## 4 Simulation and Verification
### Experimental Investigation  
A group of indexable inserts is grinded on Agathon 400 combi shown in Figure 7. Figure 8 shows the geometry of this batch 35\({}^{\lx@math@degree}\) diamond indexable insert, and the material is YT15, a kind of tungsten-cobalt-titanium hard alloy. The diameter of the inscribed circle for the indexable insert is the controlled dimension (D in Figure 8), which is measured by a grating displacement sensor. The displacement sensor used in experiment is HEIDENHAIN MT60K. Its measurement range is 60 mm, and the system accuracy is up to \(\pm\)0.5 um. The measurement diagram is shown in Figure 9. After the indexable insert is grinded, the displacement sensor measured the diameter of inscribed circle and transmitted it to the controller to decide whether the compensation is needed or not. The grinding conditions for the experiments are shown in Table 1. For the experiment, first, the machine tool runs without grinding any workpiece for half an hour. Next, it begins to grind indexable inserts. For this group experiment, the desired dimension of indexable insert is 9.435 mm. For the compensation module, the error limit E\({}_{\text{lim}}\) is set to 0.003 mm also in consideration of the  
Figure 8: Geometry diagram of the indexable insert. Figure 7: Agathon 400 combi grinding machine tool. Figure 9: Dimension measurement diagram. (**a**) The displacement sensor. (**b**) Actual measurement diagram. error acceptance of A level. The maximum compensation value C\({}_{\max}\) is chosen as 0.01 mm in order to ensure the safety. Furthermore, the initial value of updated index \(\varphi_{1}=1\). For the experiment, under the grinding conditions shown in Table 1, a group of indexable inserts have been grinded on the machine tool without compensation. The dimension data are listed in columns 2 of Table 2, which are the baseline data. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Dimensions of the Group** & **Dimensions of the Group** \\
**No. ** & **Workpieces without** & **Workpieces with** \\  & **Compensation (mm)** & **Compensation (mm)** \\ \hline
1 & 9.422 & 9.422 \\
2 & 9.424 & 9.434 \\
3 & 9.428 & 9.438 \\
4 & 9.428 & 9.434 \\
5 & 9.43 & 9.436 \\
6 & 9.43  
Figure 10 shows the comparison of dimensional variation with different sets of compensation parameters, which is a group of compensation experiment. The first set compensation parameters are: \(\rho=1,\ \mu=0.5,\ \beta=1,\ \alpha=1\). The second set compensation parameters are: \(\rho=1,\ \mu=0.5,\ \beta=10,\ \alpha=10\). The third set compensation parameters are: \(\rho=1,\ \mu=0.5,\ \beta=10,\ \alpha=20\). It is evident that under the first set compensation parameters, the dimensions fluctuated largely compared with the other two sets. The compensation effectiveness of the second set of compensation parameters is much better. Figure 10: Dimensional variation with three sets of compensation parameters for the compensation experiment. Figure 11 shows the comparison of dimensional variation before and after compensation. The blue line represents the baseline data without compensation. It is shown that the dimensional error is non monotonic. At the beginning of batch grinding, the dimensional error is relatively large. Then the dimensional error tends to within error tolerance. After a certain time grinding, the dimensional error increases largely again. Column 3 of Table 2 shows the data of the dimensions using this compensation method, which is indicated by the black line. The compensation parameters for this group experiment are: \(\rho=1,\ \mu=0.5,\ \beta=10,\ \alpha=10\). It can be seen from Figure 11 that the dimensions fluctuate around the desired value after error compensation, and the compensation effectiveness is really good. From Figure 11 and Table 2, it is evident that there are only 33 workpieces out of 68 within the error tolerance (\(-0.005,\ 0.005\)) before error compensation, while after error compensation, the amount increases to 65.