## 4 Results and Discussions
### Measurement Error Analysis  
Any measuring system will have its own error. In a measuring process, some measurement errors come from multiple aspects, including operations, external environment, the measuring device itself, and other factors [33; 34], thus finally affecting the measurement accuracy. Therefore, it is significant to examine the measurement error helping in determining the confidence level. In this work, the designed surface spacing and flatness measurement scheme would have certain errors from the data acquisition device, mechanical processing and assembly, data processing scheme, and error evaluation algorithm. The main errors are listed in Table 5. The eddy current displacement sensor used has a range of 20 mm, a resolution of 1.0 \(\upmu\)m, and a linear error of \(\pm\)1% FS (see Table 2). The standard metal plates of the calibration platform are finely manufactured using abrasive machining with a flatness of \(\pm\)0.005 mm. During installation, the two plates of the calibration platform are connected by long bolts, fixed by gaskets and nuts, and the spacing can be manually adjusted. The assembly error is also evaluated. Based on the GUM method for flatness uncertainty analysis, the uncertainty of a single point measurement is \(u_{0}\), and its uncertainty is mainly composed of the following aspects:  
(1) \(u_{ev}\) from the environment. During the experiment, the laboratory is kept dry and at a constant temperature, so the uncertainty caused by the humidity and temperature of the environment is negligible. The value \(u_{ev}=0\) is considered in further calculations. (2) \(u_{res}\) due to the sensor resolution. Based on the performance of the eddy current displacement sensors employed in this work, a resolution of 1.0 \(\upmu\)m could be obtained, resulting in an uncertainty of \(u_{rev}=0.29\)\(\upmu\)m through the calculation. (3) \(u_{rep}\) resulted from the repeatability measurement. Under the same conditions, the Class A uncertainty evaluation method [35] could be adopted for the uncertainty of repeated measurements at the same measuring point. The uncertainty caused by repetitive measurement is prominently caused by the data in the z-axis. We could randomly choose several measuring points for repeated measurement, and then the repeatability error could be calculated according to the Bessel formula [36]. For the sake of conservation, the maximum error value is selected as the uncertainty caused by repetitive measurement and set to \(u_{0}\). The uncertainty derived from the Bessel formulais given by  
\[u_{i}=\sqrt{\frac{\sum_{j=1}^{n}{(Z_{j}-Z)^{2}}}{n(n-1)}} \tag{19}\]  
where \(u_{i}\) is the uncertainty of the five repeated measurements at the point \(i\), \(n\) is the number of repeated measurements, and \(z\) is the averaged value of the five measurement results. In this work, measuring points 3, 5, and 9 are selected for five repeated measurements. The measured data are shown in Table 6. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Error Source** & **Error Value** \\ \hline
1 & Eddy current displacement sensor & \(\pm\)1\%FS \\
2 & Calibration board & \(\pm\)0.005 mm \\
3 & Parts processing and assembly error & \(\pm\)0.02 mm \\ \hline \end{tabular}
\end{table} 
Table 5: Statistical table of error sources. Taking the data in the table into Equation (19), we can obtain \(u_{3}=0.5099\)\(\upmu\)m, \(u_{5}=0.5831\)\(\upmu\)m, and \(u_{9}=0.3742\)\(\upmu\)m. Then, summing up, the \(u_{rep}=0.5831\)\(\upmu\)m is obtained. As a consequence, the uncertainty of a single point measurement could be received as  
\[u_{0}=\sqrt{({u_{{{\it{rev}}}}^{2}+{u_{{{\it{res}}}}}^{2}+{u_{{{\it{rep}}}}}^{2} })}=0.6512\)\(\upmu\)m.  
Finally, bring the coordinate values of the measuring points in Table 4 into the equations from Equations (8)-(18), and the measured data can be obtained from the planeness measuring device.