## 4 Coarse-to-Fine Free-Form Surface Machining Error Decomposition and Compensation Experiments
### Data Analysis  
In comparative group (1), the maximum absolute machining error of the free-form surface part before compensation was 0.0575 mm, and the mean absolute machining error was 0.0435 mm, which was a 24.3% reduction. However, the maximum absolute machining error after compensation was 0.0201 mm, and the mean absolute machining error was 0.0124 mm, which represents a reduction of 71.49% in the mean absolute machining error. In comparative group (2), the maximum absolute machining error of the free-form surface part before compensation was 0.0594 mm, and the mean absolute machining error was 0.0449 mm. On the other hand, the maximum absolute machining error after compensation was 0.0186 mm, and the mean absolute machining error was 0.0082 mm, which was a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Detection Method** & **Before Machining Error Compensation** & \multicolumn{2}{c}{**After Compensation for Machining Error**} \\ \hline  & Maximum absolute machining error & Mean absolute machining error & Maximum absolute machining error & Mean absolute machining error \\ On-machine measurement & 0.0580 & 0.0472 & 0.0159 & 0.0059 \\ CMM measurement & / & / & 0.0148 & 0.0055 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of two different measurement methods (mm). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode of** & & & & \\
**Compensation** & **Before Machining Error Compensation** & \multicolumn{2}{c}{**After Machining Error Compensation**} \\ \hline  & Maximum absolute machining error & Mean absolute machining error & Maximum absolute machining error & Mean absolute machining error & Maximum absolute machining error \\ Machining error direct & 0.0575 & 0.0435 & 0.0201 & 0.0124 \\ compensation method & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \multicolumn{1}{c}{} & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Machining common error compensation using different compensation methods (mm). 24.4% reduction in the maximum absolute machining error and an 81.73% reduction in the mean absolute machining error. In the compensation method proposed in this paper, the maximum absolute machining error of the free-form surface part before compensation was 0.0580 mm and the mean absolute machining error was 0.0472 mm, while the maximum absolute machining error after compensation was 0.0159 mm and the mean absolute machining error was 0.0059 mm, which represents a 72.5% reduction of the maximum absolute machining error and an 87.50% reduction of the average absolute machining error. After compensating for the machining error of the free-form surface part by using the three compensation methods, the machining accuracies greatly improved. The compensation method proposed in this chapter was the most effective, the second most effective was comparative group (2) and comparative group (1) was the least effective in terms of compensation. The main reasons for this were as follows:  
1. All three methods compensate for the error of each measurement point sequentially, and use the surface reconstruction method to make the tool trajectory in the compensated machining smoother, and thereby improve the machining accuracy of the part. 2. Because the random error was more random, it was not easy to compensate for, and, because the processing error was relatively small, direct processing error compensation rather than only the system error compensation effect was good. 3. According to the experimental data in Tables 3 and 7, the more complete and accurate the decomposed system error was, the better the compensation effect. Compared with the VMD algorithm, the coarse-to-fine machining error decomposition method proposed in this paper was superior in its ability to decompose and compensate for machining errors.