# A Polishing Processes Optimization Method for Ring-Pendulum Double-Sided Polisher
## 4 Simulation and Experiment Verification
### Experiment  
To evaluate the effectiveness of the proposed method, we conducted experiments on the radial-feed motor speed and actual polishing process using MPC and STRMPC methods.  
(4) Radial-feed motor speed  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ITAE** & **f1** & **f2** & **f3** & **f4** & **f5** \\ \hline Load & 1.9932 & 2.0238 & 2.0566 & 2.0916 & 2.1291 \\ Noise & 3.9183 & 3.9439 & 3.9727 & 4.0065 & 4.0438 \\ \hline \hline \end{tabular}
\end{table}
Table 6: ITAE indices with \(\pm\)1 noise.  
Figure 17: Convergence of system state and terminal constraint set with \(\pm\)1 noise disturbance.  
Figure 18: Step response of different models with load disturbance.  
During the experiment on radial-feed motor speed, the sampling period was set to 0.002 s, and the radial-feed motor speed of the system with MPC method and system with no controller were given as comparisons to the proposed method. The radial-feed motor speed was set to 30 rpm, and, after the speed stabilized, we changed the reference to 18 rpm at a time of 20.84 s. The experiment's results are present in Figures 20 and 21.  
The experimental results indicate that the STRMPC displays outstanding robustness in Figure 21; the speed of the system with STRMPC has fewer steady errors than the speed outputs of comparison, and hardly produces the overshoot. The better control performance can be attributed to the STRMPC's capability to restrict the output error through the single-tube terminal constraint set, as confirmed by the simulation results. The current curve in Figure 20 also verifies the robustness of the proposed method. Figure 22 demonstrates that when the speed changes abruptly, the proposed method has the more stable speed curves, which reflects its advantage in terms of robustness.  
(5) Actual polishing process  
In the surface processing experiment, we chose a fused silica optical glass with dimensions of 430 \(\times\) 430 mm and a thickness of 10 mm. The polishing pressure was set at 200 N, and the upper polishing disk had a radial displacement range of 20 mm relative to the center of the fused silica optical glass. Both the polishing disk and the glass were rotated at a speed of 15 rpm. Each piece of fused silica optical glass was then polished for a duration of 30 min.  
Figure 19: Convergence of system state and terminal constraint set of different models with \(\pm\) 0.5 noise disturbance.  
Figure 20: Step response of different models with \(\pm\)0.5 noise disturbance.  
Surface shape detection was performed before and after processing; the peak-to-valley (PV) and root-mean-square (RMS) indices are presented in Figures 23 and 24.  
The application of the MPC method yielded a 23.78% improvement in surface shape as evidenced by the reduction of the PV value from 1.43 \(\lambda\) to 1.09 \(\lambda\). However, the reduction of RMS was reduced slightly from 0.218 \(\lambda\) RMS to 0.216 \(\lambda\) RMS, with a convergence rate of 0.9%. In contrast, the STRMPC method significantly enhanced both surface shape and roughness, reducing PV and RMS values from 1.49 \(\lambda\) PV and 0.257 \(\lambda\) RMS to 0.99 \(\lambda\) PV and 0.163 \(\lambda\) RMS, respectively. Compared with the MPC method, the STRMPC method achieved a higher convergence rate 33.56% and 36.57% for both PV and RMS. Specifically, the PV convergence rate was improved by 9.78%, while the RMS convergence rate increased by 35.6%. These findings suggest that the STRMPC method is more effective in improving surface shape and roughness accuracy than the conventional MPC method. Therefore, the STRMPC method's stable radial feed enables rapid and stable processing of the ring-pendulum double-sided polisher, resulting in a more uniform surface shape.  
Figure 21: Step response of the RDP radial-feed system [ref. speed = 30 rpm].  
Figure 22: Step response of the RDP radial-feed system [ref. speed = 30 rpm and 18 rpm].