## 4 Simulation Results
### Case (1): Holographic Optical Tweezers Manipulate Control for Single Cells
As time goes by, the cell trajectories gradually coincided with the expected trajectories of the unconstrained conditions. Figure 6 is a comparison between the actual trajectories and expected trajectories of cells when different neural networks are used as the main controller. It can be seen from Figure 6 that when the PI is used as the main controller, there is a large error in both the \(X\)-axis and \(Y\)-axis directions, and it cannot be well applied in the single-cell manipulation control. When the BP, the RBF, the FCMNN or the NSOFCMNN is used as the main controllers, it has great advantages over the PI main controller in the trajectory tracking of single-cell manipulation control. Figure 5: Single-cell manipulation control trajectory using NSOFCMNN. Figure 7 shows the trajectory error using different neural networks as the main controller when holographic optical tweezers manipulate single cells. It can be seen more clearly from Figure 7 that the manipulation control error of the latter three neural networks as the main controller is obviously closer to 0. In order to further compare the control effect between the proposed NSOFCMNN and other controllers, the method of calculating RMSE and MAE to compare the specific errors between them was used, and the corresponding formulas of the two errors are Equations (62) and (63), respectively. \[RMSE=\sqrt{\sum_{t=0}^{t_{\max}}\big{(}q^{d}(t)-q(t)\big{)}^{2}/(t_{\max}+1)} \tag{62}\]  
\[MAE=\sum_{t=0}^{t_{\max}}\big{|}q^{d}(t)-q(t)\big{|}/(t_{\max}+1) \tag{63}\]  
According to Equations (62) and (63), Table 2 is obtained. Among these two kinds of errors, RMSE can evaluate the degree of error using different controllers, and MAE can better reflect the actual situation of the error. Figure 6: Comparison between the actual trajectory and the expected trajectory. The RMSE and MAE of single-cell manipulation for different main controllers are shown in Table 2. It can be seen that when FCMNN is used as the main controller, RMSE and MAE only float up and down 5.0 \(\times\) 10\({}^{-15}\) and 2.0 \(\times\) 10\({}^{-16}\), respectively, which are smaller than the RMSE and MAE when PI, BP and RBF are used as the main controllers, respectively, and the control accuracy is relatively high. Compared with the FCMNN main controller, the RMSE and MAE of the proposed NSOFCMNN are only 0.1-1% of the former. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Type** & **Main Controller** & **X-Axis** & **Y-Axis** \\ \hline \multirow{5}{*}{RMSE of single-cell manipulation} & PI & \(7.0\times 10^{-6}\) & \(1.8\times 10^{-6}\) \\  & BP & \(3.2\times 10^{-6}\) & \(8.4\times 10^{-8}\) \\ \cline{1-1}  & RBF & \(6.2\times 10^{-10}\) & \(4.0\times 10^{-12}\) \\ \cline{1-1}  & FCMNN & \(5.3\times 10^{-15}\) & \(4.2\times 10^{-16}\) \\ \cline{1-1}  & This work & \(7.4\times 10^{-18}\) & \(3.3\times 10^{-18}\) \\ \hline \multirow{5}{*}{MAE of single-cell manipulation} & PI & \(5.5\times 10^{-5}\) & \(3.7\times 10^{-6}\) \\ \cline{1-1}  & BP & \(1.3\times 10^{-5}\) & \(5.8\times 10^{-8}\) \\ \cline{1-1}  & RBF & \(2.5\times 10^{-11}\) & \(2.1\times 10^{-13}\) \\ \cline{1-1}  & FCMNN & \(2.0\times 10^{-16}\) & \(2.1\times 10^{-17}\) \\ \cline{1-1}  & This work & \(3.0\times 10^{-19}\) & \(1.3\times 10^{-19}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Root mean square error (RMSE) and mean absolute error (MAE) of single-cell manipulation. Figure 7: Trajectory error using different controllers (single-cell manipulation). It can be concluded that the single-cell manipulation control of the NSOFCMNN main controller has better control performance than the other main controller, and even in the presence of interference factors, it can also have better robustness. Figure 8 shows the changes in the number of layers and blocks of the NSOFCMNN proposed in this paper during the manipulation of single cells by holographic optical tweezers.