## 3 Research on the Effect of Path Spacing and Bulk Material Removal Depth on Residual Error
### Simulation Results and Analysis  
#### 3.2.1 Influence of Path Spacing on Residual Error  
By introducing regularization and sparse matrix operation methods, the dwell time solution algorithm based on the linear equations can effectively control the negative ratio (number of dwell points with a negative dwell time calculated/total number of dwell points) of dwell time calculated to be less than 1%, as shown in Figure 2a. Note that the influence of path spacing on the RMS and PV values of the residual error is almost consistent when bulk material removal depth is constant. As the path spacing increases, the RMS and PV values of the residual error first increase slowly and approximately linearly within the range of 0.1 mm to 0.6 mm, then rapidly increase within the range of 0.6 mm to 2.0 mm and finally stabilize in the range of 2.0 mm to 2.5 mm. This is because the deterministic polishing removal is caused by the convolution effect formed by the overlap of tool influence functions. When the distance between the paths is small, i.e., less than 3/10 of the diameter of the TIF, convergence of the deterministic polishing removal process can be effectively ensured. From the residual error profile in Figure 2a, it can be seen that as the path spacing increases, it becomes increasingly difficult to ensure convergence of the regions between adjacent paths. When the distance between the paths is larger than the diameter of the TIF, the material between the adjacent paths cannot be removed. In this case, the PV values of the residual error will be slightly larger than the original surface form error and remain basically constant. However, it should be pointed out that the RMS curve is fluctuating more at these later stages than the PV curve. This may be because the RMS value of the residual error is more susceptible to dramatic changes in convergence of the surface form error.  
Figure 2b shows that as path spacing increases, the total length of the path is drastically shortened, but the total polishing time is only slightly reduced within the path range of 0.1 mm to 1.0 mm. The polishing time for deterministic polishing is mainly determined by the TIF and volume of material removal. More polishing time means more volume removal. The error of a certain control point basically determines the total dwell time at that point. As the path spacing decreases, although the path length increases sharply, the number of dwell points increases at the same time, and the average dwell time allocated to each dwelling point will be reduced accordingly, so that the total polishing time remains basically unchanged. In addition, considering that as the path spacing increases, the degree of convergence of deterministic polishing decreases, the overall material removal will be reduced, and thus, the polishing time will tend to decrease. When the path spacing is greater than 1.0 mm, the degree of convergence of the polishing removal sharply changes and shows a downward trend, so that the total polishing time is drastically reduced with a certain degree of fluctuation.  
\begin{table}
\begin{tabular}{c c c} \hline
**Experiment Set** & **Path Pitch (mm)** & **Bulk Material Removal Depth (\(\upmu\)m)** \\ \hline Exp. 1 & 0.1, 0.2, 0.3,..., 2.5 & 0.5 \\ \hline Exp. 2 & 0.2 & 0, 0.1, 0.2,..., 1.0 \\ \hline Exp. 3 & 0.1, 0.2, 0.3,..., 2.5 & 0, 0.1, 0.2,..., 1.0 \\ \hline \end{tabular}
\end{table}
Table 2: Simulation experiment conditions.  
Figure 3a shows the power spectral density curve of the residual error calculated along the \(y\)-direction, which is the perpendicular direction of the planned path when the path distance is 0.1 mm, 0.4 mm, 0.7 mm and 1.0 mm, respectively. It can be seen that as the path spacing increases, the MSF error becomes larger and larger, which is mainly due to waviness error caused by the tool path. It can be concluded that for deterministic polishing removal, a small path spacing should be preferred, which can effectively reduce the RMS and PV values of the residual error and effectively suppress the MSF.