Equivariant Filtering Framework for Inertial-Integrated Navigation

This paper proposes a equivariant filtering (EqF) framework for the inertial-integrated state estimation problem. As the kinematic system of the inertial-integrated navigation can be naturally modeling on the matrix Lie group $SE_2(3)$, the symmetry of the Lie group can be exploited to design a equivariant filter which extends the invariant extended Kalman filtering on the group affine system.

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

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.

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

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 proposed 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 local-level navigation frame and the earth-centered earth-fixed frame, respectively, the corresponding extending Kalman filter (EKF), terms as $SE_2(3)$-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 propose a $SE_2(3)$-based smoothing algorithm based on the $SE_2(3)$-based EKF.

$SE_2(3)$ based Extended Kalman Filter for Inertial-Integrated Navigation

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 proposed 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 local-level navigation frame and the earth-centered earth-fixed frame, respectively, the corresponding extending Kalman filter (EKF), terms as $SE_2(3)$-EKF has also been derived. It provides a new perspective on the geometric EKF with a more sophisticated formula for the inertial navigation system.