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This paper proposes an $SE_2(3)$ based extended Kalman filtering (EKF)framework for the inertial-integrated state estimation problem. The errorrepresentation using the straight difference of two vectors in the inertialnavigation system may not be reasonable as it does not take the directiondifference into consideration.  Therefore, we choose to use the $SE_2(3)$ matrix Lie group to represent thestate of the inertial-integrated navigation system which consequently leads tothe common frame error representation.  With the new velocity and position error definition, we leverage the groupaffine dynamics with the autonomous error properties and derive the error statedifferential equation for the inertial-integrated navigation on thenorth-east-down (NED) navigation frame and the earth-centered earth-fixed(ECEF) frame, respectively, the corresponding EKF, terms as $SE_2(3)$ based EKFhas also been derived. It provides a new perspective on the geometric EKF witha more sophisticated formula for the inertial-integrated navigation system.Furthermore, we design two new modified error dynamics on the NED frame and theECEF frame respectively by introducing new auxiliary vectors. Finally theequivalence of the left-invariant EKF and left $SE_2(3)$ based EKF have beenshown in navigation frame and ECEF frame.

$SE_2(3)$ based Extended Kalman Filtering and Smoothing Framework for Inertial-Integrated Navigation