# Design of Multi-Cell Cooperative Control Algorithm Based on Fuzzy Brain Emotional Learning
## 2 Cell Manipulation Based on Holographic Optical Tweezers
### Dynamic Model  
Yeast cells are captured by light pinching in the culture environment and are mainly subjected to five forces: optical force, viscous resistance, buoyancy, gravity, and Langevin force. The object of this paper is on the x-y plane. Capture cells move in the x-y plane. The  
Figure 1: Schematic diagram of optical tweezers capturing particles. forces in the \(z\)-axis direction, including the optical force, buoyancy, and gravity in the \(z\)-axis direction, are balanced. The force on the \(z\)-axis does not affect the x-y plane, so it can be ignored [34]. In the process of cell transportation, the Langevin force can also be ignored because the Brownian motion caused by the Langevin force is relatively small [35; 36; 37]. The cells captured by optical tweezers are mainly affected by the optical trapping force and viscous resistance [38; 39]. The expression is:  
\[m\ddot{\textbf{X}}=\textbf{F}_{\textbf{t}}-\textbf{F}_{d} \tag{1}\]  
where \(m\) represents cell mass, \(\textbf{X}\in R^{2\times 1}\) represents cell position, \(\textbf{F}_{\textbf{t}}\) represents the optical trapping force, and \(\textbf{F}_{d}\) represents drag force. In this paper, yeast cells were taken as the control object, and their diameters were generally about 5 \(\upmu\)m to 6 \(\upmu\)m. The captured cell velocity is not more than 15 \(\upmu\)m/s. Therefore, the maximum Reynolds number \(Re\ll 1\). In a fluid environment with a low Reynolds number, the inertial force on the object can be neglected [38; 40; 41; 42]. Equation (1) can be expressed as follows:  
\[0=\textbf{F}_{\textbf{t}}-\textbf{F}_{d} \tag{2}\]  
In general, the wavelength of the laser for optical tweezers is smaller than the diameter of the cell. In this case, the value of the optical trapping force can be calculated by the ray optical model:  
\[\textbf{F}_{\textbf{t}}=k(\textbf{C}-\textbf{X}),\ ||\textbf{C}-\textbf{X}|| \leq r_{0} \tag{3}\]  
where \(k\) represents the optical trap stiffness. \(\textbf{C}\in R^{2\times 1}\) represents the position of the optical trap center. \(||\textbf{C}-\textbf{X}||\) represents the distance between the optical trap center and the cell center. \(r_{0}\) represents the distance between the optical trap center and the cell center when the optical trap force \(\textbf{F}_{\textbf{t}}\) reaches its maximum. The optical trap stiffness is usually a positive constant number. The magnitude of the optical trapping force \(\textbf{F}_{\textbf{t}}\) is related to the distance from the center of the optical trap to the center of the cell. The distance between the center of the optical trap and the center of the captured cell is zero when the optical trap is at rest, and the force of the optical trap acting on the captured cell is also zero. The distance from the center of the optical trap to the center of the cell increases with the change of the position of the optical trap, so the optical trap force also increases. The optical trapping force reaches the maximum value when the distance between the two is \(r_{0}\). When the distance between the centers of the optical trap and the cell is greater than \(r_{0}\), the magnitude of the optical trap force decreases with the increase of the distance. When the optical trap force decreases to zero, the cell will escape from the trap. In conclusion, ensuring that the cell will not escape from the control of holographic optical tweezers is important. Therefore, it is necessary to restrict the distance between the centers of the optical trap and the cell. The general practice is to limit the distance between the two to \(r_{0}\). In particular, it should be noted that the value of \(r_{0}\) is related to the radius r of the captured cell. For the captured cells with different radii, the corresponding \(r_{0}\) values can be determined when they are trapped by the optical trap and the optical trap force reaches the maximum value [43]. Figure 2 showed the relationship between the optical trap force and \(\frac{r_{0}}{r_{1}}\) for cells with different radii, where \(i\) = 1, 2, 3, 4, 5.