# The Tractor-Cart System Controller with Fuzzy Logic Rules
## 2 Tractor and Cart System Modeling
### Kinematic Bicycle Model  
The equation of motion of a vehicle is a combination of different assumptions. Engineering challenges in designing a control system are to choose the simplest possible model whose assumptions are sufficiently valid to allow accurate, robust, and/or adaptive control system design [23]. In this study, the tractor-cart behavior was studied in two different situations: (1) the cart is a follower and there is no active steering control over its movements; (2) the cart is actively guided. The proposed tractor-cart kinematic model presented in Figure 1 is composed of two parts: a tractor and cart, and the state variables of the system model. The information is presented in Figure 1 and given in Table 1.  
At the end of rows, while turning the two front wheels of a tractor, different angles relative to tractor's body are created. The first assumption in this kinematic bicycle model approach was to combine the effects of the two steering wheels and consider them parallel to each other by assuming \(\delta_{l}\cong\delta_{r}\). In other words, the front wheels were considered as a single unit. The same assumption was also made for rear wheels. Figure 2 shows the proposed kinematic bicycle model under the following assumptions: (1) the front wheels are the only steering wheels, and rear wheels (rear axle) are followers and the velocity vectors, at the front and rear axles of tractor, are in the direction of the orientation of the front and rear wheels, respectively; (2) the second assumption is that the center of the gravity (\(CG_{c}\)) of the cart resides on the axle of the cart. The definition of all terms used in Figure 2 are given in Table 1.  
According to the assumptions stated above, the kinematic bicycle model of the tractor-cart system is shown in Figure 2. The equations of motion of the tractor are expressed in (1) and (2):  
\[\left[ \begin{array}{c}\dot{X}_{t}\\ \dot{Y}_{t}\\ \dot{\varphi}_{t}\end{array} \right]=\left[ \begin{array}{c}V_{t}\cos[\varphi _{t}+\beta]\\ V_{t}\sin[\varphi_{t}+\beta]\\ \frac{V_{t}\cos\beta\tan\delta}{L_{t}}\end{array} \right] \tag{1}\]  
\[\beta=\arctan\Bigl{(}\frac{L_{r}\tan\delta}{L_{t}}\Bigr{)} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Definition of Parameters** \\ \hline \(CG_{t}\), \(CG_{c}\) & Center of gravity of tractor and cart \\ \(\delta_{f}\), \(\delta_{r}\) & Steering angles of the front wheels of the tractor \\ \(V_{t}\) & Velocity of tractor \\ \(\varphi_{t}\), \(\varphi_{c}\) & Heading angle of tractor and cart \\ \(\dot{\varphi}_{r}\), \(\dot{\varphi}_{c}\) & Yaw rate of tractor and cart \\ \(\alpha\) & Angular displacement of the hitch arm \\ \(\dot{\alpha}\) & Angular velocity of the hitch arm \\ \(\beta\) & Slip angle at the center of gravity (\(CG\)) of the tractor \\ \(L_{f}\) & Distance from the \(CG\) to the front axle of the tractor \\ \(L_{r}\) & Distance from the \(CG\) to the rear axle \\ \(L_{t}\) & Tractor wheelbase distance (\(L_{f}+L_{r}\)) \\ \(L_{c}\) & Distance from the hitch to the cart’s \(CG\) \\ \(R\) & Hitch’s arm length \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tractor and cart kinematic bicycle model parameters definition. Figure 1: Schematic representation of tractor-cart system. The angular velocity of the cart without a controller is expressed in Equation (3), and with a controller in Equation (4):  
\[\dot{\varphi}_{c}=\frac{V_{t}\cos\beta\sin(\varphi_{t}-\varphi_{c})-R\dot{\varphi }_{t}\cos(\varphi_{t}-\varphi_{c})}{L_{c}} \tag{3}\]  
\[\dot{\varphi}_{c}=\frac{V_{t}\cos\beta\cos\alpha\sin(\varphi_{t}-\alpha- \varphi_{c})-R\dot{\alpha}\cos(\varphi_{t}-\alpha-\varphi_{c})}{L_{c}} \tag{4}\]  
In this model, the front wheels were assumed to be the steering wheels, taking the input control signal \(u_{1}(\delta)\) to control (steer) the motion of the tractor. The main task of the anticipated controller was to make the tractor follow the reference path as closely as possible.