## 4 Experiment
### High-Weight Error Measurement and Compensation  
The straightness error is a position-dependent geometric error, while the magnitude of the error varies with changes in the coordinate position. Based on the weight analysis of the previous section, the positioning error and the straightness motion error are the main errors affecting the measurement accuracy, necessitating accurate measurement and compensation. The linear guide adopts the Renishaw RGSZ20-S grating ruler as the position reference, with a linearity of \(\pm\)3 \(\upmu\)m/m. The positioning and straightness errors of the linear axis were obtained by measuring the Y- and Z-axes using a Renishaw XK-10 laser interferometer, as shown in Figure 10. The measurement results are shown in Figures 11 and 12; 1+ indicates traveling in the positive direction forwards, 1\(-\) indicates traveling in the opposite direction \begin{table}
\begin{tabular}{c c} \hline \hline
**Term** & **Specifications** \\ \hline Stroke of translational axes (Y- and Z-axes) & 300 mm \(\times\) 80 mm \\ Stroke of A- and C-axes & \(\pm\)60 deg, 360 deg \\ Maximum speed of C-axis & 500 r/min \\ Positioning accuracy of linear axis & \textgreater{}2 \(\upmu\)m \\ Repositioning accuracy of linear axis & \textgreater{}1 \(\upmu\)m \\ Positioning accuracy of turntable & \textgreater{}1 \(\upmu\)m \\ Slideway straightness & \textgreater{}1 \(\upmu\)m \\ Radial runout of turntable & \(\pm\)100 nm \\ Axial runout of turntable & \(\pm\)75 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The specifications and main error requirements of measuring machine. Figure 9: Static noise test. (**a**) Noise measurement of the sensor. (**b**) Noise measurement of the laser interferometer. forwards, and the number indicates the number of repetitions. The Y-axis positioning accuracy is 10.1 \(\upmu\)m, while the Z-axis straightness error is 2.3 \(\upmu\)m per 300 mm. The Z-axis positioning accuracy is 11 \(\upmu\)m, and the Y-axis straightness error is 1.5 \(\upmu\)m per 80 mm. The rotary table is a self-developed high-precision air-bearing model [32; 33]. The positioning accuracy and repeatability of the rotary table were tested using a 24-plane prism with collimator. The results are shown in Figure 13. The positioning accuracy of the A rotary table is \(\pm\)2.52 arc sec and the repeatability is about \(\pm\)0.44 arc sec, and the positioning accuracy of the C rotary table is \(\pm\)1.69 arc sec and the repeatability is about \(\pm\)0.51 arc sec. The axial and radial runouts of the rotary table were tested using a standard sphere with a roundness of 50 nm and an inductive micrometer. As shown in Figure 14, the rotation of the tested air-bearing rotary table is controlled to 360\({}^{\circ}\). The radial runout error of the air-bearing rotary table and the roundness error of the standard sphere were  
Figure 11: Positioning accuracy of Y-axis (**a**) and Z-axis (**b**). Figure 12: Straightness error on the Y-axis (**a**) and Z-axis (**b**). Figure 10: Positioning accuracy and straightness measurement of Y-axis (**a**) and Z-axis (**b**). separated using a reverse method [34]. The radial runout error of the air-bearing rotary table was obtained by subtracting the positive and negative data and averaging them. The axial runout error of the air-bearing rotary table was obtained using the error separation technique of mathematical statistics [35]. The radial runout of the A-axis air-bearing rotary table was greater than 100 nm, while the axial runout was greater than 75 nm. Similarly, the radial runout of the C-axis air-bearing rotary table was greater than 100 nm and the axial runout was greater than 50 nm, which meets the needs of the experiment. To verify the compensation ability of the measuring frame, a picometer-level precision laser interferometer (quDIS) was used for high-precision positioning of the measuring machine. A standard flat ruler was also used to compensate for the straightness error in real-time.