## 4 Simulation Results
### Case (2): Holographic Optical Tweezers Manipulate Control for Multiple Cells  
Similar to Case (1) described in Section 4.1, it is assumed that there are four yeast cells in another cell environment, and the expected trajectory \(\boldsymbol{q}^{d2}=\left[q_{x}{}^{d2},q_{y}{}^{d2}\right]^{T}\) that satisfies the multi-cell collision-free arrival at the expected position is given by Equation (64). \[\left\{ \begin{matrix}q_{x}{}^{d2}=10+\frac{2}{8}t\\ q_{y}{}^{d2}=10+5\sin(0.05t)\sin(0.02t)\end{matrix} \right. \tag{64}\]  
The initial position of each yeast cell is shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Cell 1** & **Cell 2** & **Cell 3** & **Cell 4** & **Virtual Cell** \\ \hline initial position & \([-15,15]\) & \([20,10]\) & \([-10,10]\) & \([5,5]\) & \([15,20]\) \\ (\(\mu\)m) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial position of each yeast cell. Figure 8: Changes in NSOFCMNN structure (single-cell manipulation). As the core cell of multi-cell cooperative control, the virtual cell will move from the initial position to the expected trajectory, still in accordance with the requirements of the cell speed limit 10 \(\upmu\)m/s. Then, it will keep moving on the expected trajectory. The spacing of each cell is set, as shown in Figure 9:  
With the virtual cell numbered 5 in Figure 9 as the center, four controlled yeast cells are scattered in four directions numbered 1-4. The relative distance between each two controlled yeast cells is 15 \(\upmu\)m, and the relative distance between the virtual cell and each controlled yeast cell is \(\frac{15\sqrt{2}}{2}\)\(\upmu\)m.  
Here, the self-organizing structure adjustment parameter of NSOFCMNN is \(T_{n}=0.1\), \(T_{m}=0.001\), \(T_{k}=0.35\). Other parameter settings are the same as in Section 4.1. The simulation results of the cooperative control of holographic optical tweezers to manipulate multiple cells are shown in Figure 10.  
In Figure 10, red, blue, cyan, black and violet represent controlled yeast cells 1-4 and virtual cells, respectively. From the initial position, each controlled yeast cell can quickly  
Figure 10: Trajectory of multicellular cooperative control. Figure 9: Distance between each cell. reach a position that maintains a preset relative distance from the virtual cell. Then the virtual cells move along an expected trajectory, and each controlled yeast cell moves at an expected position that maintains a certain preset distance from the virtual cells. Figure 11 shows the relative distances between the controlled yeast cells 1-4 and the virtual cells in the \(X\)-axis direction and the \(Y\)-axis direction during 0-5 s. Each controlled cell moves under the control of an optical trap created by a holographic optical tweezers system. At the initial time, the controlled yeast cells start from different initial positions and gradually approach the virtual cells. Cell movement is limited due to the existence of a maximum distance \(r_{0}\) that limits the movement of the optical trap. Then the optical tweezers manipulate each cell and gradually adjust it to a preset position at a certain distance from the virtual cell. It is observed from Figure 11 that within the first second of manipulated the cells, the relative distances of each controlled cell in the \(X\)-axis direction and the \(Y\)-axis direction can converge to the expected distance, and finally achieve \(\varepsilon_{i}(t)\to 0\). This means that each controlled cell can maintain a certain formation and continue to move together. This verifies that the holographic optical tweezers based on the NSOFCMNN proposed in this paper is effective in the cooperative control of multiple cells. Figure 12 shows the comparison between the actual trajectories of the four controlled cells with their expected trajectories when the NSOFCMNN proposed in this paper is used as the main controller. It can be seen that under the control of the optical trap, the controlled cells can precisely move along the expected trajectory with almost zero error.