. With the new velocity and position error definition, we leverage the group affine dynamics with the autonomous error properties and derive the error state differential equation for the inertial-integrated navigation on the north-east-down local-level navigation frame and the earth-centered earth-fixed frame, respectively, the corresponding extending Kalman filter (EKF), terms as $SE_2(3)$-EKF has also been derived. It provides a new perspective on the geometric EKF with a more sophisticated formula for the inertial navigation system.

中文翻译：

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基于$ SE_2（3）$的扩展卡尔曼滤波器，用于惯性集成导航

在惯性导航系统中使用两个向量的直线差的误差表示可能不合理，因为它没有考虑方向差。因此，我们建议使用$ SE_2（3）$矩阵李群表示惯性集成导航系统的状态，从而导致常见的帧误差表示。利用新的速度和位置误差定义，我们利用了仿射动力学和自主误差属性，并推导出了误差模型的微分方程，用于东北向下局部水平导航框架和以地球为中心的惯性积分导航还推导了固定在地球上的框架，相应的扩展卡尔曼滤波器（EKF），术语为$ SE_2（3）$-EKF。