## 2 System Model  
A tractor-trailer system is a mode of transportation where a heavy truck pulls a trailer, mainly to carry freight. The tractor in this particular study has a hitch located off-center and the trailer has only one axis. We considered a kinematic model that captures the positional capabilities and constraints of the system to design a smooth sliding mode controller. In this paper, we utilize a control-oriented model for the tractor-trailer, conceptualized as a double bicycle model, drawing inspiration from the models presented in [8; 37]. The tractor's design emulates a light-duty truck, with its hitch positioned at the rear, just past the back bumper, connecting to a trailer with a single axle. The dynamics of this model are outlined in the subsequent equations:  
\[\dot{x}_{1}=v\cos\theta_{1} \tag{1a}\] \[\dot{y}_{1}=v\sin\theta_{1}\] (1b) \[\dot{\theta}_{1}=v\tan\varphi/L_{1} \tag{1c}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**References** & **Methods** & **Remarks** \\ \hline
[1; 3; 4] & Path following for forward motion & The problem of stability in forward motion arises at high speed. \\ \hline
[9; 10; 11; 12] & Neural network techniques & For reverse (backward) motion. \\ \hline
[16; 17; 18] & Fuzzy Control and Systems & It does not depend on precise mathematical models \\ \hline
[19; 20] & Model Predictive Control & Compensates for the uncertainty of the system \\ \hline
[25; 26; 27; 28; 29; 30; 33; 34] & Sliding Mode Control & Robust Control for systems subject to uncertainties and disturbance. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature summary. \[\dot{\psi}=-\frac{v\tan\varphi}{L_{1}}-\frac{v(L_{1}\sin\psi+\text{sgn}(v)l\tan \varphi\cos\psi)}{L_{1}L_{2}} \tag{1d}\] \[\dot{\theta}_{2}=-\frac{v(L_{1}\sin\psi+\text{sgn}(v)l\tan\varphi\cos\psi)}{L_{1}L_ {2}} \tag{1e}\]  
The parameters pertinent to path tracking control in the tractor-trailer vehicle model are illustrated in Figure 1. Here, \(\theta_{1}\) and \(\theta_{2}\) signify the heading angles of the tractor and the trailer, respectively, as defined by the Earth coordinate frame. The articulation angle is expressed as \(\psi=\theta_{2}-\theta_{1}\), representing the difference between the two. The front wheel steering angle of the tractor, denoted as \(\varphi\), serves as the control variable in this scenario and is restricted to \(\varphi\in\left[\underline{\varphi},\overline{\varphi}\right]\). In this study, all angular state variables are regarded as positive in the counterclockwise direction. In a control-oriented model focused on low-speed scenarios, the steering kinematics of a tractor-trailer vehicle are determined by four distinct parameters: the wheelbase of the tractor vehicle, \(L_{1}\), spanning from the front to the rear axle; the wheelbase of the trailer vehicle, \(L_{2}\), extending from the hitch to the trailer's axle; the hitch length, \(l\), measured from the tractor's rear axle to the hitch; and the longitudinal speed of the tractor, \(v\), with negative values representing reverse motion. The used parameters and symbols are summarized in Table 2. The rear axle center location of the tractor, represented as \((x_{1},y_{1})\), in the Earth coordinate frame, summarizes the control system model, as denoted in (1). It should be noted that the \(sgn\) operator is utilized to alternate the model between forward and backward driving modes. **Definition 1**. : _The system is said to be maneuverable within a maneuverability range \((\underline{\psi}_{M},\overline{\psi}_{M})\) if \(\forall\psi\in(\underline{\psi}_{M},\overline{\psi}_{M})\), \(\exists\varphi_{1},\varphi_{2}\in[\underline{\varphi},\overline{\varphi}]| \dot{\psi}(\varphi_{1})>0\), \(\dot{\psi}(\varphi_{2})<0\)._  
**Proposition 1**.