Uncertainty in LiDAR measurements, stemming from factors such as range sensing, is crucial for LIO (LiDAR-Inertial Odometry) systems as it affects the accurate weighting in the loss function. While recent LIO systems address uncertainty related to range sensing, the impact of incident angle on uncertainty is often overlooked by the community. Moreover, the existing uncertainty propagation methods suffer from computational inefficiency. This paper proposes a comprehensive point uncertainty model that accounts for both the uncertainties from LiDAR measurements and surface characteristics, along with an efficient local uncertainty analytical method for LiDAR-based state estimation problem. We employ a projection operator that separates the uncertainty into the ray direction and its orthogonal plane. Then, we derive incremental Jacobian matrices of eigenvalues and eigenvectors w.r.t. points, which enables a fast approximation of uncertainty propagation. This approach eliminates the requirement for redundant traversal of points, significantly reducing the time complexity of uncertainty propagation from $\mathcal{O} (n)$ to $\mathcal{O} (1)$ when a new point is added. Simulations and experiments on public datasets are conducted to validate the accuracy and efficiency of our formulations. The proposed methods have been integrated into a LIO system, which is available at https://github.com/tiev-tongji/LOG-LIO2.

翻译：激光雷达测量的不确定性（源于测距感知等因素）对LIO（激光雷达-惯性里程计）系统至关重要，因为它影响损失函数中的精确加权。尽管现有LIO系统已考虑了测距感知引发的不确定性，但学界常忽视入射角对不确定性的影响。此外，现有不确定性传播方法存在计算效率低下的问题。本文提出一种综合点不确定性模型，该模型同时考虑激光雷达测量和表面特征引起的不确定性，并针对基于激光雷达的状态估计问题，提出一种高效的局部不确定性分析方法。我们采用投影算子将不确定性分解为射线方向及其正交平面，进而推导特征值与特征向量关于点的增量雅可比矩阵，从而实现不确定性传播的快速近似。该方法避免了对点的冗余遍历，当新增点时，可将不确定性传播的时间复杂度从$\mathcal{O} (n)$显著降低至$\mathcal{O} (1)$。通过仿真实验和公开数据集验证了所提公式的准确性与效率。所提方法已集成至LIO系统中，相关代码开源地址为：https://github.com/tiev-tongji/LOG-LIO2。