## 2 Materials and Methods
### Experimental Principles
To establish the relative coordinate system, we used the plasma bag centerline as the \(y^{1}\)-axis; then, we translated the previously calculated line AC to the contour corner mutation point, using this translated line as the \(x^{1}\)-axis, and set the supporting plane as the origin plane to obtain the relative coordinate system \(\left[x^{1},y^{1},z^{1}\right]\). The established coordinate system is shown in Figure 2.  
After the coordinate system was established, we used the floating coordinate method to design and analyze it. The endpoint position of the end input tube was point A, and its position in the global coordinate system \([X_{1},Y_{1},Z_{1}]\) was calculated using Equation (1), where D is the direction cosine matrix \(C_{A^{1}}^{A}\) of the relative coordinate system with respect to the global coordinate system; \(u_{0}\) is the position of the input tube endpoint A in the relative coordinate system before deformation; \(A1\left[x^{1},y^{1},z^{1}\right]\) is the position of the deformed input tube endpoint in the relative coordinate system, as shown in Figure 3; \(u_{f}\) is the deformation displacement of the input tube; and R is the position of point A when the input tube is undeformed in the global coordinate system. We then differentiated Equation (1) to obtain the motion velocity of point A, as shown in Equation (3), and differentiated Equation (3) again to obtain the motion acceleration of point A, as shown in Equation (4). The velocity of the plasma bag movement was known, and the time required for the residual-input tube to move without relative displacement was obtained using Equation (5). Once the time \({}^{\prime}t^{\prime}\) was determined, the system calculated the average of the images captured after time \({}^{\prime}t^{\prime}\) to obtain more accurate positional information. After the system obtained the initial residual-input tube endpoint information, the lead screw slide moved accordingly. When the averaged data were finally obtained, the  
Figure 3: Diagram of the plasma-bag input-tube movement. lead screw slide moved to the calculated position and waited for the residual-input tube endpoint to reach that point before initiating the cutting process. \[r=R+D\Big{(}u_{0}+u_{f}\Big{)} \tag{1}\]  
\[u_{f}=\sqrt{\Big{(}X_{1}-X_{1}^{1}\Big{)}^{2}+\Big{(}Y_{1}-Y_{1}^{1}\Big{)}^{2}+ \Big{(}Z_{1}-Z_{1}^{1}\Big{)}^{2}} \tag{2}\]  
\[\dot{r_{A}}=\ddot{R}+\dot{D}u_{f}+\dot{D}\dot{u}_{f} \tag{3}\]  
\[\ddot{r_{A}}=\ddot{R}+\ddot{D}u_{f}+2\dot{D}\dot{u_{f}}+D\ddot{u_{f}} \tag{4}\]  
\[t=\frac{rpm\times\pi\times r}{180^{\circ}\times\dot{r_{A}}} \tag{5}\]  
To calculate the waiting time required for the cutting blade, we determined the angle between line segments OA and OC. The radius of the plasma bag tray, which is known to be 650 mm, was used to calculate the additional rotation angle \(\alpha\) needed by the bottom motor, as shown in Equation (6). The bottom stepper motor rotated at a constant speed, and the number of steps required for the additional rotation angle was used to calculate the motor's forward steps. The step count of the stepper motor was set to n/360\({}^{\circ}\), and the additional rotation steps were determined using Equation (7). The system counted the pulse steps of the stepper motor, and when the required step count was reached, the electromagnet was activated to initiate the cutting process. \[\alpha=arcsin\frac{a}{650-b} \tag{6}\]  
Figure 2:0