## 4 Experimental Verification
### Results Analysis  
Firstly, the four-axis linkage machining trajectory of the impeller was simulated using the UG12 software's simulation module for machine tool processing. The purpose of this simulation was to check for any tool interference issues. After the simulation, there were no issues of cutting tool interference or other problems in the four-axis machining trajectory, and practical processing tests could be carried out. Secondly, to analyze and confirm the accuracy of the impeller blade machining, the adjacent five-axis five-linkage points were randomly selected for testing purposes. Table 5 displays the tool position data for the adjacent tool center points prior to interpolation, while Table 6 shows the interpolated tool position data. Analyzing the data in Tables 5 and 6, it can be calculated that the maximum non-linear errors before and after transformation of the five-axis five-linkage trajectory were \(\varepsilon_{max1}=1.4\) um and \(\varepsilon_{max2}=0.36\) um, respectively. It can be deduced that the maximum \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**X** & **Y** & **Z** & **I** & **J** & **K** \\ \hline \(-18.0272\) & \(-9.2689\) & \(18.6089\) & \(0.3593827\) & \(-0.2294031\) & \(0.9045542\) \\ \(-18.066833\) & \(-9.300400\) & \(18.616133\) & \(0.365500\) & \(-0.229973\) & \(0.901955\) \\ \(-18.106467\) & \(-9.331900\) & \(18.623367\) & \(0.359382\) & \(-0.230561\) & \(0.904260\) \\ \(-18.1461\) & \(-9.3634\) & \(18.6306\) & \(0.3655032\) & \(-0.2311342\) & \(0.9016565\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Tool position data after interpolation. Figure 10: Impeller machining trajectory. (**a**) Multi-blade rough; (**b**) splitter blade finishing; (**c**) blade finishing; (**d**) wheel hub finishing. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**X** & **Y** & **Z** & **I** & **J** & **K** \\ \hline \(-18.0272\) & \(-9.2689\) & \(18.6089\) & \(0.3593827\) & \(-0.2294031\) & \(0.9045542\) \\ \(-18.1461\) & \(-9.3634\) & \(18.6306\) & \(0.3655032\) & \(-0.2311342\) & \(0.9016565\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Tool position data before interpolation. nonlinear error of the adjacent points after interpolation is less than the maximum nonlinear error prior to interpolation, and the result is \(\varepsilon_{max1}>\varepsilon_{max2}\). Therefore, this algorithm demonstrated its ability to effectively enhance the machining quality of parts. However, it is essential to further improve the method. Since the addition of tool position occurs after the interpolation of the adjacent five-axis tool position, the processing time is increased, and processing efficiency is sacrificed as a result. The results of 3D IPW were compared between the five-axis path (finishing path) calculated by the impeller programming module and the path generated by the transformation algorithm. Figure 11 shows that the machining accuracy of the blade (1) and hub finishing is relatively similar, with slightly better accuracy observed on the blade. When the path transformed by the algorithm is used to process the root of the blade (3) and the splitter blade (2), there are many places where the machining accuracy deteriorates. This may be due to significant changes in the tool vector, resulting in missing or over-cutting. From the results of Figure 11b,d, it is clear that, prior to using the algorithm, the maximum machining error position in the impeller machining was located at the root of the splitter blade, with an error value of 0.201 mm. After the algorithm was applied, the maximum machining error position remained located at the root of the splitter blade, with a modified error value of 0.479 mm. Table 7 summarizes the error report obtained by the deviation gauge. Using the five-axis five-linkage trajectory, a total of 614,723 sampling points were analyzed. Among them, 47,215 points were found to be out of tolerance, accounting for 7.68% of the total sampling points.