## 4 Coarse-to-Fine Free-Form Surface Machining Error Decomposition and Compensation Experiments
### Coarse-to-Fine Decomposition of the Original Machining Error Processes  
First, the VMD algorithm was utilized for coarse decomposition of the original machining error, and the number of decomposition layers \(K\) was set to 7 according to the instantaneous frequency mean method. The decomposition results are shown in Figure 12.  
The correlation coefficients between each IMF component in Figure 12 and the original machining error were calculated separately, and the corresponding correlation coefficients for each IMF component are shown in Table 4. The correlation coefficient thresholding was set to 0.3, and the corresponding correlation coefficients of each component were compared with the set thresholding to filter out the eligible IMF components, which were the IMF components containing systematic errors. As shown in Table 4, the components IMF1, IMF2, IMF4 and IMF7 met the screening conditions, which proves that they contained systematic errors, which were reorganized to obtain the coarse systematic error \(X_{1}\).  
Second, the coarse systematic error \(X_{1}\) was used as the error to be decomposed, and it was adaptively decomposed using the EMD algorithm to obtain nine IMF components. Its decomposition results are shown in Figure 13.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline IMF Component & IMF1 & IMF2 & IMF3 & IMF4 & IMF5 & IMF6 & IMF7 \\ \hline correlation coefficient & 0.4113 & 0.3893 & 0.2431 & 0.3775 & 0.2406 & 0.1158 & 0.3662 \\ \hline \end{tabular}
\end{table}
Table 4: Correlation coefficients between each IMF component in VMD coarse decomposition and the raw machining error.  
Figure 12: Coarse decomposition of the original machining error via VMD.  
The correlation coefficients between each IMF component in Figure 13 and the original machining error were calculated separately, and the corresponding correlation coefficients for each IMF component are shown in Table 5.  
By comparing the correlation coefficients corresponding to each of the IMF components with the thresholding set, it can be seen that the IMF components IMF1, IMF2, IMF3, IMF4, IMF5 and IMF9 were suitable for the screening conditions, and they were reorganized to obtain the systematic error \(X_{2}\).  
From the above decomposition process, it can be seen that components such as IMF3, IMF5 and IMF6 in the coarse decomposition process of the VMD algorithm were weakly correlated IMF components, and components such as IMF6, IMF7 and IMF8 in the decomposition process of the EMD algorithm were weakly correlated IMF components. All weakly correlated IMF components from the two decomposition processes were reorganized and decomposed using the wavelet thresholding method to obtain the residual systematic error \(X_{3}\) contained in the weakly correlated IMF components. The residual systematic error \(X_{3}\) and the systematic error \(X_{2}\) obtained in the EMD process were reorganized to obtain the fine systematic error \(X_{4}\) decomposed from the original machining error from coarse to fine, and its value and distribution are shown in Figure 14.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline IMF Component & IMF1 & IMF2 & IMF3 & IMF4 & IMF5 & IMF6 & IMF7 & IMF8 & IMF9 \\ \hline correlation coefficient & 0.4688 & 0.3410 & 0.3160 & 0.3500 & 0.3369 & 0.1652 & 0.1511 & 0.1236 & 0.3650 \\ \hline \end{tabular}
\end{table}
Table 5: Correlation coefficients corresponding to each IMF in EMD.  
Figure 13: EMD algorithm for the decomposition of coarse systematic error.