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Through assembling the navigation parameters as matrix Lie group state, the corresponding inertial navigation system (INS) kinematic model possesses a group-affine property. The Lie logarithm of the navigation state estimation error satisfies a log-linear autonomous differential equation. These log-linear models are still applicable even with arbitrarily large initial errors, which is very attractive for INS initial alignment. However, in existing works, the log-linear models are all derived based on first-order linearization approximation, which seemingly goes against their successful applications in INS initial alignment with large misalignments. In this work, it is shown that the log-linear models can also be derived without any approximation, the error dynamics for both left and right invariant error in continuous time are given in matrix Lie group SE_2 (3) for the first time. This work provides another evidence for the validity of the log-linear model in situations with arbitrarily large initial errors.

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This paper proposes a unified mathematical framework for inertial measurement unit (IMU) preintegration in inertial-aided navigation system in different frames under different motion condition. The navigation state is precisely discretized as three part: local increment, global state, and global increment. The global increment can be calculated in different frames such as local geodetic navigation frame and earth-centered-earth-fixed frame. The local increment which is referred as the IMU preintegration can be calculated under different assumptions according to the motion of the agent and the grade of the IMU. Thus, it more accurate and more convenient for online state estimation of inertial-integrated navigation system under different environment.

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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. Furthermore, details of the analytic state transition matrices for left invariant error and right invariant error are given.

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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 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.

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Currently state estimation is very important for the robotics, and the uncertainty representation based Lie group is natural for the state estimation problem. It is necessary to exploit the geometry and kinematic of matrix Lie group sufficiently. Therefore, this note gives a detailed derivation of the recently proposed matrix Lie group $SE_K(3)$ for the first time, our results extend the results in Barfoot \cite{barfoot2017state}. Then we describe the situations where this group is suitable for state representation. We also have developed code based on Matlab framework for quickly implementing and testing.

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