## 4 Residual Error Optimization Strategy Based on Root-Mean-Square and Peak-to-Valley Maps
### Path Spacing Optimization Model Based on Surface Form Error  
Based on the RMS map of Figure 5b, the RMS contour of the residual error shown in Figure 6a is obtained. On the RMS contour, all points with the same RMS value form continuous curves in sequence. The curve reflects the correspondence between the bulk material removal depth to be corrected and path spacing for which the RMS value of residual error is controlled to be a constant value. Although corrective polishing should be preferably performed with a small path pitch, the path pitch in actual polishing cannot be set infinitely small. The variation range of expected bulk material removal depth in one polishing iteration is constant. As shown in Figure 6a, once the minimum path spacing is set, for the minimum path used in the region with largest bulk material removal depth, the theoretical optimum RMS value of the residual error after polishing is determined. Conversely, given the RMS value of the residual error we expect, we can get the corresponding minimum path spacing. Moreover, the smaller the RMS value of the desired residual error, the smaller the corresponding minimum path spacing. In the simulation experiments in this paper, the bulk material removal depth varies from 0.5 \(\upmu\)m to 1.0 \(\upmu\)m. We specify the minimum path spacing as \(p_{\text{min}}=0.2\) mm and the maximum path spacing is \(p_{\text{max}}=0.6\) mm. The P1 point determined by the minimum pitch gives the best RMS value of the residual error corresponding to key point P2, and then, the contour line of the best RMS value is located. The contour curve determines the point P3 at the position of the maximum path spacing. P3 determines the critical value of bulk material removal depth corresponding to point P4. When the bulk material removal depth of the region to be corrected is smaller than the critical value, the path spacing corresponding to the region should be set to the maximum path spacing. Finally, the relationship between the bulk material removal depth and the path spacing based on the RMS map is obtained, as shown in Figure 6b. Similarly, the PV contour of residual error and the relationship between the bulk material removal depth and the path spacing based on the PV map are obtained, as shown in Figure 6c,d. A comparison of Figure 6b,d shows that the trend of the two curves is similar, which is attributed to the similarity of the effects of bulk material removal depth, and path spacing on the RMS and PV values of the residual error. A piecewise polynomial fitting method was used to fit the two curves,  
Figure 6: (**a**) RMS contours of the residual error; (**b**) relationship curve between bulk material removal depth and the path pitch based on the RMS map; (**c**) PV contours of the residual error; (**d**) relationship curve between bulk material removal depth and the path pitch based on the PV map. spacing optimization models based on the RMS map (Equation (5)) and the PV map (Equation (6)) were obtained, respectively. \[p_{rms}\left(z_{h}\right)=\left\{ \begin{array}{ll}0.6&,z_{h}<0.3297\\ -4.46{z_{h}}^{5}+18.0{z_{h}}^{4}-28.2{z_{h}}^{3}+22.8{z_{h}}^{2}-9.95{z_{h}}+2.2 2&,z_{h}\geq 0.3297\end{array} \right. \tag{5}\]  
\[p_{pv}\left(z_{h}\right)=\left\{ \begin{array}{ll}0.6&,z_{h}<0.3194\\ -9.44{z_{h}}^{5}+35.2{z_{h}}^{4}-52.4{z_{h}}^{3}+39.2{z_{h}}^{2}-15.3{z_{h}}+2. 85&,z_{h}\geq 0.3194\end{array} \right. \tag{6}\]  
in which \(z_{h}\) stands for the bulk material removal depth (\(\upmu\)m); \(p_{rms}\), \(p_{pv}\) represent the resulting path spacing (mm) based on the RMS-map optimization model and the PV-map optimization model, respectively.