  This paper proposes an $SE_2(3)$ based extended Kalman filtering (EKF)
framework for the inertial-integrated state estimation problem. 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 choose 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 (NED) navigation frame and the earth-centered earth-fixed
(ECEF) frame, respectively, the corresponding EKF, terms as $SE_2(3)$ based EKF
has also been derived. It provides a new perspective on the geometric EKF with
a more sophisticated formula for the inertial-integrated navigation system.
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.
