## 5 Comparative Study of Optimization Strategies
### Case 2: Aspheric Workpiece  
To further verify the feasibility of the proposed optimization strategy for corrective polishing of complex surfaces, in this case, we selected the aspherical surface shown in Figure 13a as the test sample. Table 4 gives the specific parameters of the aspheric surface, which is expressed by Equation (14) as follows:  
\[z=\frac{c\left(x^{2}+y^{2}\right)}{1+\sqrt{1-c^{2}\left(1+k\right)\left(x^{2} +y^{2}\right)}}+a_{2}\left(x^{2}+y^{2}\right)^{2}+a_{3}\left(x^{2}+y^{2} \right)^{3}+a_{4}\left(x^{2}+y^{2}\right)^{4}+\cdots \tag{14}\]  
where \(R_{0}\) represents the radius of curvature of the aspherical vertex; \(c\) represents the curvature of the aspherical vertex, with \(c=1/R_{0}\); \(D\) represents the aspherical aperture; \(k\) represents the quadratic constant; \(a_{2}\), \(a_{3}\), \(a_{4}\) represent the coefficients of higher order terms in the aspheric equation, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(R_{0}\) (mm) & \(c\) (mm\({}^{-1}\)) & \(D\) (mm) & \(k\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) \\ \hline
20 & \(1/R_{0}\) & 20 & \(-0.32\) & \(-1.5\times 10^{-14}\) & \(-1.5\times 10^{-14}\) & \(-3.0\times 10^{-16}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Detailed parameters of the aspheric surface. Figure 12: PSD analysis results of residual error in the (**a**) \(x\)-direction and (**b**) \(y\)-direction. Figure 13d shows the slope distribution corresponding to the aspherical surface. It can be seen that the slope of the surface is rotationally symmetric and gradually increases from the center to the periphery. Figure 13b,c,e,f respectively shows the isometric and top view of the aspherical surface form error and of the extended surface form error obtained after extending a distance of \(1.5R_{\mathit{lif}}\). Since the material removal depth variation in all regions in this case is small (about 0.75 \(\upmu\)m to 1 \(\upmu\)m), the chosen minimum path in Equation (7) and its variation range are narrow. The path obtained in the Cartesian space is too dense to observe the effect of pitch change by the naked eye; thus, Figure 13g,h gives the RMS-map-based optimization path in polar coordinate and its pitch-variation along the path. Similar to Case 1, Figure 14 shows the distribution of residual error and dwell time, respectively. It can be seen that the residual error distribution obtained by the optimization strategy is hardly affected by the distribution of parent form error, and the approximately uniform distribution of dwell time indicates that the polishing speed for the whole process is maintained at a stable level, such that polishing dynamics are optimized. Figure 15a-c shows the profile changes of the two section lines corresponding to three residual errors, respectively. The imprint of the parent form error is almost invisible in the optimized section profile curve, which further verifies the usefulness of the proposed optimization strategy. For the circular workpiece, the sub-domain evaluation strategy proposed in Section 5.1 is also used to obtain the RMS and the PV curves in sub-regions of the residual error, as shown in Figure 15d,e. It can be seen from the variation of the curve that the optimization strategy can significantly improve the spread of RMS and PV values across the residual error and that different optimization models result in different optimization effects. It can be concluded that for the corrective polishing of the surface, the proposed optimization strategies are both effective, and the distribution of residual error is significantly improved. Figure 13: (**a**) Aspheric surface; (**b**) isometric view of the original form error distribution; (**c**) isometric view of the extended form error distribution; (**d**) slope distribution of the aspheric surface; (**e**) top view of the original form error distribution; (**f**) top view of the extended form error distribution; (**g**) RMS-map optimization path in polar coordinates; (**h**) variation of path pitches of the RMS-map optimization path.