## 2 Problems
By improving the  
Figure 1: The shape error and attitude error of the 3D point cloud: (**a**) a photo of the environment for collecting the 3D point cloud; (**b**) top view of a 3D point cloud collected by a rotating 2D lidar; (**c**) top view of another 3D point cloud collected by a rotating 2D lidar. manufacture and assembly accuracy of the mechanical parts of our prototype, we minimized the impact of mechanical errors on the accuracy of the 3D point cloud. We used computer numerical control (CNC) machine tools to manufacture the key parts of our prototype, with a tolerance level of IT5 (Chinese standard, GB/T1184-1996). In Appendix B, we verified that the self-reasons of our prototype (which include the mechanical error) has a much smaller impact on the accuracy of the 3D point clouds than the error of the 2D lidar itself (\(\pm\)40 mm). Therefore, we can exclude this item and assume that other reasons are causing the resulting observed error. 2. The acceleration and deceleration of the motor shaft will cause error. In a 3D scan of a rotating 2D lidar, we assume roughly that the shaft of the stepper motor rotates at angular velocity \(\omega\) for time \(T\), and it rotates 180\({}^{\circ}\) in total. However, in practice, the motion of the motor shaft is more complex. The motor shaft accelerates from standstill to angular velocity \(\omega\), and then it keeps rotating at this speed, after which it decelerates to standstill. We have ignored the acceleration and deceleration of the motor shaft, and this can lead to the shape error of the 3D point cloud, which is especially obvious at the closed position of the point cloud. The so-called closed position here refers to the area where the points collected by the 2D lidar when the motor shaft starts to rotate or stops rotating. The areas of these two types of points are adjacent, and these points are most directly affected by the acceleration and deceleration of the motor shaft. 3. The use of photoelectric switches may cause error. Since no encoder or servo motor is used, it is necessary to define the initial position of motor shaft by photoelectric switch. When finding the initial position, the motor shaft will rotate back and forth by a large margin and make the shading sheet which rotates with the motor shaft triggering the photoelectric switch. After the photoelectric switch is triggered, the controller sends a stop command to the motor to stop its shaft. The position of the motor shaft after it stops is considered to be the initial position. This process can be described in more detail, the shading sheet blocks the light beam of the photoelectric switch--the photoelectric switch is triggered--the controller sends the stop command to the stepper motor--the shaft of stepper motor decelerates until it stops. The accuracy of the initial position of the motor shaft is affected by many factors such as the response time of the photoelectric switch, the rotation direction and the speed of the motor shaft. For each 3D scan of a rotating 2D lidar, the motor shaft starts rotating from its initial position. Therefore, if the initial position of the motor shaft is not accurately defined, the attitude error of the 3D point cloud will be caused. 4. The uncertain time deviation may cause error. In the actual engineering operation, we found that although we require the 2D lidar and the stepper motor to start to work at the same time, there is an uncertain time deviation between the starting time of the 2D lidar and the starting time of the stepper motor. The value of this time deviation is very small, but it is enough to have a significant impact on the accuracy of the 3D point cloud. The reason for the uncertain time deviation is that the response time of the 2D lidar and stepper motor to the command is inconsistent and not constant, as mentioned in [16].