# Design and Test of Seeding-Picking Mechanism of Fully Automatic Transplanting Machine
## 3 Theoretical Calculation
### Analytical Calculations of the Seeding Extraction Mechanism  
During the seedling extraction process, the expansion and contraction of the electric actuator enable the forward and backward movement of the entire seedling extraction arm. Furthermore, the rotation of the electric rotary table enables the swinging motion of the seedling extraction arm. A mathematical model was established, as depicted in Figure 5, to derive the geometric relationships between the structural components of the seedling extraction system and the movement dynamics of the seedling claw [21]:  
This model primarily focuses on the expansion and contraction processes of the electric actuator, as well as the rotational dynamics of the electric rotary table. We establish a coordinate system with point \(B\), the center of the electric rotary table, as the origin. Point \(A\) represents the installation location of the seedling-picking claw, allowing the seedling-picking arm \(AB\) to rotate around point \(B\). During the seedling extraction operation, the electric actuator extends to position. \(D_{1}\), resulting in the seedling extraction arm being in the configuration \(A_{1}B_{1}\). At this stage, the seedling extraction arm is rotated \(\gamma_{1}\) degree clockwise, moving to position \(A_{2}B_{1}\), after which the electric actuator retracts from \(D_{1}E_{1}\) to position \(DE\). When the seedling extraction arm reaches the seedling casting position horizontally, the entire arm rotates \(\gamma_{3}\) degrees counterclockwise, positioning it at \(A_{3}B\), with point \(A_{3}\) directly above the seedling cup. In accordance with the established coordinate system, we define \(A_{1}B_{1}=A_{2}B_{1}=AB=A_{3}B=a\), and \(B_{1}B=b\), where the angle between the seedling arm and the horizontal direction during extraction is denoted as \(\gamma\).  
Figure 5: Model of the movement of the seedling-picking mechanism.  
\[\gamma=180^{\circ}-\alpha \tag{3}\]  
where \(\gamma\) is the initial angle of the seedling arm, and \(\alpha\) is the angle between the bottom of the cavity tray and the ground surface, Taking the seedling arm as the object of study, the coordinates of point \(A_{1}\) can be expressed as:  
\[\left\{\begin{array}{l}x_{A1}=l_{1}\cos\gamma+b\\ y_{A1}=l_{1}\sin\gamma\end{array}\right. \tag{4}\]  
where \(l_{1}\) denotes seedling arm length (mm). Taking the seedling arm rotation angle \(\gamma\) the clamping jaws clamp the hole tray seedlings out of the hole tray, to reach the \(A_{2}B_{1}\) position; at this time, \(A_{2}\) coordinates can be expressed as:  
\[\left\{\begin{array}{l}x_{A2}=l_{1}\cos\gamma_{2}+b\\ y_{A2}=l_{1}\sin\gamma_{2}\end{array}\right. \tag{5}\]  
where \(\gamma_{2}\) is the angle between taking the seedling arm and the horizontal direction after rotating the \(\gamma\) angles. When the overall contraction of the seedling-picking arm is backward, to reach the pre-projection position, the overall angle of the seedling-picking arm remains unchanged; at this time, the coordinates of point \(A\) can be expressed as:  
\[\left\{\begin{array}{l}x_{A}=l_{1}\cos\gamma\\ y_{A}=l_{1}\sin\gamma\end{array}\right. \tag{6}\]  
when the seedling feeding operation is carried out, the seedling-picking arm turns the angle counterclockwise and reaches the \(A_{3}B\) position; at which time, the \(A_{3}\) coordinate can be expressed as:  
\[\left\{\begin{array}{l}x_{A3}=l_{1}\cos(\gamma_{3}-\gamma_{2})\\ y_{A3}=l_{1}\sin(\gamma_{3}-\gamma_{2})\end{array}\right. \tag{7}\]  
Knowing the displacement equation for the end pickup point of the pickup arm, the second-order derivatives of Equations (4)-(7) are carried out at time \(t\), respectively, to obtain the expression for the acceleration at the end pickup point as:  
\[\left\{\begin{array}{l}\ddot{x}_{A1}=-l_{1}\cos\gamma\cdot\left(\dot{\gamma }\right)^{2}-l_{1}\sin\gamma\cdot\ddot{\gamma}\\ \ddot{y}_{A1}=-l_{1}\sin\gamma\cdot\left(\dot{\gamma}\