# The Tractor-Cart System Controller with Fuzzy Logic Rules
## 7 Conclusions  
The equation of motion of the tractor-cart system was modeled using a kinematic bicycle model approach. Furthermore, to simplify the system models for small angular and low speed motions, Taylor Series expansion was introduced to substitute trigonometric functions in the system model. An 81 Fuzzy Logic (FL) rules-based controller for the tractor steering and cart hydraulic hitch was developed. The FL rules were designed to compute an adequate value of the steering angle and required flow rate from pump to hydraulic cylinders of cart's hitch to change the position of cart and control its direction with respect to the actual row pattern of plants. The steering algorithm and designed controller were validated with different trajectory patterns via simulations in MATLAB/Simulink. The simulation results showed that the proposed fuzzy logic controller improves the behavior of cart up to 96% in lower driving speeds up to 5 m/s and up to 72% in higher running speeds 5 to 10 m/s in comparison to a passively controlled system. The best performance of the controller was observed in 1-4 m/s in a \"sinusoidal\" pattern, which is a running speed in most agricultural operations. Therefore, it was concluded that the computed values of controller parameters for \(u_{1}(\delta)\) and \(u_{2}(\alpha)\) with the FL system were appropriate (for low speed and simple path) with respect to the computed error margins of the system trajectory. However, the main disadvantage of the fuzzy logic system is, for higher accuracy, more fuzzy rules are required to be designed, which require evaluation of the optimal rule values before hardware design.  
The simulation results of the new hydraulic drive design showed that the retracting and extending speed of the hydraulic cylinder with respect to the provided flow rate from tractor's hydraulic pump was adequate to turn the hitch arm with respect to the FL based controller output. The selected cylinder size was sufficient to meet the system requirements and taking the action with acceptable small-time delays in responses.  
One of our future studies will be dedicated to implementing our proposed controller of the hitch arm to design a hydraulic cylinder and pump system with respect to the overall system parameters and field conditions. In the future studies, terrain and environment aspects will be considered in the system model equations. Another crucial study is to compare the result of designed fuzzy inference system (FIS) controller in this study with other common adaptive and model predictive controllers with applications of evolutionary algorithms, ANN, ANFIS-PSO, etc. or combination of these methods to validate and improve the performance of the proposed controller.  
[PERSON] conducted the experiments and simulation, designed experiments, and developed control algorithm and control strategy, analyzed data, wrote the manuscript, and edited it further. [PERSON] was responsible for overall supervision, prepared the structure of the study, developed the control algorithm and control strategy, system equations of motion and simulation models, reviewed the results, and edited the manuscript. [PERSON] validated the results and edited of manuscript. [PERSON] obtained the funding, wrote the project proposal, and corrected and edited the manuscript. [PERSON] corrected and edited the manuscript. All authors have read and agreed to the published version of the manuscript.  
Coordinated Agricultural Program, award No. 2016-69004-24784, and the North Dakota Soybean Council.  
This research was possible thanks to the funding provided by the North Dakota State University Agricultural Experimental Station project ND 01557, the USDA-NIFA award No. 2016-69004-24784, \"CropSys--A novel management approach to increase productivity, resilience, and long-term sustainability of cropping systems in the northern Great Plains\".  
The authors declare no conflicts of interest.