## 3 Methodology
### Tightly Coupled Pose Compensation  
Mechanical Lidar detects the environment by rotating the laser emitter during data collection. Data is generally collected and stored in the form of equal-angle fan-shaped sectors \(fan_{i}\) starting from the Y axis in a clockwise direction. For example, each laser frame contains 75 equal-angle fan-shaped sectors. The acquisition time for each fan-shaped sector in a Lidar frame with an operating frequency of 10 Hz is 1.33 ms. However, during movement, the vehicle carries the mounted Lidar along with it, resulting in an additional superimposed change in the Lidar's movement direction besides the change in the scanning direction between the various fan angles in the laser scanning frame. As shown in Figure 5, after the superposition change of the scanning and movement directions, the fan angle \(\mathbf{P}_{fan_{i}}^{n}\) is transformed into \(x^{\prime}y^{\prime}z^{\prime}\) in the ego body coordinate system. IMU can output high-frequency pose data through dead reckoning (DR). Therefore, the pose-transformation matrix between adjacent Lidar frame nodes can be calculated by coupling the pose information output by IMU pre-integration [38]. The fan-shaped sector can then be compensated for pose using a timestamp-based linear interpolation method. The initial pose value of the current laser frame can be estimated by combining the pose of the previous laser historical keyframe. In Figure 5's trajectory nodes' queue, \(n\) represents the current laser frame, and the pose of the \((n-1)\)th laser historical key frame \([\mathbf{R}_{n-1},\mathbf{t}_{n-1}]\) is obtained from Section 3.2. \(\mathbf{R}_{n-1}\) denotes the rotation matrix of the pose relative to the origin coordinate, and \(\mathbf{t}_{n-1}\) represents the translation vector. This section focuses on standardizing the pose of the current \(n\)th laser frame, which includes calculating the reference pose of the current laser frame and compensating for the fan-shaped sector pose based on this reference pose of laser frame. First, the reference pose of the current laser frame is calculated. This reference pose is obtained through IMU coupling. IMU pre-integration [27] outputs the pose-transformation matrix \(\mathbf{\Gamma}_{n,n-1}^{\text{IMU}}\) from the adjacent node \(n-1\) to \(n\):  
\[\mathbf{\Gamma}_{n,n-1}^{\text{IMU}}= \begin{bmatrix}\Delta\mathbf{R}_{n,n-1}^ {\text{IMU}}&\Delta\mathbf{t}_{n,n-1}^{\text{IMU}}\\ 0^{\text{T}}&1\end{bmatrix} , \tag{1}\]  
\[\Delta\mathbf{v}_{n,n-1}^{\text{IMU}}\boldsymbol{\xi}= \begin{pmatrix}\mathbf{ R}_{n-1}^{\text{IMU}}\\ \end{pmatrix} ^{\text{T}}(\mathbf{v}_{n}-\mathbf{v}_{n-1}-g\Delta t_{n,n-1}), \tag{2}\]  
Figure 5: Schematic diagram illustrating posture compensation with tightly coupled Lidar and IMU. \[\Delta\mathbf{t}_{n,n-1}^{\text{IMU}}=\left(\mathbf{R}_{n-1}^{\text{IMU}}\right)^{ \text{T}}\!\left(\mathbf{t}_{n}-\mathbf{t}_{n-1}-\mathbf{v}_{n-1}\Delta t_{n,n- 1}-\frac{1}{2}g(\Delta t_{n,n-1})^{2}\right), \tag{3}\]  
\[\Delta\mathbf{R}_{n,n-1}^{\text{IMU}}\&=\left(\mathbf{R}_{n-1}^{\text{IMU}} \right)^{\text{T}}\mathbf{R}_{n}^{\text{IMU}}. \tag{4}\]  
Among these variables, \(\mathbf{v}\) represents the vehicle speed, and \(\mathbf{g}\) denotes the acceleration due to gravity. Then, the initial posture of the current laser frame can be determined using the IMU posture transformation formula:  
\[\tilde{x}_{n}=\left[\mathbf{R}_{n}^{\text{init}},\mathbf{t}_{n}^{\text{init}} \right]=\mathbf{R}^{\text{IMU}}\mathbf{T}_{n,n-1}^{\text{IMU}}[\mathbf{R}_{n- 1},\ \mathbf{t}_{n-1}], \tag{5}\]  
where \([\mathbf{R}_{n-1},\mathbf{t}_{n-1}]\) is obtained from Section 3.2; \(\mathbf{R}^{\text{IMU}}\) represents the external parameter relationship between IMU and Lidar. Secondly, perform posture compensation on the fan-shaped sector of the current laser frame. Posture compensation corrects the offset caused by the movement of the ego vehicle during a laser scan, aligning all fan-shaped sectors' coordinate systems within a frame to the coordinate system of the first fan sector.