## 4 Measurement Methods and Data Analysis
### Data Analysis
#### 4.2.1 Analysis of the Data Collected at Different Distances with Equal Spacing  
In this study, the distance of the laser tracker from the circular trajectory generator was set to 1 m, 2 m, 3 m, 5 m, and 10 m; the equal spacing of the laser tracker was set to 5 mm, 10 mm, and 20 mm; and the equal sampling frequency of the laser tracker was set to 10 Hz, 20 Hz, and 50 Hz. The performance parameters of the laser tracker under dynamic state were analyzed in different scenarios. The schematic diagram of measurement is shown in Figure 3. Four transfer stations are arranged around the circular trajectory generator, and the measurement accuracy of the transfer stations is 0.0155 mm, 0.0177 mm, 0.0199 mm, and 0.0264 mm, respectively. Figure 2: Flow chart of the measurement. Table 1 shows the dynamic performance of the laser tracker when measuring at equal spacing. The projection position of sampling points for different measurement intervals is shown in Figure 4. It means that the measurement points were projected in the \(Y\)-axis direction for the measurement distance of 1 m. And it is obvious that there is a large amount of data for the measurement intervals of 5 mm and 10 mm. Through data analysis, we drew the following four conclusions:  
Figure 4: Projection position of sampling points with different measurement intervals—(D = 1 m). Figure 3: Schematic diagram of the measurement. Within 10 m, the dynamic limit speed increases with the increase of the measurement distance. Figure 5 shows the relationship between dynamic limit speed and measured distance. As the measurement distance increases, the swing amplitude of the measuring head of the laser tracker decreases. When the measurement distance is 10 m, the maximum rotational velocity of the circular trajectory generator is 23.42 rad/s. At this time, the dynamic limit rotational velocity V\({}_{\rm L}\) of the laser tracker reaches 10.57 m/s. With the increase of measurement distance, the maximum rotational velocity of the circular trajectory generator rises, but it may cause safety issues. Under the dynamic state, after setting the equal spacing for the laser tracker, the actual spacing is greater than the set one. Figure 6 shows the relationship between the dynamic limit speed and the actual measured interval. When the circular trajectory generator is at low speed, the difference between the two is insignificant, but the difference becomes great with the speed increase. The larger the actual measurement spacing, the fewer the measurement points. It can be seen in Table 1 that the equal spacing of the laser tracker is 5 mm. The actual spacing is 7.17 mm at the dynamic limit speed of 3.40 m/s. When the speed is set to 4.60 m/s, the actual spacing is 6.47 mm. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Dynamic limit**} & \multicolumn{2}{c}{**Equal Spacing Measurement (5 mm)**} & \multicolumn{2}{c}{**Equal Spacing Measurement (10 mm)**} & \multicolumn{2}{c}{**Equal Spacing Measurement (20 mm)**} \\ \cline{2-13}
**Measurementment** & **Rotational** & **Dynamic** & **Indica-** & **Actual** & **Number of** & **Dynamic** & **Indica-** & **Actual** & **Number of** & **Dynamic** & **Indica-** & **Actual** & **Number of** \\
**Distance** & **Velocation** & **Action** & **Spacing** & **Measurement** & **Error** & **Variability** & **Size** & **Mean-** & **Mean-** & **Mediation** & **Spac-** & **Mediation** & **Spac-** & **Mediation** \\
**trace** & **(m/s)** & **Error** & **(mm)** & **Angle** & **(mm)** & **Error** & **Angle** & **(  
The dynamic indication error is smallest when the measurement distance is 3 m, indicating that the fitted diameter at a distance of 3 m is closest to the diameter of the circular trajectory generator. Figure 7 shows the relationship between measurement distance and dynamic indicating value error.