# Robotic Manipulation of Cumulus-Oocyte Complexes for Cumulus Cell Removal
## 3 Methods
### Design of Controller  
After assessing the oocyte's maturity, the COC is repeatedly aspirated and dispensed by the micropipette to remove the cumulus cells. During the micropipette aspiration, the COC is required to be accurately positioned at the target position, aiming to avoid oocyte loss inside the micropipette. However, as the cumulus cells are removed during aspiration and dispensing, the mass of the COC changes. If a fixed external force is applied to it, the acceleration of the COC will change, leading to the large positioning errors. Therefore, the mass change in the COC cannot be ignored and should be updated in real time to compensate for the positioning errors based on the dynamics of the COC's motion inside the micropipette. The dynamics of the COC's motion shown in Figure 4 are described as  
\[M\ddot{x}=F_{d}=6\pi\mu r_{o}\big{(}\gamma v_{p}-\dot{x}\big{)} \tag{5}\]  
where \(M\) and \(x\) represent the mass and position of the COC inside the micropipette, respectively. \(F_{d}\) is the drag force [27], \(\mu\) is the viscosity of the medium, \(r_{o}\) is the radius of the oocyte, \(\gamma\) is the ratio of the cross-sectional areas of the pump piston and the micropipette, and \(v_{p}\) is the velocity of the pump piston. The \(M\) is  
\[M=\frac{4}{3}\pi\Bigg{(}r_{o}{}^{3}+\sum\limits_{i=1}^{N}r_{i}^{3}\Bigg{)}\rho \tag{6}\]  
where \(N\) is the number of cumulus cells, \(r_{i}\) is the radius of the \(i\)-th cumulus cell, and \(\rho\) is the density of the cell. Substituting Equation (6) into Equation (5) obtains  
\[(\beta r_{o}{}^{2}\ddot{x}+\frac{1}{\gamma}\dot{x})+\frac{\beta\sum\limits_{i =1}^{N}r_{i}{}^{3}}{r_{o}}\ddot{x}=v_{p} \tag{7}\]  
where  
\[\beta=\frac{2\rho}{9\mu\gamma}\]  
In Equation (7), \(\beta\), \(r_{o}\), and \(\gamma\) are constants, but \(N\) varies during the micropipette aspiration, leading to changes in the total mass of cumulus cells. Figure 4: Schematic diagram of the COC dynamics during the micropipette aspiration. To address the varying mass of cumulus cells, an adaptive controller has been proposed to control the piston velocity for accurate COC positioning inside the micropipette. In the adaptive control law, \(u_{1}\) and \(u_{2}\) represent the feedback and compensate terms, respectively. \[u_{1}=k_{p}e+k_{d}\dot{e},\ u_{2}=\frac{\sum\limits_{i=1}^{N}{{r_{i}}^{3}}}{{{r_ {o}}^{3}}}(k_{p}e+k_{d}\dot{e}-\frac{1}{\gamma}\dot{x}) \tag{8}\]  
where \(k_{p}\) and \(k_{d}\) are the proportional and derivative gains, respectively, \(e=x-x_{d}\) is the positioning error, and \(x_{d}\) is the target position. \(u_{2}\) is used to update \(N\) and the corresponding \(r_{i}\), aiming to compensate for the positioning errors caused by the varying mass of cumulus cells. \(N\) and \(r_{i}\) can be estimated by the network for cumulus cell detection in Section 3.3. Finding the control law \(u=u_{1}+u_{2}\) is equivalent to minimizing the cost function. \[J=\int_{t_{0}}^{t_{f}}{(X^{T}QX+{u_{1}}^{T}Ru_{1})}dt \tag{9}\]  
where \(t_{0}\) and \(t_{f}\) are the time for the beginning and end, respectively; \(Q\) and \(R\) are the weights for the state and control, respectively; \({u_{1}}^{T}Ru_{1}\) and \(X^{T}QX\) represent the input energy and position error, respectively. The minimum of Equation (9) is solved based on the linear quadratic regulator (LQR) theory, and the \(u_{1}\) is  
\[u_{1}=-Kx=-R^{-1}BPx \tag{10}\]  
where \(K=[k_{p},k_{d}]\) is the feedback matrix, and \(A\) and \(B\) are coefficient matrices. Based on the Riccati equation, \(P\) is  
\[A^{T}P+PA-PBR^{-1}B^{T}P+Q=0 \tag{11}\]  
As shown in Figure 5, the adaptive control, including an LQR and adaptive compensation, is proposed to adjust the piston velocity based on the positioning errors of the COC. The LQR is used to control the main body of the COC (i.e., oocyte), and the adaptive compensation is used to further adjust the piston velocity based on the mass changes of cumulus cells. During the aspiration, the position \(x\) of the COC, which serves as the control output, is obtained from visual tracking.