## 5 Experimental Analysis
### Result Analysis and Optimizing Uncertainty
\[u_{\text{R}}^{\prime}=\sqrt{\frac{1}{(9-1)}\sum\limits_{j=1}^{9}{(\overline{t}_ {j}-\overline{\overline{t}})}^{2}}=0.518\;\mu\text{m} \tag{43}\]  
It can be seen that the standard uncertainty introduced by reproducibility has reduced from 0.915 \(\mu\)m to \(u_{\text{R}}^{\prime}=0.518\;\mu\)m by optimizing the experimental scheme. Considering that the repeatability indexes have been updating by the latest measurement information continuously arising during the process of the measurement process, regard the repeatability uncertainty component \(u_{\text{r}}=0.357\;\mu\)m of the flatness measurement in Section 3.4.2 as prior information \(u_{0}\), and the number of measurements contained in the prior information \(n_{0}=10\). Table 10 shows the daily measurement data of the flatness for three workpieces in the same batch. Take workpiece A as the sample information and the repeatability uncertainty component can be calculated as follows after information fusion:  
\[u_{\text{r1}}=\sqrt{\frac{(n_{0}-1)u_{\text{r0}}{}^{2}+\sum\limits_{j=1}^{n_{ 1}}{(t_{\text{lj}}-\overline{t}_{1})}^{2}}{n_{0}+n_{1}-3}}=0.339\;\mu\text{m} \tag{44}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline The three times measurement mean of & Group I & Group II & Group III \\ surveyor A \(\overline{t}_{j}\) (mm) & 0.0053 & 0.0062 & 0.0045 \\ \hline The three times measurement mean of & Group IV & Group V & Group VI \\ surveyor B \(\overline{t}_{j}\) (mm) & 0.0041 & 0.0058 & 0.0065 \\ \hline The three times measurement mean of & Group VII & Group VIII & Group IX \\ surveyor C \(\overline{t}_{j}\) (mm) & 0.0047 & 0.0066 & 0.0049 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The first reproducibility measurement data of flatness. \begin{table}
\begin{tabular}{c c c c} \hline \hline The three times measurement mean of & Group I & Group II & Group III \\ surveyor A \(\overline{t}_{j}\) (mm) & 0.0051 & 0.0058 & 0.0048 \\ \hline The three times measurement mean of & Group IV & Group V & Group VI \\ surveyor B \(\overline{t}_{j}\) (mm) & 0.0047 & 0.0058 & 0.0055 \\ \hline The three times measurement mean of & Group VII & Group VIII & Group IX \\ surveyor C \(\overline{t}_{j}\) (mm) & 0.0049 & 0.0061 & 0.0049 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The optimized reproducibility measurement data of flatness. That is to say, after incorporating the daily measurement information, the repeatability index has been updated from 0.357 \(\upmu\)m to 0.339 \(\upmu\)m; replace the prior information with the result of information fusion to prepare for the next update; then:  
\[{{u_{\text{r0}}}^{*}}={{u_{\text{r1}}}}=0.339\ \upmu\text{m} \tag{45}\]  
\[{{n_{\text{0}}}^{*}}={{n_{\text{0}}}}+{{n_{\text{1}}}}-2=11 \tag{46}\]  
Based on the updated prior information, further integrate the measurement results of part B, then the latest production and measurement information can be continuously integrated to achieve real-time and continuous updates of the repeatability index. By fusing the daily measurement data of flatness, the uncertainty components introduced by the repeatability of the flatness measurement of workpieces B and part C can be as follows:  
\[{{u_{\text{r2}}}}=\sqrt{\frac{{{{({{n_{\text{0}}}^{*}}-1){({{u_{\text{r0}}}^{ *}})}^{2}}}+\sum\limits_{j=1}^{{{n_{\text{1}}}}}{{{({{t_{2j}}}-{{{{{t_{2}}}}} )}^{2}}}}}}{{{{n_{\text{0}}}}^{*}}+{{n_{\text{2}}}}-3}}=0.323\ \upmu\text{m} \tag{47}\]  
\[{{u_{\text{r3}}}}=\sqrt{\frac{{{{({{n_{\text{0}}}^{**}}-1){({{u_{\text{r0}}}^{ **}})}^{2}}}+\sum\limits_{j=1}^{{{n_{\text{1}}}}}{{{({{t_{3j}}}-{{{{t_{3}}}} )}^{2}}}}}}}{{{{n_{\text{0}}}^{**}}+{{n_{\text{3}}}}-3}}=0.309\ \upmu\text{m} \tag{48}\]  
If the repeatability index has not been updated, the uncertainty component caused by repeatability will always keep the evaluation results of the previous repeatability experiments. If only daily measurement data is adopted to evaluate the repeatability, then the sample data will be small in number and less representative.