Before revision:  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.
After revision:  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 (NED) 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.