## 3 System Modeling
### Coordinate Conversion  
The principle of our prototype is to combine the data collected by the 2D lidar with the rotation angle of the motor shaft, and then calculate the 3D coordinates of the sampling \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mode** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(T/\)s & 1 & 2 & 4 & 5 & 8 & 10 & 16 \\ \(\gamma/^{\circ}\) & 4.5 & 2.25 & 1.125 & 0.9 & 0.5625 & 0.45 & 0.28125 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Seven scanning modes of our prototype. Figure 3: A photo of our prototype. The 2D lidar Hokuyo UST-10 LX is installed on a rotating unit, and a triangular plate is used to calibrate the attitude error of the 3D point cloud. points. After that, the attitude of the point cloud is corrected according to the known position of the triangular plate. This process involves the conversion among three Cartesian coordinate frames, namely the coordinate frame of the 2D lidar (\(L\)), the coordinate frame of rotating unit before attitude correction (\(O\)) and the coordinate frame of rotating unit after attitude correction (\(O\)), as shown in Figure 4.  
The definition of the 2D lidar coordinate frame \(L\)-\(X_{L}Y_{L}Z_{L}\) is as follows. The origin point \(L\) of this coordinate frame is the center of the 2D lidar scanning sector. The plane \(Y_{L}LZ_{L}\) is coplanar with the scanning sector. The axis \(Z_{L}\) is the middle line of the scanning sector, and the axis \(X_{L}\) is perpendicular to the scanning sector. The position of coordinate frame \(L\) relative to the 2D lidar is fixed, but its position relative to rotating unit is not fixed, because the 2D lidar is rotational relative to rotating unit. The definition of coordinate frame \(O\)'-\(X\)'\(Y\)'\(Z\) is as follows. This is the coordinate frame of the rotating unit before attitude correction. The origin point \(O\)' of the coordinate frame is located on the rotating axis of the motor shaft, and the line composed of point \(L\) and point \(O\)' is perpendicular to the rotating axis of the motor shaft. The axis \(Z\)' coincides with the rotation axis of the motor shaft and is parallel to the axis \(Z_{L}\) of coordinate frame \(L\). The axis \(Y\)' coincides with the starting and ending positions of a 3D scan, as shown in Figure 5. Since the starting and ending positions of a 3D scan is uncertain (as described in Section 2), the position of coordinate frame \(O\)' relative to the rotating unit is uncertain, too. The definition of coordinate frame \(O\)-\(XYZ\) is as follows. This is the coordinate frame of the rotating unit after attitude correction, its origin point \(O\) coincides with the origin point \(O\)' of coordinate frame \(O\)'-\(X\)'\(Y\)'\(Z\)'. Its axis \(Z\) coincides with the axis \(Z\)' of coordinate frame \(O\)'-\(X\)'\(Y\)'\(Z\)'. Although the position of coordinate frame \(O\)' relative to the rotating unit  
Figure 4: Three Cartesian coordinate frames. The coordinate frame of the 2D lidar (\(L\)) is shown in green, the coordinate frame of rotating unit before attitude correction (\(O\)’) is shown in blue, and the coordinate frame of rotating unit after attitude correction (\(O\)) is shown in red. is uncertain, the position of coordinate frame \(O\) relative to the rotating unit is certain, and the positive direction of axis \(X\) parallel to the front direction of the prototype (as shown in Figure 4). In our prototype, the scanning sector of the 2D lidar does not coincide with the rotation axis of the motor shaft, and they are parallel to each other, the distance between them is 13.9 mm (as shown in Figure 4). The reason for this design can be explained in Figure 5. Through this design, we can collect the 3D point cloud with strip blank area and strip overlap area (from the top view). This is an important mark. The parts of the 3D point cloud on both sides of this mark are collected when the rotating 2D lidar starts or stops a 3D scan.