## 2 Materials and Methods
### Innovative Design of Transplanting Mechanism
#### 2.2.2 Transplanting Manipulator  
As shown in Figure 6a, a combination of synchronous belt linear actuators was used to create the transplanting manipulator labeled X-axis and Y-axis matrixes in the Cartesian coordinate system. Two linear actuators of the same construction were mounted side by side in the Y-axis direction, connected through a long connecting shaft. Another linear actuator was installed at the X-axis direction, which was straightly hung onto the sliders of two Y-axis linear actuators. The gripper moving horizontally and longitudinally was fixed onto the slider of the horizontal linear actuator. Each linear actuator was supported by lightweight aluminium alloy wires, and two limit travel ends were installed with the flexible rubber cushion for the sake of reducing vibration. Moreover, two limit positioning sensors were allocated, and one start positioning sensor was used to detect the zero position. For the precise motion, each linear actuator was driven by a servo motor allocated with a high-performance servo driver. Besides, the upright direction was allocated with a self-locking commutator. After power-off, the Y-axis linear actuators could not spontaneously rise or fall. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & \multicolumn{5}{c}{\(F_{GA}\) **(N)**} & \multicolumn{2}{c}{\(F_{S1}\) **(N)**} \\ \cline{2-7}  & **Minimum Value** & **Maximum Value** & **Mean Value** & **Standard Deviation** & **\# (128-Cell)** & **\# (72-Cell)** \\ \hline
1 & 1.80 & 2.75 & 2.19 & 0.24 & 0.62 & 0.87 \\
2 & 1.55 & 2.68 & 2.21 & 0.32 & 0.63 & 0.88 \\
3 & 1.67 & 2.82 & 2.03 & 0.28 & 0.57 & 0.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of seedling pulling testing and computational analysis in picking seedlings. Figure 5: Mechanical tests of pulling seedlings under the quasistatic loading conditions. According to the working space of the robotic transplanting workcell, the total effective length of the transplanting manipulator should meet the following design conditions. \[\left\{ \begin{array}{l}L_{HD}\geq l_{1}+l_{2}+l_{3}+2\Delta l\\ L_{VD}\geq l_{4}+l_{5}+2\Delta l\end{array} \right. \tag{6}\]  
where \(L_{\rm HD}\) and \(L_{\rm VD}\) are the total lengths of the linear actuators in the horizontal direction and the vertical direction, respectively, mm; \(l_{1}\), \(l_{2}\), and \(l_{3}\) are the center distance of the corresponding group of flowerpots, the distance from the flowerpot center to the first stray, and the largest distance of the total ray cells, respectively, mm; \(l_{4}\) and \(l_{5}\) are the total height of plug seedlings and the height of a flowerpot exceeding the plug seedling, respectively, mm; \(\Delta l\) is the shared dimensions of the single-end installation and fixation in each actuator, mm. To meet the high-speed operation, the selection design of the linear actuator should be carried out under the calibration load of its slider (Figure 6b). The force-checking formula was expressed as follows. \[\lambda=\frac{F_{Y}}{F_{YMAX}}+\frac{F_{Z}}{F_{ZMAX}}+\frac{M_{X}}{M_{XMAX}}+ \frac{M_{Y}}{M_{YMAX}}+\frac{M_{Z}}{M_{ZMAX}}\leq 1 \tag{7}\]  
where \(\lambda\) is the maximum load coefficient of operation life in the linear actuators; \(F_{Y}\) and \(F_{Z}\) are the loads of the linear actuator sliders at different directions, respectively, N; \(F_{YMAX}\), \(F_{ZMAX}\) are the calibrated loads of the linear actuator sliders at different directions, respectively, N; \(M_{X}\), \(M_{Y}\) and \(M_{Z}\) are the load torques of the linear actuator sliders at different directions, respectively, N.m; \(M_{XMAX}\), \(M_{YMAX}\) and \(M_{ZMAX}\) are the calibrated load torques of the linear actuator sliders at different directions, respectively, N.m. It was assumed that the plug tray was transported along its length direction, and there were two growth pots waiting to be planted. Moreover, the outer diameter of the widely used growth pot was about 200 mm, and its vertical height was up to 300 mm.