## 2 Establishment of Measurement Control Network Model
### Error Characteristics Analysis of Measuring Instruments  
The horizontal and vertical angle encoders of laser tracker and total station provide azimuth angle \(\alpha\) and zenith angle \(\beta\) relative to their own center spheroid coordinate system. \(D\) is the return oblique distance. The schematic diagram of the coordinate system for such devices is shown in Figure 2. By projecting Point (\(X\), \(Y\), \(Z\)) onto the xoy and yoz planes,  
Figure 1: Grading of measurement control network. respectively, the conversion relationship from spherical coordinates to cartesian coordinates can be obtained as follows:  
\[ \begin{cases}X&=D\cdot\text{cosa}\cdot\text{sin}\beta\\ Y&=D\cdot\text{sina}\cdot\text{sin}\beta\\ Z&=D\cdot\text{cos}\beta\end{cases}  \tag{1}\]  
Taking the Leica AT402 absolute laser tracker as an example, the device offers a measurement range from 1.5 m to 80 m, a vertical range of \(\pm\)45\({}^{\circ}\). It employs ADM technology to perform absolute distance measurement and has a constant error of \(\pm\)10 \(\upmu\)m in distance measurement, as well as an angular measurement accuracy of \(\pm\)(15 \(\upmu\)m + 6 \(\upmu\)m/m). As depicted in Figure 3, angular measurement error decreases as the measurement distance increases, and finally stabilizes to about 1.28\({}^{\prime\prime}\), which is inversely proportional to the measured distance. A point at (\(-\)30,000, 500, \(-\)1500) mm is chosen to be repeatedly observed 1000 times using the laser tracker, and the repeatability cloud is produced. Figure 4a-c displays the projections of the three-dimensional repeatability cloud on the xoz, xoy, and yoz planes, respectively. The coordinate measurement errors of the laser tracker are greater in the y and z directions, indicating that errors in the horizontal direction are smaller than those in the vertical direction. The repeatability cloud projected on the xoz and xoy plane is upward at an angle, forming an elliptical distribution; on the yoz planes, the projections appear as irregular circular distributions. Simulation analysis confirms that, within an effective measuring range, the laser tracker can achieve a single-point coordinate measurement accuracy better than 266 \(\upmu\)m. However, due to the limited range of the laser tracker, using it alone to measure target points across an entire range necessitates multiple station transfers, leading to error accumulation. Figure 3: Angular measurement error of the laser tracker. Figure 2: Schematic of the coordinate system of the measuring instrument. Moreover, an error analysis is performed using the Leica MS60 total station as an example, which offers angular measurement precision of 0.5\({}^{\prime\prime}\), distance measurement range extends from 1.5 m to more than 10,000 m with a prism and from 1.5 m to 2000 m without a prism with a distance measurement accuracy of 1.2 mm + 2 mm/km. The total station performs repeated observations on a point at (\(-\)15,000, 2500, \(-\)2500) mm in space, resulting in the repeatability cloud as shown in Figure 5. The analysis shows that the total station's measurement errors mainly concentrate longitudinally along the x and y directions. As shown in Figure 6a-c, measurement errors predominantly occur in the x direction, with significantly smaller errors in the y and z directions. From the above simulation analysis, it is evident that the total station's extensive measurement range and high angular precision significantly compensate for the limitations of the laser tracker. Using a total station to repeatedly measure the simulated point, the maximum error in single-point coordinate measurement within a range of 15 m can reach  
Figure 4: Repeatability cloud for single\(-\)point coordinate measurement by laser tracker: (**a**) projection on the xoy plane; (**b**) projection on the xoz plane; and (**c**) projection on the yoz plane. Figure 5: Repeatability cloud for single\(-\)point coordinate measurement by total station.