## 2 Problems
Because both the transmission of the command and the response of the motor or the 2D lidar take time, there is an uncertain time deviation between the time when the controller starts to send the command to the motor and the time when the motor starts to work. Similarly, there is an uncertain time deviation between the time when the controller starts to send the command to the 2D lidar and the time when the 2D lidar starts to work. The uncertainty of the transmission time of the command and the response time of the device is a common problem which is difficult to solve. Because of this problem, we can mark the time when the controller starts to send the command to the motor or the 2D lidar in the program, we cannot know the actual working time of the motor or 2D lidar. This is the cause of the uncertain time deviation between the starting time of the 2D lidar and the starting time of stepper motor. For each 3D scan of a rotating 2D lidar, this time deviation is uncertain, so the shape and attitude error of the 3D point cloud for each scan is also uncertain (see Figure 1). The reason for the shape error of the 3D point cloud can be analyzed as follows. Due to the uncertain time deviation, there may be two situations. One is that the motor starts to rotate after the 2D lidar has started scanning continuously for a short period of time. The 3D point cloud collected in this case is shown in Figure 1b. At the position where the motor starts to rotate (the beginning of the yellow circular arrow), there is a shape error marked by red line. The other situation is that the motor starts working earlier than the 2D lidar, so that the motor has stopped rotating before the 2D lidar stops scanning. The 3D point cloud collected in this case is shown in Figure 1c. At the position where the motor stops rotating (the end of the yellow circular arrow), there is a shape error marked by red line. In either case, a shape error of the 3D point cloud will result. In addition, the attitude error can also be caused. Therefore, the uncertain time deviation between the starting time of the 2D lidar and the starting time of stepper motor is one of the reasons for the uncertain error of the 3D point cloud in shape and attitude. In addition to item 1 of the above 4 items, it is our work to exclude the errors in the other three items. Unfortunately, these three errors mentioned above may be uncertain, for the acceleration and deceleration curve of the motor shaft may be uncertain, the response time of the photoelectric switch may be uncertain, especially, the time deviation between the moment when the 2D lidar starts to work and the moment when stepper motor starts to work may be uncertain, too. Due to the above uncertainties, the shape and attitude error of the 3D point clouds may also be uncertain. The effect of item 2, 3 and 4 on the 3D point cloud can be divided into two categories, one is shape error and the other is attitude error, as shown in Figure 1. For simplicity, we first only analyze the shape error of the 3D point cloud. According to Formulas (1)-(3), we can find that if the shape of the 3D point cloud is correct, the necessary and sufficient condition is that the stepper motor rotates at constant angular velocity \(\omega\) during the continuous scanning of the 2D lidar. The acceleration and deceleration of motor shaft in item 2 will cause the shape error of the 3D point cloud. We have tried to estimate the approximate speed curve of the motor shaft, and to eliminate the shape error of the 3D point cloud. It turns out that this is not feasible. First, the acceleration and deceleration curve of the motor shaft may not be constant. Second, the shape error of the 3D point cloud is not entirely attributed to the acceleration and deceleration of motor shaft. The uncertain time deviation in item 4 also has a significant impact on the shape of the 3D point cloud. Therefore, the shape error of 3D point cloud cannot be effectively eliminated by estimating the approximate speed curve of the motor shaft.