# The Tractor-Cart System Controller with Fuzzy Logic Rules
## 6 Results and Discussion
[35] indicated that controllers of tractor-cart systems are quite effective at low speeds, i.e., up to 10 km/h (2.77 m/s) in maize, where the course is limited to straight rows, and the system could only be used on curved rows in slower speeds. Figure 13: Performance of tractor-cart position control system under a Zigzag pattern at 5 [m/s]. The accuracy of the controller in finding the right path was calculated. The chosen range for the \(x\)-axis was \(-10\) cm to \(10\) cm and \(-1\) cm to \(1\) cm for the \(y\)-axis. Any reading beyond these ranges was not counted as an accurate reading (Table 6). The proposed control model performances were compared with the results of other studies in a similar domain. Table 7 reports the comparison results of the studies reported by other researchers and the present work. Most of the proposed controllers in other studies for off-road vehicles were only able to control the tractor-trailer system up to 4.47 m/s. The proposed controller in this study could correct the position of cart accurately up to 5-7 m/s depending on the complexity of the reference pattern, which compared to other studies in Table 7, is a very good improvement. In most of the previous studies, the running speed of the tractor was considered as a major limitation of the controller, while the proposed fuzzy logic controller in this study is able to control the tractor-cart system up to 5-7 m/s depending on the complexity of the reference pattern. Figure 14: Performance of tractor-cart position control system under a Mixed pattern in 2 [m/s]. Figure 15: Performance of tractor-cart position control system under a Mixed pattern in 4 [m/s]. to 7 m/s. Another advantage of the proposed model is that it is simple and efficient in terms of the computation and processing time for optimal and adaptive control. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controlizer** & **Max Speed** & **Reported Accuracy (cm) and Errors** & **Reference** \\ \hline Controlling tractor-trailer & & Maximum displacement in position in & \\ with feed-forward + & & backward motion: 14 cm for & \\ feed-back based on & 0.33 m/s & pole-placement, 9 cm for optimal control. & [PERSON] et al., 2005 [96] \\ pole-placement technique & & No significant difference in final positional & \\ and optimal control & & error between the two feedback types. & \\ \hline Controlling tractor with & & Maximum lateral and heading error in & \\ feed-back rapid prototyping & & Straight line trajectory: 0.1026 (m), & \\ controller based on Iterative & 2.23 m/s and 4.47 m/s & 1.1023 (deg). And for curve trajectory: & \\ Learning identification (ILI) & & 0.8384 (m), 0.4178 (deg) at 4.44 m. Model & \\ model of tractor & & identified is accurate enough for the & \\ \hline Controlling tractor-trailer & & Standard deviation of the errors smaller & \\ with Nonlinear Model & & than 10 cm. Active joint in trailer drawbar & \\ Predictive Control (NMPC) & 2.22 m/s & was found to be valuable and NMPC was & [PERSON] et al., 2009 [88] \\ based on a full kinematic & & able to smoothly control curves. & \\ model of the tractor-trailer & & Mean value of Euclidian error with respect to the straight-line trajectory is 3.33 cm and & \\ \hline Controlling tractor-trailer & & 3.22 cm for tractor and trailer respectively: & \\ nonlinear model predictive & & Capable of driving tractor–trailer system to & \\ control algorithm & & any desired trajectory with high accuracy & \\  & & and robustness. & \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline Controling tractor-trailer & & Controls the cart motion with the accuracy \\ with an adaptive fuzzy logic & & of up to 90\% of the trajectory in lower \\ controller using the & & driving speeds up to 5 m/s with the \\  & & maximum offset of 3 cm and accuracy of up \\  & & to 72\% of trajectory in higher running speeds from 5 to 10 m/s with the maximum \\  & & error of 10 cm in magnitude. \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Coverage percent (without controller/with controller).