# IMU-Aided High-Frequency Lidar Odometry for Autonomous Driving
## 3 The Proposed Approach  
### Coordinate Systems and Notations  
The aim of this work is to accurately estimate the pose of the vehicle. Three coordinate systems are used in our approach, which are defined as follows:  
* World Coordinate System \(\{W\}\). In \(\{W\}\), we define the coordinates of a point by latitude, longitude, and altitude, the x-axis points to the east, the y-axis points to the north, and the z-axis points to the sky, following the right-hand rule. This coordinate system is defined in a global scale and will never change. * Body Coordinate System \(\{B\}\). This coordinate is fixed with the IMU, which is installed in the center of two rear wheels of the vehicle. The x-axis points to the right, the y-axis points to the forward, and the z-axis points to the upward, following the right-hand rule. * Lidar coordinate system \(\{L\}\). The lidar is mounted on top of the vehicle, and the origin of \(\{L\}\) is located at the center of the lidar. The three axes of this coordinate system are the same as in \(\{B\}\). Figure 2 shows the coordinate system \(\{W\}\) and \(\{B\}\) in the 2D plane and the relationship between them. The body coordinate system \(\{B\}\) changes with the movement of the vehicle. \(\mathbf{T}_{\mathbf{k}}^{\text{WB}}\) represents the transformation between the coordinate system \(\{W\}\) and \(\{  
transformation of the vehicle between time \(t_{\rm k-1}\) and \(t_{\rm k}\). The relationship between \(\triangle\mathbf{T}_{\rm k-1,k}^{\rm B}\) and \(\mathbf{T}_{\rm k}^{\rm WB}\) can be expressed as:  
\[\triangle\mathbf{T}_{\rm k-1,k}^{\rm B}=\left(\mathbf{T}_{\rm k-1}^{\rm WB} \right)^{-1}\mathbf{T}_{\rm k}^{\rm WB}\,. \tag{1}\]  
In addition, the relationship between the coordinate system \(\{B\}\) and \(\{L\}\) is shown in Figure 3.  
We use \(\mathbf{T}_{\rm k}^{\rm WL}\) to denote the pose of the lidar at time \(t_{\rm k}\). \(\triangle\mathbf{T}_{\rm k-1,k}^{\rm L}\) represents the relative transformation of the lidar between time \(t_{\rm k-1}\) and \(t_{\rm k}\). The transformation \(\mathbf{T}^{\rm BL}\), which transforms the coordinate system \(\{B\}\) to \(\{L\}\), can be obtained from the hand-eye calibration. The transformation \(\mathbf{T}\) can be expressed by:  
\[\mathbf{T}= \begin{bmatrix}\mathbf{R}&\mathbf{p}\\ \mathbf{0}^{T}&1\end{bmatrix} \,, \tag{2}\]  
where \(\mathbf{R}\in\mathbb{R}^{3\times 3}\) represents the rotational matrix and \(\mathbf{p}\in\mathbb{R}^{3}\) is the translational vector. In _Lie Group_[19], \(\mathbf{T}\) belongs to the _Special Euclidean group_\(\rm SE(3)\) and the rotational matrix \(\mathbf{R}\) belongs to the _Special Orthogonal group_\(\rm SO(3)\). Correspondingly, the \(\mathfrak{se}(3)\) and \(\mathfrak{so}(3)\) are two kinds of _Lie Algebra_. \(\mathbf{\theta}\in\mathfrak{so}(3)\) indicates the rotation angle which can be expressed as \(\mathbf{\theta}=\mathbf{\theta}\mathbf{n}\), where \(\theta\) represents the  
Figure 3: The relationship between the body coordinate system \(\{L\}\) and the lidar coordinate system \(\{B\}\), where the yellow axis indicates \(\{L\}\) and the green axis indicates \(\{B\}\). Figure 2: The relative relationship between the world coordinate system \(\{W\}\) and the body coordinate system \(\{B\}\) in the 2D plane, where the black axis indicates \(\{W\}\) and the red axis indicates \(\{B\}\). magnitude of the rotation angle and \(\mathbf{n}\) denotes the rotation axis. The pose can thus be represented by \(\mathbf{\xi}= \begin{bmatrix}\mathbf{\theta}&\mathbf{p}\end{bmatrix} ^{T}\in\mathfrak{se}(3)\). The exponential map and logarithmic map can be used to derive the relationship between the _Lie Group_ and the _Lie Algebra_. For example, \(\mathbf{T}\) can be represented by the exponential map of the \(\mathbf{\xi}\), namely, \(\mathbf{T}=\exp\left(\mathbf{\xi}^{\wedge}\right)\). Conversely, \(\mathbf{\xi}\) can be represented by the logarithmic map of the \(\mathbf{T}\), namely, \(\mathbf{\xi}=\log\left(\mathbf{T}\right)^{\vee}\).