  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.
