# An Improved Self-Born Weighted Least Square Method for Cylindicity Error Evaluation
## 5 Experimental Verification and Literature Verification
### Experimental Verification
#### 5.1.2 Error Evaluation after Inserting Gross Error Points  
Considering the influence of gross differences on the error assessment, a numerical experiment of inserting gross differences was designed. According to the existing measurement data, the column surface points of one measurement were selected, and three gross error points were inserted randomly in the 1 st, 2 nd, and 3 rd layer measurement sections to observe the change of the axis position. The insertion of the gross difference data is shown in Table 2. The ISWLS method was used to calculate the axial parameters and cylindricity error values, and the results are shown in Table 3.  
In order to reflect the influence of the gross difference value on the error results before and after the iteration of the ISWLS method, the true error value \(v^{(i)}\) of Equation (16) in the ISWLS method was selected as the variable, and the true error values \(v^{(i)},(i=1,\ldots,240)\) of 240 points on the column before the iteration were compared with the calculated results after the iteration, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **X** [**mm**] & **Y** [**mm**] & **Z** [**mm**] \\ \hline
1 & 46.28403 & 10.00008 & 22 \\
2 & 34.63978 & 12.22836 & 19 \\
3 & \(-\)55.36314 & 9.22029 & 16 \\ \hline \end{tabular}
\end{table} 
Table 2: Insertion point data. \begin{table}
\begin{tabular}{c c c c} \hline
**Insert Gross Point** & **Axi Parameters** & \\
**a** [**mm**] & **b** [**mm**] & **p** [**mm**] & **q** [**mm**] \\ \hline
0 & \(-\)0.0087 & \(-\)0.0010 & 0.0004 & 0.0000 \\
1 & \(-\)0.0467 & \(-\)0.0092 & 0.0033 & 0.0007 \\
2 & 0.0786 & 0.0356 & \(-\)0.0090 & \(-\)0.0038 \\
3 & \(-\)0.0027 & 0.0488 & \(-\)0.0090 & \(-\)0.0038 \\ \hline
**Insert Gross Point** & **Cylinder Radius r** [**mm**] & **Cylindricity Error** [**s**] \\ \hline
0 & 46.2909 & & 0.1336 \\
1 & 46.2953 & & 0.1475 \\
2 & 46.2966 & & 0.1875 \\
3 & 46.2998 & & 0.2038 \\ \hline \end{tabular}
\end{table} 
Table 3: Evaluation results of ISWLS method when inserting gross error points. According to Figure 7a, the true error value \(v^{(i)}\) did not change much before and after the iteration when the gross error point was not inserted, and the result was more stable. As shown in Figure 7b, the true error value \(v^{(i)}\) at the beginning of the iteration was 0.95 mm larger than that at the rest of the points after inserting a gross point, and the deviation of the true error value \(v^{(i)}\) at the gross point was reduced to normal after the end of the iteration. As shown in Figure 7c, after the insertion of 2 gross points, the maximum true error value \(v^{(i)}\) at the 2 gross points inserted at the beginning of the iteration was 9.5 mm compared to the remaining points, and the true error values \(v^{(i)}\) at the gross points can converge to the normal value after the end of the iteration. According to Figure 7d, after the insertion of three gross points, the maximum true error value \(v^{(i)}\) at the three gross points inserted at the beginning of the iteration was 9.8 mm larger than the deviation of the remaining points, and the maximum deviation of the true error value \(v^{(i)}\) at the gross points returned to 0.2 mm after the end of the iteration. It can be seen that the true error values of cylindricity at each point are larger under the influence of inserting gross error points. The true error parameter values of the gross difference part tended to be smooth after iteration. It reflected that the ISWLS method was very effective in reducing the influence of gross errors. According to Table 3, it can be seen that the radius of the cylindrical fit of the ISWLS method was 46.2953 mm for the insertion of one gross point. The radius of the cylindrical fit was 46.2966 mm for the insertion of two gross points. The radius of the cylindrical fit was 46.2998 mm when three gross points were inserted. Compared to when gross error  
Figure 7: Statistics chart of true error value when inserting gross error points. (**a**) No gross error point insertion.