# An Improved Self-Born Weighted Least Square Method for Cylindicity Error Evaluation
## 5 Experimental Verification and Literature Verification
### Experimental Verification
#### 5.1.1 Evaluation of Experimental Data  
The least square objective function was constructed from the sampled point data. Then the least square (LS), self-born weighted least square (SWLS), improved self-born weighted least square (ISWLS), standard particle swarm algorithm (standard PSO), and standard genetic algorithm (standard GA) were used to calculate the cylindricity error, respectively. The selection of the irrelevance group of the SWLS method followed the actual situation, and the 1 st, 57 th, 112 th, 167 th, and 222 nd error equations were selected as the 5 error equations that make up the maximum linear independence group. The number of updates of the independent system of equations was 30 when the experiment uses the ISWLS method. The population size of the genetic algorithm was 200, the number of evolutions was 2000, and the crossover probability and mutation probability were 0.7 and 0.2. The particle swarm optimization population size was 500, the number of evolutions was 1000, and the learning factors were c1 = c2 = 2. The results of the first group of measurement data processing are shown in Table 1. At the same time, the fitting result of the ISWLS method is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Evaluation Method** & \multicolumn{4}{c}{**Axis Parameters**} \\  & **a [mm]** & **b [mm]** & **p [mm]** & **q [mm]** \\ \hline LS & 0.0004 & 0.0000 & \(-\)0.0009 & \(-\)0.0007 \\ SWLS & \(-\)0.0087 & \(-\)0.0010 & 0.0004 & 0.0000 \\ ISWLS & 0.0267 & 0.0275 & \(-\)0.0010 & \(-\)0.0010 \\ PSO & 0.0003 & 0.0010 & 0.0142 & 0.0943 \\ GA & \(-\)0.0013 & \(-\)0.0008 & 0.0256 & 0.0141 \\ \hline
**Evaluation Method** & \multicolumn{4}{c}{**Cylindricity Error [mm]**} & **Running Time [s]** \\ \hline LS & 0.1955 & 0.074023 \\ SWLS & 0.1346 & 0.177704 \\ ISWLS & 0.1336 & 0.082950 \\ PSO & 0.1415 & 7.467153 \\ GA & 0.1514 & 15.34568 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The first set of data evaluates the results. Figure 4: Cylindricity error measurement. In Table 1, the cylindricity axis parameters, cylindricity error value and calculation time were calculated, and it can be seen that the axis positions calculated by the five methods are slightly different. The cylindricity error value of the LS method was 0.1955 mm, which was significantly larger than the error of other algorithms. The standard PSO method and the standard GA method were global optimization algorithms with cylindricity error results of 0.1415 mm and 0.1514 mm, respectively, and the computing time was more than 7 s. The results were more accurate than those of the LS method, but the computation time was longer. The cylindricity error results of the ISWLS method and SWLS method were 0.1346 mm and 0.1336 mm, respectively. Their operation time were 0.0977 s and 0.08295 s, respectively. They both had higher accuracy compared to the LS method and shorter operation time than the standard PSO and standard GA methods. Since the operation time of the ISWLS method did not include the selection time of the independent set of equations, and the influence of gross errors on the selection of the independent set of equations was guaranteed to be excluded in each operation, the algorithm is generally better than the SWLS method. Under the same conditions, the same sampling method was used to measure the same measurement object 10 times, and the error data were obtained by using the four methods of processing in Section 5.1.1, as shown in Figure 6.  
As can be seen from Figure 6, the cylindricity error values calculated by the ISWLS method were consistent with those of the SWLS method, LS method, standard PSO method, and standard GA method in the evaluation of 10 measurement results, which indicated  
Figure 5: ISWLS method cylindrical fitting. Figure 6: Cylindricity error evaluation comparison. the effectiveness of the ISWLS method. The value of cylindricity error found by the LS method is larger than the value of other algorithms.