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.

翻译：针对惯性里程计的滤波器是一种递归方法，用于通过自运动测量和相对位姿测量估计位姿。目前，尚无已知滤波器能保证针对非线性测量模型计算全局最优解。本文证明，一种创新滤波器——以状态空间为$SE_2(3)$且以$\sqrt{n}$一致位姿作为初始化的滤波器——能够在均方误差最小意义上高效实现渐近最优性。该方法专为实时SLAM和惯性里程计应用设计。我们的第一个贡献是提出了一种基于李群上高斯-牛顿法的迭代滤波方法，该方法通过数值手段从先验信息和非线性测量中求解状态估计。该滤波器的突出之处在于其迭代机制与自适应初始化。其次，在处理周围环境测量时，我们采用$\sqrt{n}$一致位姿作为单次迭代更新步骤的初始值。该解具有闭式形式，计算复杂度为$O(n)$。第三，我们从理论上证明，该方法能够基于先验和虚拟相对位姿测量（见问题~\ref{prob:new update problem}）在均方误差最小意义上实现渐近最优性。最后，为验证我们的方法，我们开展了广泛的数值实验和实际评估。结果一致表明，我们的方法在精度和运行时间上均优于其他先进滤波器方法，包括迭代扩展卡尔曼滤波器和不变扩展卡尔曼滤波器。