Before revision:  Furthermore, we propose a SE_2(3)-based smoothing algorithm based on the SE_2(3)-based EKF .
After revision:  Furthermore, we design two new modified error dynamics on the NED frame and the ECEF frame respectively by introducing new auxiliary vectors. Finally the equivalence of the left-invariant EKF and left SE_2(3) based EKF have been shown in navigation frame and ECEF frame .