## 5 Results
### Optimization Results Verification  
#### 5.3.1 Parameters Reverse Calculation and Optimization of BPNN  
The verification experiment process is shown in Figure 9. Initially, 20 target surface-roughness values were randomly selected, along with their corresponding experimental sample sizes. The threshold for the experimental sample size was then determined, and a reverse calculation was performed to derive the corresponding robotic grinding parameters. Using these optimized parameter combinations, robotic grinding experiments were conducted on the test workpiece 20 times individually.  
Figure 8: (**a**). Scatter plot and (**b**). line chart of RSM and BPNN model with predictive and true values.  
Figure 7: Model \(\text{MSE}\) and \(\text{R}^{2}\) on the training set, verification set, and test set during the training of BPNN.  
In Figure 10, the symbol square represents the random target surface-roughness values, and the symbol triangle represents the true surface-roughness values obtained from grinding experiments using the above parameter combinations. It can be observed that in the random target surface-roughness prediction task using the BPNNprediction model, the MAPE and RMSE between the target values and the actual values is 6.60% and 0.410 \(\upmu\)m, which demonstrates the accuracy and reliability of the BPNNprediction model.  
30 and 60 groups of small-sample data are used as the test sample size. Through threshold detection by RSM-BPN, optimization methods, including RSM-numerical optimization and BPNN-genetic algorithm optimization, are employed. The optimal robotic control parameter combinations are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample Size** & \(\mathbf{F(N)}\) & \(\mathbf{v_{f}(mm/s)}\) & \(\mathbf{v_{s}(rpm)}\) & \(\mathbf{N}\) & \(\mathbf{R_{a}(\upmu)}\) \\ \hline
30 & 4.1474 & 3.0000 & 7312.7826 & 400 & 0.3472 \\
60 & 5.4593 & 3.4935 & 7122.5485 & 600 & 0.3449 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Combinations of predicted and optimized surface-roughness values.  
Figure 10: Comparison between target and actual surface roughness on test workpiece.  
Figure 9: Reverse calculation verification experiment of BPNNon the test workpiece.  
For the application of robotic grinding, this subsection shows a comparison of the surface-roughness values based on 20 measurements (each value measured three times)under the same robotic path when the optimal grinding parameters are used versus when they are not (when not used, the central experimental point parameters from DCCD are applied). While the optimal robotic control parameter combinations are applied, the average \(R_{a}\) values are 0.3582 \(\upmu\)m and 0.3558 \(\upmu\)m for the 30-sample and 60-sample group sizes, respectively. Compared to the average \(R_{a}\) values obtained without using the optimized robotic parameters and the theoretical optimal \(R_{a}\) value, the relative improvement percentages for the 30-sample and 60-sample experimental datasets are 30.91% and 30.69%, respectively. Additionally, the overall APE and RMSE of the actual experimental results compared to the theoretical values after optimization are 3.15% and 0.0288 \(\upmu\)m, respectively.