## 5 Comparative Study of Optimization Strategies
### Case 1: Planar Workpiece  
In this case, a planar test piece with a length of 10 mm and width of 10 mm is again selected. Figure 8 shows the initial typical form error distribution and the extended form error distribution, respectively. It can be seen that the distribution of form error is a typical convex surface form with a height variation range of 0.5 \(\upmu\)m to 1 \(\upmu\)m. The TIF specified in Section 3 is again used in this simulation. Therefore, we can directly use the RMS-map-based (Equation (5)) and PV-map-based path-pitch model (Equation (6)) obtained in Section 3 to optimize the initially planned polishing paths. In addition, since it is a planar specimen, the slope compensation factor in Equation (11) is null, i.e., \(k_{slope}=0\). Table 3 shows the key parameters for planning the spiral-polishing path with optimized variable pitch. According to the kind of optimization model, we get three kinds of polishing paths, namely the unoptimized path, RMS-map-based optimized path and PV-map-based optimized path. Figure 9a shows the initially planned spiral-polishing path with boundary constraints. Using the two proposed optimization models respectively to optimize the initial path given in Figure 9a, the corresponding optimized path distributions are obtained, as shown in Figure 9b,c. It can be seen that the polishing path pitch before optimization is always 0.2 mm and is evenly distributed. Although there are subtle differences in paths optimized by different optimization models, their path pitches are continuously changed with the variation of the surface form error in the range of 0.2 mm to 0.6 mm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p_{\text{min}}\) (mm) & \(p_{\text{max}}\) (mm) & \(p_{0}\) (mm) & \(\Delta\theta\) (deg) & \(r_{0}\) (mm) \\ \hline
0.2 & 0.6 & 0.2 & \(\pi/6\) & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for planning the initial spiral-polishing path. Figure 8: (**a**) Original form error distribution; (**b**) extended form error distribution. Figure 9: (**a**) Boundary-defined unoptimized path with constant pitch; (**b**) RMS-map-based optimized path with varying pitch; (**c**) PV-map-based optimized path with varying pitch; (**d**) dwell time map along the unoptimized path; (**e**) dwell time map along the RMS-map-based optimized path; (**f**) dwell time map along the PV-map-based optimized path; (**g**) residual error corresponding to the unoptimized path; (**h**) residual error corresponding to the RMS-map-based optimized path; (**i**) residual error corresponding to the PV-map-based optimized path. By solving the dwell times corresponding to the three polishing paths, the dwell time distribution maps corresponding to the three polishing paths are obtained, as shown in Figure 9d-f. It can be seen from Figure 9d that the dwell time distribution corresponding to the initial equally-spaced spiral polishing path is consistent with the initial form error distribution. It further verifies the experimental results obtained in Section 3.2.2, that is, when the path spacing is constant, there is a positive linear relationship between the dwell time of each dwell point and the amount of form error. For the optimized paths, especially in the regions we care about, the dwell time of each dwell point does not change very much. From Figure 3b in Section 3.2.1, it can be seen that there is also a positive linear relationship between the path spacing within a certain range (0.2 mm-0.6 mm) and the dwell time of each point when bulk material removal depth is constant. It is under the combined influence of bulk material removal depth and path spacing that the dwell time at each point on the optimized path remains mostly the same. To some extent, the optimized path is a good way to improve dwell time distribution. Since spacing of the planned dwell points along the path is basically the same, the optimized polishing path will run under a more stable polishing speed, which is crucial for obtaining a stable polishing process [24].