# An Improved Self-Born Weighted Least Square Method for Cylindicity Error Evaluation
## 5 Experimental Verification and Literature Verification
### Cylindricity Error Uncertainty Experiment  
According to the first group of cylindricity error evaluation data in Table 1, the cylindricity error uncertainty was calculated by the ISWLS method. The combined standard uncertainty of the output was calculated, and the inclusion interval of the 95% inclusion probability was obtained for the interval factor \(k=1.96\). The data are shown in Table 8.  
The MCM method, as a supplementary method to the GUM method, can perform an uncertainty assessment of cylindricity error. The results can be compared with the GUM method. The results are shown in Table 9 and Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Evaluation Method**} & \multicolumn{4}{c}{**Axis Parameters**} & \multirow{2}{*}{**Cylindricity Error \([\mathbf{mm}]\)**} \\ \cline{2-2} \cline{4-5}  & \(\mathbf{a}\)\([\mathbf{mm}]\) & \(\mathbf{b}\)\([\mathbf{mm}]\) & \(\mathbf{p}\)\([\mathbf{mm}]\) & \(\mathbf{q}\)\([\mathbf{mm}]\) \\ \hline MZC & \(0.0024\) & \(-0.0040\) & \(-0.0140\) & \(0.0070\) & \(0.00279\) \\ MCC & \(0.0025\) & \(-0.0042\) & \(-0.0140\) & \(0.0070\) & \(0.00282\) \\ MIC & \(0.0037\) & \(-0.0039\) & \(-0.0140\) & \(0.0070\) & \(0.00279\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: The data evaluation results in the literature [4]. Reprinted with permission from Ref. [4]. 2004, [PERSON]; [PERSON].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Evaluation Method**} & \multicolumn{4}{c}{**Axis Parameters**} \\ \cline{2-4}  & \(\mathbf{a}\)\([\mathbf{mm}]\) & \(\mathbf{b}\)\([\mathbf{mm}]\) & \(\mathbf{p}\)\([\mathbf{mm}]\) & \(\mathbf{q}\)\([\mathbf{mm}]\) \\ \hline LS & \(-0.0140\) & \(0.0070\) & \(0.0036\) & \(-0.0046\) \\ PSO & \(0.0024\) & \(-0.0250\) & \(-0.0139\) & \(0.0069\) \\ GA & \(0.0045\) & \(-0.0068\) & \(-0.0140\) & \(0.0070\) \\ ISWLS & \(0.0037\) & \(-0.0034\) & \(-0.0140\) & \(0.0070\) \\ \hline
**Evaluation Method** & **Cylindricity Error \([\mathbf{mm}]\)** & **Running Time \([\mathbf{s}]\)** \\ \hline LS & \(0.00288\) & \(0.034277\) \\ PSO & \(0.00286\) & \(6.508752\) \\ GA & \(0.00279\) & \(12.568197\) \\ ISWLS & \(0.002788\) & \(0.041420\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: ISWLS method data evaluation results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \(\mathbf{Value}\)\([\mathbf{\mu}\mathbf{m}]\) & \(\mathbf{Name}\) & \(\mathbf{Value}\)\([\mathbf{\mu}\mathbf{m}]\) \\ \hline \(u_{1}\) & \(0.0052\) & \(u_{a}\) & \(0.3274\) \\ \(u_{2}\) & \(1.556\) & \(u_{b}\) & \(0.3274\) \\ \(u_{3}\) & \(0\) & \(u_{p}\) & \(0.0012\) \\ \(u_{4}\) & \(0\) & \(u_{q}\) & \(0.0012\) \\ \(u_{0}\) & \(1.560\) & \(u_{\delta}\) & \(2.2216\) \\ \hline
**Parameter** & \multicolumn{4}{c}{**Value \([\mathbf{mm}]\)**} \\ \hline best estimate & \multicolumn{4}{c}{\(0.1338\)} \\ \(95\%\) probability & \multicolumn{4}{c}{\([0.1294,0.1381]\)} \\ inclusion interval & \multicolumn{4}{c}{\([0.1294,0.1381]\)} \\ \hline \hline \end{tabular}
\end{table}
Table 8: GUM method evaluation results of the first set of data.