## 5 Experimental Analysis
### Result Analysis and Optimizing Uncertainty  
After the measurement task is determined, it is necessary to select the measuring instruments and methods according to the accuracy requirements. During measurement, see Table 7 [35] for the requirements of the geometric parameters for measurement accuracy, which is particularly important for geometrical tolerances detection; the tolerance level is Level 5-6 for the measurement object as shown in Figure 3, so the measurement uncertainty should be less than 20% of the tolerance value as required and then the uncertainty should be less than 2.0 \(\upmu\)m and 3.0 \(\upmu\)m respectively in flatness and perpendicularity detections. Obviously, the measurement uncertainty given in Formula (31)-Formula (40) has the evaluation results not in conformity with the accuracy requirements of the measurement tasks. There are two major reasons for such inconformity as follows: Firstly, the production-oriented CMM is not the preferred measuring instrument for geometrical tolerances. Its ability for measuring geometrical tolerance is slightly inferior to that of the roundness meter, autocollimator and other special instruments; secondly, the CMM task-oriented uncertainty evaluation model described in Section 2.2 features \"overestimation\" to ensure the safety and reliability of the evaluation results. When CMM is selected as the measurement instrument to complete the measurement tasks, it is bound to optimize the conventional measurement scheme and its uncertainty evaluation model. Therefore, here follows the research on the method for optimizing and evaluating the uncertainty of CMM task-oriented measurement in this paper. Take the flatness measurement in Section 3.4.2 as an example to illustrate how to optimize and evaluate the CMM task-oriented measurement uncertainty and to determine the target measurement uncertainty \(U_{\text{o}}=2.0\ \upmu\text{m}\) of the measurement task according to the requirements in Table 7. Therefore, only if the uncertainty of flatness measurement \(U\) after optimal evaluation should be smaller than \(U_{\text{o}}\) can the accuracy requirements of the measurement be satisfied. Use \(P=1.2\ \upmu\text{m}\) calibrated in Table 1 to substitute as the upper limit of the indication error and then the uncertainty component caused by the indication error in flatness measurement is as follows:  
\[u_{\text{E}}^{\prime}=\frac{P}{\sqrt{3}}=\frac{1.2}{\sqrt{3}}=0.69\ \upmu\text{m} \tag{41}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Standard Uncertainty** & **Source of Uncertainty** & **Evaluation Result** \\ \hline \(u_{\text{E}}\) & Indication error & 2.449 \(\upmu\text{m}\) \\ \(u_{\text{r}}\) & Repeatability & 0.394 \(\upmu\text{m}\) \\ \(u_{\text{R}}\) & Reproducibility & 1.060 \(\upmu\text{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The uncertainty budget for perpendicularity measurement. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline Tolerance Grade & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
**Proportion of** & & & & & & & & & & & & \\
**Uncertainty to** & & & & & & & & & & & & \\
**Tolerance** & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Accuracy requirements of geometric sense measurement. In the reproducibility experiment of the flatness measurement in Section 3.4.2, try to have the number and distribution of the sampling points on the plane to be measured and other factors kept somewhat different. See Table 8 for the data of reproducibility measurement. \[u_{\text{R}}=\sqrt{\frac{1}{(9-1)}\sum\limits_{j=1}^{9}{(\overline{t}_{j}- \overline{\overline{t}})}^{2}}=0.915\;\mu\text{m} \tag{42}\]  
In time for secondary optimal evaluation, distribute the sampling points evenly, use 12 sampling points in all and adopt 4 nm probes and 20 mm measuring rods uniformly; see Table 9 for the reproducibility experimental data after the experimental scheme has been optimized.