## 5 Comparative Study of Optimization Strategies
### Case 1: Planar Workpiece
Figure 9g-i respectively shows the residual error distribution obtained by figuring the extended form error using the three polishing paths, respectively. It can be seen that the strategy of extending the path and the surface form error effectively prevents material removal of the region of interest from being affected by the edge-warping phenomenon. Compared to the residual error distribution before optimization, the imprint of the initial surface form the error distribution can hardly be found visually in the optimized residual error distributions. As shown in Figure 10a,c,e, in order to further clearly compare and quantify the difference of the residual error distribution before and after optimization and the effects of two optimization models on the residual error distribution, the residual error distribution of the region of interest is extracted. At the same time, two cross-sectional profiles passing through the center origin and parallel to the x-axis and y-axis are respectively given for each residual error distribution. From Figure 10a, it can be seen that the residual error distribution corresponding to the unoptimized polishing path is highly similar to the distribution of the parent surface form error shown in Figure 8a, that is the error shows a decreasing trend from the center to the periphery. However, the residual error distribution after optimization shown in Figure 10c,e, respectively, is more flattened and is almost unaffected by the distribution of parent form error. Considering the use of RMS and PV values of the overall residual error to evaluate the residual error distribution, there are two disadvantages: (i) the optimization effect of the proposed residual error optimization strategy on residual error distribution cannot be evaluated intuitively and comprehensively; (ii) it is difficult to compare subtle differences in the residual error distribution obtained by the two optimization models proposed. In this paper, a method of sub-domain evaluation is proposed. Firstly, the residual error of the region of interest is evenly divided into 25 sub-regions, as shown in the Figure 10b,d,f. Then, the RMS and PV values of the residual error corresponding to each sub-region are calculated, and the corresponding RMS and PV curves are plotted. Figure 11a,b shows the RMS and PV curves obtained from the simulation results, respectively. It can be seen that, using the sub-domain evaluation method, the RMS curves obtained by the three polishing paths all show both obvious and inconspicuous peaks, and the occurrence frequency of the peaks has a certain regularity. The curve peaks appear in the ranges A1-A5, A6-A10, A11-A15, A16-A20 and A21-A25, respectively. For the range A1-A5, if the average value of the form errors corresponding to each sub-area is plotted as a curve, a similar peak curve is obtained. From the results obtained in Section 3, we know that the RMS value of the residual error increases with the material removal depth; therefore, the resulting RMS curve also exhibits the peak. This can also explain the formation of the corresponding peak in other ranges. In addition, the average amplitude and corresponding amplitude fluctuation range of the RMS curve with optimization are both smaller than that without optimization. This is because adjustment of the path spacing effectively compensates for the influence of different material removal depths caused by the surface form error distribution on the residual error, which greatly reduces the influence of the distribution of parent surface form error on the child residual error distribution. There are multiple outlier points on the PV curve given in Figure 11b, which appear at the positions of A1, A5, A13, A21 and A25, respectively. Comparing the residual error distribution shown in Figure 10, we can see that these outlier points appear at the center of the residual error region and  
Figure 11: (**a**) RMS and (**b**) PV curves for sub-regional evaluation.