## 3 Research on the Effect of Path Spacing and Bulk Material Removal Depth on Residual Error
### Simulation Experiment Design  
In submerged jet polishing (for narrow sapphire nozzles with a diameter less than 0.3 mm) and other deterministic polishing, the TIF usually presents a Gaussian distribution with a maximum removal at the center. In the simulation study of this paper, the TIF established by the Gaussian mathematical model given in Equation (4) is used, where \(R_{x}\) and \(R_{y}\) represent the material removal area radius (mm) of the TIF in the \(x\) direction and the \(y\) direction, respectively. \(a\) represents the maximum material removal depth per unit time (\(\upmu\)m/min). Figure 1a shows the rotationally-symmetric three-dimensional TIF profile with a removal radius of 1 mm and a maximum removal depth of 0.1 \(\upmu\)m/min. Table 1 gives the parameter values of the TIF used in this paper.  
Without loss of generality, a scanning path was planned on a planar specimen with a length of 10 mm and a width of 10 mm, and the deterministic polishing simulation was implemented according to the given TIF, as shown in Figure 1b. Since it is difficult for the computer to achieve continuous removal in simulations, in order to make the discrete removal simulation more closely approximate the continuous removal commonly used in actual polishing, the number of dwell points along the planned path was increased. Here, the distance between adjacent dwell points along the path was set to 0.05 mm. The simulation results in our test show that the actual continuous polishing removal can be well approximated. The RMS and PV values of the residual error obtained after correction were used as evaluation indexes for the deterministic polishing convergence accuracy and the removal effect. In addition, the power spectral density (PSD) method was used to analyze the MSF error component in the residual error and study the effect of path spacing and surface form error on the component. In order to avoid the influence of edge effect in the simulation process, the central region of the specimen with a length of 6 mm and width of 6 mm served as a residual error evaluation region.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parms** & \(a\) (\(\upmu\)/min) & \(R_{x}\) (mm) & \(R_{y}\) (mm) \\ \hline Values & 0.1 & 1.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameter values of tool influence function (TIF).  
Figure 1: (**a**) Tool influence function (TIF) with Gaussian shape; (**b**) tool path design.  
In order to deeply study the influence of path spacing and bulk material removal depth on the residual error obtained after polishing, three groups of basic simulation experiments were carefully designed. The first set of experiments studied the effect of path spacing on residual error under fixed bulk material removal depth. The second set of experiments investigated the effect of bulk material removal depth on the residual error in the case of fixed path spacing. The third set of experiments investigated the effect of path spacing variations on the residual error resulting from different bulk material removal depths. Table 2 gives details of the experimental conditions for the three simulation experiments.