The Geometry and Kinematics of the Matrix Lie Group $SE_K(3)$
This work provides foundational mathematical derivations for robotics researchers working on state estimation, though it appears to be an incremental extension of existing theory.
This paper provides a detailed derivation of the $SE_K(3)$ matrix Lie group, extending previous work by Barfoot. It also describes the suitable applications for this group in state estimation.
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.