## 3 Experimental Study  
The compensation experiment was carried out to verify the effectiveness of the proposed method, where an optical freeform surface was employed to test the feasibility of the compensation processing. The designed surface is given as Equation (7). The surface was machined by STS based on Precitech Nanoform 700 ultra single point diamond turning machine, and was measured by the newview8200 measurement system from Taylor Hobson Co., Ltd., in UK. The workpiece material is Al, and the tool parameters and machining parameters are shown in the Table 4.  
The freeform surface was measured after the first processing. Figure 6 shows the machined surface and its form error after the matching process. As shown in Figure 6b, the form error between the machined surface and the designed surface is relatively small, and the PV and RMS value were 1.4 um and 173 nm respectively. Such precision could not meet the requirements of ultra-precision fabrication, the compensation machining is a necessary process to enhance the form accuracy of the freeform surface. The compensation machining process is as follows. After the surface is measured, take the measurement result as the surface after coordinate transformation of ideal freeform surface. Then the measured surface is matched with the ideal surface by matching algorithm. The form error is derived by the algorithm and taken as the surface to be removed in the next step. The Z axis data of the surface shape error is inverted, and then added to the Z axis data of the ideal freeform surface. It is used as the surface shape to be processed in compensation machining. Then reprogram to generate a new processing program and start compensation machining. The compensation process is designed according to the form error calculated, and the cutting parameters are the same as the first processing, compensation result is shown in Figure 7a. The form error was reduced to PV value of 560 nm and RMS value of 55 nm. It is obvious that the surface quality was improved effectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{**Cutting Tool Parameters**} & \multicolumn{3}{c}{**Machining Parameters**} \\ \hline Radius/mm & Rake/\({}^{\circ}\) & Clearance angle/\({}^{\circ}\) & Rotate speed/rpm & Feedrate/mm/r & Cut depth/um \\ \hline \(1\) & \(0\) & \(20\) & \(40\) & \(0.005\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cutting tool parameters and machining parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **tx/mm** & **ty/mm** & **tz/mm** & **a/rad** & **\#frac** & **\(\gamma\)_trad_ \\ \hline \multirow{4}{*}{Translation} & \(1\) & \(4.94\times 10^{-9}\) & \(-4.32\times 10^{-10}\) & \(-5.51\times 10^{-9}\) & \(3.99\times 10^{-10}\) & \(-1.55\times 10^{-9}\) \\  & \(1.94\times 10^{-9}\) & \(1\) & \(3.70\times 10^{-9}\)  
In addition, there are some factors affecting the performance of compensation processing, such as random errors in the machining process. Only a one-time compensation failed to achieve the desired result, so the second compensation processing was carried out. As shown in Figure 7b, in the second compensation, PV value is reduced to 352 nm, and RMS value is reduced to 23 nm, the surface quality is further improved. That is the same as the process of polishing. It can be said that the form error compensation is a process of continuous convergence. As a result, it is necessary to carry out compensation machining cyclically until the form error meets the requirement. The third compensation processing was conducted, and the result is shown in Figure 8. The PV value is 335 nm and the RMS value is 34 nm. This time the PV didn't change much, but the RMS increased slightly. As described above, the form error is fluctuant in the process of compensation so this phenomenon is normal in the error convergence process, for the random error such as vibration of the machine or tool wear is likely to affect performance in the submicron accuracy range.