# Research on Multiple-Axis Contour Error Suppression Method Based on Composite Layered Control
## 3 Design of Composite Layered Cross-Coupling Controller
### Experimental Verification of Cross-Coupling Controller Based on Composite Layering
Upon inspecting Table 4, it is evident that although the introduction of the fuzzy PID cross-coupling control did not significantly improve the tracking accuracy of a single axis, Table 5 reveals that, following the implementation of the fuzzy PID cross-coupling control, compared to the traditional PID method, there was an 84.39% enhancement in the XY plane contour accuracy; similarly, the accuracy of the three-dimensional spatial contour increased by 84.45%. This observation underscores that the design of the hierarchical controller markedly elevated the contour performance, holding significant effects for both individual planes and the overall spatial contexts. This reaffirms the proposition made in Section 3: the quality of the single-axis tracking performance, though impacting the final machining precision to a certain degree, was overshadowed by the influence of contour errors. The presented composite hierarchical contour control method effectively enhanced  
Figure 26: Self-tuning curves of Kp, Ki, and Kd in fuzzy PID control system. Figure 29 portrays the comparative outcomes between the composite layered contour control algorithm (CLCC) and the results of incorporating solely the lower-level cross-coupling controller. Contrasting with the observations from Figure 29, the CLCC algorithm demonstrated a substantial enhancement in terms of the spatial contour curvature upon the integration of the upper-level cross-coupling controller. The space between the desired contour and the actual contour evolved from having a discernible gap to near elimination. Computational assessment revealed that the ultimate contour error achieved by the CLCC algorithm measured 0.0115 mm. This constitutes a 61.9% improvement over the preceding XY lower-level control algorithm's 0.0302 mm and a 94% augmentation compared to the traditional cross-coupling control algorithm's \(-\)0.1935 mm. Following the comprehensive series of algorithm performance test analyses, it is evident that the designed CLCC algorithm is efficacious. It not only addresses the precision of individual planes' contours but also significantly elevates the overall contour precision. Figure 30 illustrates the simulation results of the method proposed in this paper compared to those of reference [34] and reference [35] when tracking spatial curves with multiple curvatures. The tracking performance of the spatial curves with multiple curvatures effectively highlights the differences among the algorithms, as frequent changes in the curvature demand higher real-time requirements for the algorithms. As observed from Figure 30, reference [34] and reference [35] exhibited instances of non-conformance and vertical fluctuation when tracking spatial curves with multiple curvatures. However, the method proposed in this paper demonstrated the ability to accurately track the target curve. The input parameters of spatial multi-curvature lines are shown in Table 6.  
According to Table 7, it can be observed that the proposed method in this paper significantly reduced the contour errors of spatial curves with multiple curvatures compared to references [34] and [35]. In comparison to reference [34], the maximum contour error, average contour error, and integrated absolute error (IAE) of the proposed method decreased by 32.2%, 80.2%, and 39.3%, respectively. Compared to reference [35], the proposed \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Input Function**} & \multirow{2}{*}{**Axis**} & \multicolumn{4}{c}{**Parameters**} \\ \cline{3-6}  & & \(A\) & \(\omega\) & \(\phi\) & \(d\) \\ \hline \multirow{3}{*}{\(f(x)=A\cdot\sin(\omega\cdot x+\phi)+d\)} & \(X\) & 1 & \(\pi\) & \(\pi/2\) & 0 \\ \cline{2-6}  & \(Y\) & 1 & \(\pi\cdot 1.25\) & 0 & 0 \\ \cline{1-1} \cline{2-6}  & \(Z\) & 1 & \(\pi/2\) & 0.2 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Input parameters for multi-curvature spatial contour lines. Figure 29: Comparison of lower-layer control algorithm curves between CLC and XY.