## 2 Structural Design
### Designing to Prevent Damage Seedlings  
After calculating the transplanting manipulator's movement trajectory, the vertical heights of the three key points are used to confirm the movement strokes at various phases of the transplanting manipulator, as shown in Figure 7. The three key points include the location of clamping seedlings, the a value, and the b value.  
Figure 6: The part of seedling’s stalk in the soil.  
Figure 7: The trajectory corresponding to each movement stage of the cam.  
The cam only controls the opening and closing of the transplanting manipulator and has no effect on the transplanting manipulator's moving direction. Before designing the cam contour curve, according to the transplanting manipulator's movement stroke, the cam edge contour curve is designed with four zones combined with the crank rotation angle: push zone, far rest zone, return zone, and near rest zone. The blue and red strokes correspond to the far rest zone and near rest zone, respectively. The black stroke corresponds to the return zone. The yellow stroke corresponds to the push stroke. The design of the return zone is the most important [25]. To prevent seedlings from being damaged by the gripping levers, the impact force at the moment of clamping seedlings must be zero. Therefore, at the end of the return zone, the moving speed and acceleration of the follower are zero, and the clamping force is only provided by the spring. To summarize, the motion law equation of the follower of the cam mechanism during the cam return stroke is designed as follows.  
The displacement function of the cam mechanism's follower can be expressed as.  
\[s=s(\varphi) \tag{11}\]  
where \(s\) is the linear motion displacement of the follower motion component, mm. \(\varphi\) is the cam angle variable and \(\varphi e(0,\phi)\), rad.  
According to the displacement function, the velocity and acceleration of the follower moving part are:  
\[\left\{\begin{array}{c}\frac{d_{s}}{d_{\varphi}}=\frac{\varphi}{\omega}mm/ rad\\ \frac{d_{s}}{d_{\varphi}^{2}}=\frac{\varphi}{\omega^{2}}mm/rad^{2}\end{array}\right. \tag{12}\]  
where the angular velocity of cam rotation is \(\omega\), rad/s.  
Let the displacement function be:  
\[s=c_{0}+c_{1}\varphi+c_{2}\varphi^{2}+c_{3}\varphi^{3}+c_{4}\varphi^{4}+c_{5} \varphi^{5}+\ldots+c_{n}\varphi^{n} \tag{13}\]  
Let the constants higher than the fifth-order term in Equation (13) be zero, and the boundary condition of the cam return zone can be obtained as follows:  
\[\left\{\begin{array}{c}\varphi=0,\ \ s=0,\ \ \frac{d_{s}}{d_{\varphi}}=0,\ \ \frac{d_{s}^{2}}{d_{\varphi}^{2}}=0\\ \varphi=\phi,\ \ s=h,\ \ \frac{d_{s}}{d_{\varphi}}=0,\ \ \frac{d_{s}^{2}}{d_{ \varphi}^{2}}=0\end{array}\right. \tag{14}\]  
The equation of motion for the return zone can be obtained as:  
\[\left\{\begin{array}{c}s=h\bigg{[}10\Big{(}\frac{\varphi}{\phi}\Big{)}^{3}- 15\Big{(}\frac{\varphi}{\phi}\Big{)}^{4}+6\Big{(}\frac{\varphi}{\phi}\Big{)} ^{5}\bigg{]}\\ \frac{d_{s}}{d_{\varphi}}=\frac{30h}{\phi}\bigg{[}\Big{(}\frac{\varphi}{\phi} \Big{)}^{2}-2\Big{(}\frac{\varphi}{\phi}\Big{)}^{3}+\Big{(}\frac{\varphi}{ \phi}\Big{)}^{4}\bigg{]}\\ \frac{d^{2}z}{d_{\varphi}^{2}}=-\frac{60h}{\phi^{2}}\bigg{[}\frac{\varphi}{ \phi}-3\Big{(}\frac{\varphi}{\phi}\Big{)}^{2}+2\Big{(}\frac{\varphi}{\phi} \Big{)}^{3}\bigg{]}\end{array}\right. \tag{15}\]  
where \(h\) is the descending height of the follower when the cam is working in the return zone.  
Figure 8 shows the curve graph of the movement rule. The speed and acceleration of the quintic equation are all zero at the end of the return zone, and there is no sudden change in their curves. Therefore, this exercise law meets the requirements of low-damage seedlings.