  A filter for inertial-based odometry is a recursive method used to estimate
the pose from measurements of ego-motion and relative pose. Currently, there is
no known filter that guarantees the computation of a globally optimal solution
for the non-linear measurement model. In this paper, we demonstrate that an
innovative filter, with the state being $SE_2(3)$ and the
$\sqrt{n}$-\textit{consistent} pose as the initialization, efficiently achieves
\textit{asymptotic optimality} in terms of minimum mean square error. This
approach is tailored for real-time SLAM and inertial-based odometry
applications.
  Our first contribution is that we propose an iterative filtering method based
on the Gauss-Newton method on Lie groups which is numerically to solve the
estimation of states from a priori and non-linear measurements. The filtering
stands out due to its iterative mechanism and adaptive initialization. Second,
when dealing with environmental measurements of the surroundings, we utilize a
$\sqrt{n}$-consistent pose as the initial value for the update step in a single
iteration. The solution is closed in form and has computational complexity
$O(n)$. Third, we theoretically show that the approach can achieve asymptotic
optimality in the sense of minimum mean square error from the a priori and
virtual relative pose measurements (see Problem~\ref{prob:new update problem}).
Finally, to validate our method, we carry out extensive numerical and
experimental evaluations. Our results consistently demonstrate that our
approach outperforms other state-of-the-art filter-based methods, including the
iterated extended Kalman filter and the invariant extended Kalman filter, in
terms of accuracy and running time.
