# Control Design and Assessment for a Reversing Tractor-Trailer System Using a Cascade Controller
## 2 System Model  
A tractor-trailer system is a type of transportation system that consists of a tractor unit that is a heavy unit truck towing a trailer. Figure 1 shows a tractor-trailer system considered in this work. The tractor in this paper has a hitch located off-axis and the trailer is a single axis trailer, and the dynamical model is given by:  
\[\dot{x}=vcos(\theta_{1}) \tag{1}\]  
\[\dot{y}=vsin(\theta_{1}) \tag{2}\]  
\[\dot{\theta}_{1}=\mathit{t}\mathit{t}\mathit{a}(\varphi)/L_{1} \tag{3}\]  
\[\dot{\psi}=-(v(L_{2}\mathit{t}\mathit{a}(\varphi)+L_{1}\mathit{s}\mathit{i}( \psi)-\mathit{t}\mathit{a}(\varphi)\mathit{cos}(\psi)))/(L_{2}L_{1}) \tag{4}\]  
where \(v\) is the system longitudinal speed and \(L_{1}\) is the tractor's wheelbase that is measured from the front wheel to the rear wheel. The length from the tractor's hitch point to the rear axile is denoted by \(l\). \(L_{2}\) is the length from the trailer's axile to the hitch joint. Point M is the reference point that will be explained in Section 3. In Equations (1) and (2), \(\dot{x}\) and \(\dot{y}\) are the velocity of the model in the x and y Cartesian axis. \(\varphi\) represents the wheel steering angle while \(\theta_{1}\) is the heading angle of the tractor measured form the positive \(x\)-axis. \(\dot{\theta}_{1}\) is the tractor's heading rate and it is obtained using Equation (3). The articulated angle's rate, presented in Equation (4), represents the angle's rate between the tractor and trailer. Figure 1: tractor–Trailer System model. From Equations (3) and (4), the trailer's heading angle rate is calculated and is presented in Equation (5). In (6), \(\theta_{1c}\) and \(\theta_{2c}\) represent the angle of tractor and trailer's track, calculated from the body of the tractor and trailers to the \(x\)-axis, when moving in reverse (relative to the direction of the movement of the system) respectively. To achieve this, an assumption was made that there is no slip in the systems wheels. Using equations of rigid body motion, the velocity of point M was derived in Equations (7) and (8). \[\dot{\theta}_{2}=\dot{\psi}+\dot{\theta}_{1} \tag{5}\]  
\[\theta_{1c}=\theta_{1}-\ \pi,\theta_{2c}=\theta_{2}-\pi \tag{6}\]  
\[\dot{X}_{m}=\dot{x}\ -\dot{\theta}_{1}l\ sin(\theta_{1c})-\dot{\theta}_{2}L_{2} sin(\theta_{2c}) \tag{7}\]  
\[\dot{Y}_{m}=\dot{y}+\dot{\theta}_{1}l\ cos(\theta_{1c})+\dot{\theta}_{2}L_{2} cos(\theta_{2c}) \tag{8}\]  
Further necessary details about kinematics of the tractor and trailer system as well as the presented mathematical model can be obtained from [9; 14]. The maneuverability of the system is crucial in controller design requirements of the proposed control strategy. The system remains maneuverable if it has the ability to increase and decrease the articulated angle within the steering capability limits. Using equations developed in [9; 10], the limits in which the system remains maneuverable within the given steering limits \(\left[\underline{\varphi},\overline{\varphi}\right]\) are presented in Equations (9) and (10). \[\overline{\psi}_{M}=tan^{-1}(-L_{1})/(ltan(\overline{\varphi}))+cos^{-1}(tan( \overline{\varphi})/((l/(L_{2}tan(\overline{\varphi})))^{2}+(-L_{1}/L_{2})^{2 })^{-\frac{1}{2}}) \tag{9}\]  
\[\dot{y}=vsin(\theta_{1}) \tag{2}\]0  
Equation (9) gives the upper articulated angle limit and Equation (10) shows the lower articulated angle limit when the steering input is saturated. If the limits are exceeded the articulated angle will approach \(\pi\), in that case it will cause a jackknife incident. Theses given constraints and steering command limits are used in the development of the cascade controller especially when it comes to the tuning of the controller gains. Both the upper limit and the lower limit were imposed on the articulated angle feedback controller to ensure that the required articulation command does not exceed such limits. Table 1 shows a summary of the parameters used in Equations (1)-(10) with their units.