# A Survey of Low-Cost 3D Laser Scanning Technology
## 3 Classification of the Prototypes
### Discussion of the Classification of the Prototypes
We can calculate \(p_{L}\), which is the coordinate of the sampling point \(\mathbf{p}\) relative to the coordinate frame \(L\), by Equation (1) in Section 2 of this paper. For a rotating 2D LiDAR, next, point \(\mathbf{p}_{L}\) in coordinate frame \(L\) is converted to the corresponding point \(\mathbf{p}_{P}\) in coordinate frame \(P\)  
\[\mathbf{p}_{P}=R_{L}^{P}\mathbf{p}_{L}+T_{L}^{P} \tag{3}\]  
where \(R_{L}^{P}\) and \(T_{L}^{P}\) are the rotation matrix and the translation vector from coordinate frame \(L\) to coordinate frame \(P\), respectively. These two parameters show the position and attitude  
Figure 13: The conversions of the coordinates of the sampling points in a rotating 2D LiDAR and a push-broom 2D LiDAR respectively. Left: a rotating 2D LiDAR is mounted on a mobile platform; right: a 2D LiDAR is fixedly mounted on a mobile platform, making it a push-broom 2D LiDAR. of the 2D LiDAR relative to the prototype, depending on the rotation angle of the 2D LiDAR rotated by the motor, and the mechanical assembly position of the 2D LiDAR on the prototype. Next, point \(\mathbf{p}_{P}\) in coordinate frame \(P\) is converted to the corresponding point \(\mathbf{p}_{M}\) in coordinate frame \(M\)  
\[\mathbf{p}_{M}=R_{P}^{M}\mathbf{p}_{P}+T_{P}^{M} \tag{4}\]  
where \(R_{P}^{M}\) and \(T_{P}^{M}\) are the rotation matrix and the translation vector from coordinate frame \(P\) to coordinate frame \(M\), respectively. These two parameters show the position and attitude of the prototype relative to the mobile platform, being dependent on the installation of the prototype on the mobile platform. Usually, these two parameters are constant and do not change with time. Because for general applications, prototypes are fixedly mounted on mobile platforms. Next, point \(\mathbf{p}_{M}\) in coordinate frame \(M\) is converted to the corresponding point \(\mathbf{p}_{W}\) in coordinate frame \(W\)  
\[\mathbf{p}_{W}=R_{M}^{W}\mathbf{p}_{M}+T_{M}^{W} \tag{5}\]  
where \(R_{M}^{W}\) and \(T_{M}^{W}\) are the rotation matrix and the translation vector from coordinate frame \(M\) to coordinate frame \(W\), respectively. These two parameters show the position and attitude of the mobile platform relative to the world coordinate frame, depending on the movement of the mobile platform. Until now, for a rotating 2D LiDAR, the conversion of the sampling point \(p\) from the 2D LiDAR coordinate frame \(L\) to the world coordinate frame \(W\) has been completed. Having calculated the coordinate of the sampling point \(p_{L}\) relative to the coordinate frame \(L\) by Equation (1) in Section 2, for a push-broom 2D LiDAR, next, we will convert point \(p_{L}\) in coordinate frame \(L\) to the corresponding point \(p_{M}\) in coordinate frame \(M\)  
\[\mathbf{p}_{M}=R_{L}^{M}\mathbf{p}_{L}+T_{L}^{M} \tag{6}\]  
where \(R_{L}^{M}\) and \(T_{L}^{M}\) are the rotation matrix and the translation vector from coordinate frame \(L\) to coordinate frame \(M\), respectively. These two parameters show the position and attitude of the 2D LiDAR relative to the mobile platform, depending on the installation of the 2D LiDAR on the mobile platform. Usually, these two parameters are constant and do not change with time. Because for general applications, 2D LiDARs are fixedly mounted on mobile platforms. Next, point \(\mathbf{p}_{M}\) in coordinate frame \(M\) is converted to the corresponding point \(\mathbf{p}_{W}\) in coordinate frame \(W\)  
\[\mathbf{p}_{W}=R_{M}^{W}\mathbf{p}_{M}+T_{M}^{W} \tag{7}\]  
where \(R_{M}^{W}\) and \(T_{M}^{W}\) are the rotation matrix and the translation vector from coordinate frame \(M\) to coordinate frame \(W\), respectively. These two parameters show the position and attitude of the mobile platform relative to the world coordinate frame, depending on the movement of the mobile platform. Until now, for a push-broom 2D LiDAR, the conversion of the sampling point \(\mathbf{p}\) from the 2D LiDAR coordinate frame \(L\) to the world coordinate frame \(W\) has been completed.