## 3 LiDAR-Inertial odometry  
As early as 2005, the framework of LiDAR SLAM was initially determined, and many studies have been achieved later on LiDAR-based SLAM. For example, the early 2D LiDAR SLAM frameworks cartographer [15] and hector [16] can build environmental maps in a 2D plane, but the localization accuracy is poor. Later, 3D LiDAR SLAM was proposed, such as the LOAM series [17; 18], which significantly improved the performance of localization and mapping. Pure LiDAR SLAM relies on simple scanning and matching methods, which has poor performance in dynamic and complex environments and cannot achieve reliable localization and mapping. Therefore, the SLAM framework with fusion of LiDAR and inertial measurement unit (IMU) is proposed. IMU can measure the acceleration and angular velocity of robots, and is not easily affected by external interference. The fusion of IMU can solve the problem of localization drift of pure LiDAR SLAM in fast motion scenarios. For example, Lego-LOAM [19] eliminates unreliable feature points based on LOAM, but adopts a loosely coupled approach to fuse IMU, which does not fully utilize sensor data. The later proposed LIOM [20], LIO-SAM [21], and FAST-LIO [22] are all tightly coupled LiDAR-inertial SLAM systems based on factor graph optimization, which does not require making Gaussian distribution assumptions compared to the traditional filtering method, by calculating the conditional probability between variables and passing the results to Bayesian trees [23] for posterior inference, it can efficiently handle large scale sparse matrices, improve the accuracy and robustness of the model, and has become the mainstream SLAM optimization method.  
The LiDAR-inertial subsystem in this framework is based on the LIO-SAM, which has better localization and mapping effects. The world frame is W, and the robot body frame is  
Figure 1: Framework diagram of the system.  
B, assuming that the IMU frame and the robot body frame are consistent, then the robot state can be described as:  
\[x=\left[R^{\mathrm{T}}p^{\mathrm{T}}v^{\mathrm{T}}b^{\mathrm{T}}\right]^{\mathrm{T}}. \tag{1}\]  
\(R\in\mathrm{SO}(3)\) is the rotation matrix, which describes the orientation or attitude of the robot in 3D space. \(p\in R^{3}\) is the position vector. \(\mathrm{v}\) is the velocity of the robot. \(b\) is the IMU bias, which captures systematic errors or offsets in the IMU measurements. It can more accurately model and estimate the motion of the robot. The transformation \(\mathrm{T}\in\mathrm{SE}(3)\) from B to W is represented as \(\mathrm{T}=[R|p]\).  
As shown in Figure 2, the factor graph of the LIO subsystem is composed of four constraints, IMU preintegration constraint, LiDAR odometry constraint, loop closure constraint, and robot state. The raw measurement data of IMU is used to estimate the sensor motion during LiDAR scanning. The estimated motion can be used not only to eliminate point cloud distortion, but also as the initial guess for LiDAR odometry optimization. The obtained initial guess can then be used to estimate the IMU bias and form a loop closure constraint through closed-loop detection. Finally, the relationships among all constraints are considered to achieve global factor graph optimization.