## 3 Cell Manipulation Control
### Cell Dynamics Model  
The force on the \(X\)-\(Y\) axis of the cell is mainly composed of an optical trapping force and a viscous drag force from the liquid. In the \(Z\)-axis direction, the force acting on the cell includes the component of the optical trapping force in the \(Z\)-axis direction, the gravity acting on the cell and the buoyancy force. According to [PERSON]'s third law, the force in the \(Z\) axis will be balanced when the cell is in a stable state. This section mainly studies the motion state of the cells captured by the optical trap under the force on the plane of the \(X\)-\(Y\) axis, and the force on the cells in the \(Z\)-axis direction does not contribute to the motion on the \(X\)-\(Y\) axis plane [39]. Thus, the force in the \(Z\)-axis direction is ignored. The force analysis of the cells in the optical trap is shown in Figure 3, and the force on the cell can be expressed by Equation (28).  
\[m\stackrel{{\cdot}}{{\boldsymbol{q}}}=F_{\text{trap}}-F_{\text{ drag}} \tag{28}\]  
Because of the Reynolds number Re \(<\)\(<\) 1, the inertia force can be ignored [39]. Therefore, Equation (28) can be simplified as:  
\[0=F_{\text{trap}}-F_{\text{drag}} \tag{29}\]  
\[F_{\text{trap}}=k(I\!-\!q),\|I\!-\!q\|\leq r_{0} \tag{30}\]  
\[F_{\text{drag}}=\delta\,\dot{\boldsymbol{q}} \tag{31}\]  
Figure 3: Force analysis of cells in optical traps.  
where \(F_{\text{trap}}\) is the optical trapping force, \(F_{\text{drag}}\) is the blocking force, \(k\) is the optical trap stiffness, \(I\in\text{R}^{2\times 1}\) is the position of the optical trap produced by optical tweezers, \(\mathbf{q}=\left[q_{x};q_{y}\right]\in\text{R}^{2\times 1}\) is the position of the cell, \(\left\|I-\mathbf{q}\right\|\) is the offset between the center of the cell and the center of the optical trap, and \(\delta\) is the coefficient of viscous resistance.  
Substituting Equations (30) and (31) into Equation (29), the dynamic equation of the cell in the optical trap is obtained as:  
\[0=k(I-\mathbf{q})-\delta\,\dot{\mathbf{q}},\left\|I-\mathbf{q}\right\|\leq r_{0} \tag{32}\]  
Rewriting Equation (32) as:  
\[\frac{\delta}{k}\dot{\mathbf{q}}=I-\mathbf{q},\left\|I-\mathbf{q}\right\|\leq r_{0} \tag{33}\]  
However, in practice, the micromanipulation of cells is very sensitive to environmental disturbances, such as stochastic dynamics caused by equipment vibration, liquid flow and random Brownian motion [40]. The interference term is recorded as \(\mathbf{d}\in\text{R}^{2\times 1}\) and substituted into Equation (33). The actual cell dynamics model is shown as Equation (34):  
\[\dot{\mathbf{q}}=\frac{k}{\delta}\times(I-\mathbf{q})+\mathbf{d},\left\|I-\mathbf{q}\right\| \leq r_{0} \tag{34}\]  
It should be noted that when the force on the cell is in equilibrium, the center of the cell is completely coincident with the center of the optical trap, and the optical trapping force is zero. The optical trapping force increases linearly with the offset between the cell center and the optical trap center until the offset is greater than \(r_{0}\). When the offset is greater than \(r_{0}\), the optical trapping force decreases with the increase of the offset. When the cell is completely outside the optical trap, the optical trapping force becomes zero. Therefore, in the process of manipulating cells with optical tweezers, the displacement of cells is usually constrained within the range of \(r_{0}\).