  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.
