## 3 Research on the Effect of Path Spacing and Bulk Material Removal Depth on Residual Error
### Simulation Results and Analysis
#### 3.2.2 Influence of Bulk Material Removal Depth on Residual Error  
Figure 3b shows the influence of the bulk material removal depth on the residual error when the path spacing is constant. With the increase of bulk material removal depth, RMS and PV values of the residual error showed a linear growth trend. This also reflects the linear decreasing trend of convergence of deterministic polishing with the increase of the bulk material removal depth. With the increase of the bulk material removal depth, the linear increase of the polishing time indicates that polishing time has a positive correlation with the amount of material removal. Since the number of pre-planned dwell points remains constant, the average dwell time allocated to each dwell point increases accordingly. Figure 3c shows the power spectral density calculation results for the residual error along the vertical direction of the planned path for different bulk material removal depths (0.1 \(\upmu\)m, 0.4 \(\upmu\)m, 0.7 \(\upmu\)m, 1.0 \(\upmu\)m) when the path pitch is 0.2 mm. It can be seen that the profiles of the power spectral density curve corresponding to different bulk material removal depths are similar, but as the bulk material removal depth increases, the power density amplitude corresponding to each spatial frequency shows a significant increase. This may be because with the increase of the bulk material  
Figure 3: (**a**) \(y\)-direction power spectral density (PSD) results for different path pitches; (**b**) influence of the bulk material removal depth on residual error (path pitch = 0.2 mm); (**c**) \(y\)-direction PSD results for different path pitches and bulk material removal depths.  
Figure 2: Effect of path pitch on (**a**) residual error and (**b**) other path parameters (bulk material removal depth = 0.5 \(\upmu\)m).  
removal depth, the dwell time of each dwell point is prolonged. The increase in the dwell time of the polishing tool at a certain point will make the path effect more pronounced, thereby introducing deeper ripple errors caused by the tool path. Therefore, for a large form error, a small path spacing should be preferred, which can effectively improve the convergence of deterministic removal, reduce the RMS and PV value of the residual error and effectively suppress the MSF error component in the residual error.  
2.3 Effect of Path Spacing Variations on the Residual Error Resulting from Different Bulk Material Removal Depths  
As can be seen from Figure 4a,b, the influence law of path spacing variations on the RMS and PV values of residual error resulting from different bulk material removal depth is consistent. For deterministic polishing removal, a reasonable polishing path spacing value should be less than 3/10 of the diameter of the TIF. When the value is within this range, drastic variation of RMS and PV values of the residual error can be effectively avoided. Therefore, in this simulation, the RMS and PV curves with a path pitch of less than 0.6 mm are the parts of interest. As can be seen from the partially enlarged view, shown in Figure 4c,d, the slope of the curve increases significantly with the increase of the form error. This indicates that the larger the bulk material removal depth, the greater the effect of the change in the path spacing on the residual error, which makes the subsequent adjustment of the path spacing more useful, especially for regions with large bulk material removal depth.