# The Tractor-Cart System Controller with Fuzzy Logic Rules
## 5 Hydraulic Cylinder Design  
To design a hydraulic actuator for hitch, the required hydraulic flow rates to pull/push the hitch arm and change the angle of cart to correct its position was taken with respect to the Case IH Magnum 310 with maximum 380 hp and 44 GPM. The dimensions and weight of the cart was taken from Amity ST250 Air Cart (Amity technology, Fargo, ND, USA) with a capacity of 200 bushels (up to 7 tons). According to the tractor power and cart specifications, a chosen hydraulic cylinder should be capable of changing the direction of hitch arm within a required range (considering the distance between plant rows). The parameters of the selected hydraulic cylinder are given in Table 4. The linear relationship between the cylinder rod's linear motion and angular displacement (\(\alpha\)) of the hitch arm is shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Bore \(\times\) Stroke** \\ **(inch)** \\ \end{tabular} & \begin{tabular}{c} **Shaft Dia** \\ **(inch)** \\ \end{tabular} & \begin{tabular}{c} **Retracted** \\ **Length (inch)** \\ \end{tabular} & \begin{tabular}{c} **Extended** \\ **Length (inch)** \\ \end{tabular} & \
\begin{tabular}{c} **Rated Pressure** \\ **(PSI)** \\ \end{tabular} &
\begin{tabular}{c} **Max Pressure** \\ **(PSI)** \\ \end{tabular} \\ \hline \(2\times 6\) & \(1\) 1/8 & 16 1/4 & 22 1/4 & 2500 & 3850 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The hydraulic cylinder specifications.  
Figure 7: Membership functions of the input and output variables of the cart controller.  
A fully retracted length of the chosen cylinder is 41.2 cm, and its fully extended length is 56.51 cm. The cylinders move in parallel with the hitch arm. Therefore, the length of the cylinders in the normal situation (L\({}_{\rm H}\)) is equal to 50 cm. The maximum and minimum stretched length of the cylinders (left or right) with respect to \(\alpha\) values is \(L=[17.36 in,21.13 in]=[44.11\) cm, 53.7 cm].  
The required flow rate change with respect to the length of the cylinder was calculated from Equation (21):  
\[\text{Required GPM}= \frac{0.016\times\text{Cylinder Area }(\text{cm}^{2})\times\text{ Stroke Length }(\text{cm})}{\text{ Stroke Time}(s)} \tag{21}\]  
where Stroke Length is extension or retraction of the cylinder rod; Cylinder Area for retraction is the area of rod end (13.87 cm\({}^{2}\)) and for extension is the area blind end (20.25 cm\({}^{2}\)). Since the sampling frequency in this study was 20 Hz, the Stroke Time is 0.05 s. For stroke time = 0.05 s, the maximum flow rate was required to change \(\alpha\) from \(-0.5\) to 0.5 rad or change the cylinder length from fully extended to fully retracted. For this cylinder with a maximum flow rate of 44 GPM, the speed of moving from fully retracted to fully extended is 53.95 in/s = 1.37 m/s and for moving from fully extended to fully retracted is 78.79 in/s = 2 m/s. Figure 9 shows the relationship between the required flow rate and angular displacement rate of the hitch arm \((\Delta\alpha)\). For double acting cylinders, the left side of the plot (Figure 9) represents retraction and the fluid flows through rod's end hose. The right side of the plot (Figure 9) represents extension where the oil flows through the blind end hose.  
Figure 8: Relation between \(\alpha\) vs. cylinder’s rod translational motion.  
Figure 9: Relationship between changes of hitch arm angle vs. required flow rate.