## 2 System Model  
Considering that tractor-trailers often operate at low speeds in agricultural scenarios, which are significantly impacted by kinematic constraints, the simplified kinematic model depicted in Figure 1 is employed in this research, where an articulated angle is used to connect the tractor and the trailer, the reference path is regarded as the \(X\) axis, and \(P_{i(i=0,1,2)}\) denote the targeted point, the articulated point, and the center point of the trailer axle, respectively. The other parameters can be found in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Description** \\ \hline \((x,y,\theta)\) & Position at the center of the rear axle of the trailer \\ \(v\) & Velocity of tractor-trailer \\ \(w\) & Angular velocity of trailer \\ \(\varphi\) & Articulated angle \\ \(L_{(i=1,2,3)}\) & Vehicle lengths \\ \(\delta\) & The front wheel steering angle \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters in the tractor-trailer’s kinematic model.  
Based on the analysis of the center of the rear axle of the trailer, the kinematic model of the tractor-trailer can be obtained as follows:  
\[\left\{\begin{array}{l}\dot{x}=v\text{cos}\theta\\ \dot{y}=v\text{sin}\theta\\ \dot{\varphi}=w=\frac{\text{tan}\delta}{L_{1}}\\ \dot{\varphi}=-\frac{v}{L_{2}}\text{sin}\varphi-\frac{v}{L_{1}L_{2}}(L_{3} \text{cos}\varphi+L_{2})\text{tan}\delta\end{array}\right. \tag{1}\]  
It should be noted that the model described in (1) is a typical high-order underactuated nonlinear system that is challenging to directly control. To simplify the analysis in this study, a desired straight line was chosen, which can be represented as  
\[\alpha_{1}x+\alpha_{2}y+\alpha_{3}=0 \tag{2}\]  
where \(\alpha_{i(i=1,2,3)}\) denote the geometric constants of the desired path, and they also satisfy \(\alpha=\alpha_{1}^{2}+\alpha_{2}^{2}\
eq 0\).  
An error variable and an error transform are presented in this study to address the problem of underactuation. The authors define the new error variable as  
\[\xi=\alpha_{1}x+\alpha_{2}y+\alpha_{3} \tag{3}\]  
Taking the derivate of (3), one can obtain  
\[\dot{\xi}=v\alpha\Big{(}\frac{\alpha_{1}}{\alpha}\text{cos}\theta+\frac{ \alpha_{2}}{\alpha}\text{sin}\theta\Big{)} \tag{4}\]  
Further, based on the desired straight line, the authors define  
\[\text{cos}\theta_{d}=\frac{\alpha_{2}}{\alpha},-\text{sin}\theta_{d}=\frac{ \alpha_{1}}{\alpha} \tag{5}\]  
Based on (5), Equation (4) can be further formulated as  
\[\dot{\xi}=v\text{asin}(\theta-\theta_{d}) \tag{6}\]  
Figure 1: Schematic of tractor-trailer path tracking.  
Defining \(\theta=\theta-\theta_{d}\) and combining it with (1) yields  
\[\left\{\begin{array}{l}\dot{\frac{\dot{\zeta}}{\dot{\zeta}}}=v\alpha\text{sin} \theta\\ \theta=\frac{v}{L_{1}}u\\ \dot{\vartheta}=-\frac{v\text{sin}\theta}{L_{2}}-v\Big{(}\frac{L_{3}}{L_{1}L _{2}}\text{cos}\varphi+\frac{1}{L_{1}}\Big{)}u\end{array}\right. \tag{7}\]  
where \(u=\text{tan}\delta\) denotes the control input.  
**Remark 1**: _The tractor-trailer path tracking error model (7) was derived from the kinematic model and a small angle assumption, which is suitable for low-speed straight-line tracking control. Nevertheless, the error model incorporating path curvature into tracking control is more significant for some curved paths._