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.

翻译：目前的国家估算对机器人非常重要, 以不确定性代表为基础的 Lie 组对于国家估算问题来说是自然的。 有必要充分利用矩阵 Lie组的几何学和运动学 。 因此, 本说明对最近提议的矩阵 Lie组 $SE_ K(3) 美元进行了详细推算, 我们第一次将结果扩展为 赤脚\ cite{ barfof2017 state} 。 然后我们描述该组适合国家代表的情况。 我们还根据Matlab 框架制定了快速实施和测试的代码 。