## 2 Design of Path Tracking Controller with Constraints on Tire Slip Angles  
### Vehicle Model  
In the context of PTC, a 2-DOF bicycle model has been widely employed to represent yaw and lateral behaviors of a vehicle, as discussed in the literature [25; 30; 31; 32; 33; 34; 35; 39]. Figure 2 shows the coordinates, geometry, and variables of the bicycle model. This models the lateral and yaw motions of a vehicle under the assumption of a constant longitudinal velocity. The state variables for this model are the yaw rate, \(\gamma\), and the side-slip angle, \(\beta\). With these variables, the equations of motion (1) are obtained [32]. The tire slip angles, denoted as \(\alpha_{f}\) and \(\alpha_{r}\), are defined as Equation (2) for the front and rear wheels. The linear lateral tire forces, \(F_{\mathit{off}}\) and \(F_{\mathit{yr}}\), are calculated as Equation (3). By combining Equations (1)-(3), the state-space equation for the bicycle model is derived as Equation (4). \[\left\{ \begin{array}{l}mv_{x}\Big{\{}\dot{\beta}(t)+\gamma(t)\Big{\}}=F_{ \mathit{yr}}(t)+F_{\mathit{yr}}(t)\\ I_{z}\dot{\gamma}(t)=l_{f}F_{\mathit{yr}}(t)-l_{r}F_{\mathit{yr}}(t)+\Delta M_ {z}(t)\end{array} \right. \tag{1}\]  
\[\alpha_{f}(t)=\delta_{f}(t)-\beta(t)-\frac{l_{f}\gamma(t)}{v_{x}},\;\alpha_{r} (t)=\delta_{r}(t)-\beta(t)+\frac{l_{r}\gamma(t)}{v_{x}} \tag{2}\]  
\[F_{\mathit{yr}}(t)=C_{f}\alpha_{f}(t),\;F_{\mathit{yr}}(t)=C_{r}\alpha_{r}(t) \tag{3}\]  
Figure 2: 2-DOF dynamic bicycle model. \[ \begin{split}\dot{\beta}(t)&=\left(-\frac{C_{f}+C_{r}}{m \varpi_{x}}\right)\beta(t)+\left(-\frac{C_{f}l_{f}+C_{l}l_{r}}{m\varpi_{x}^{2}}-1 \right)\gamma(t)+\frac{C_{f}}{m\varpi_{x}}\delta_{f}(t)+\frac{C_{r}}{m\varpi_{ x}}\delta_{f}(t)\\ \dot{\gamma}(t)&=\left(-\frac{C_{f}l_{f}+C_{l}l_{r}}{l _{z}}\right)\beta(t)+\left(-\frac{\frac{C_{f}l_{f}+C_{l}l_{r}}{l_{z}\varpi_{x}}}{ l_{z}\varpi_{x}}\right)\gamma(t)+\frac{l_{f}C_{f}}{l_{z}}\delta_{f}(t)-\frac{l_{f}C_{ r}}{l_{z}}\delta_{f}(t)+\frac{1}{l_{z}}\Delta M_{z}(t)\end{split}  \tag{4}\]  
Figure 3 shows the coordinates and variables derived from the target path for path tracking. The literature on PTC introduced the concepts of lateral offset and heading errors, denoted as \(e_{y}\) and \(e_{\varphi}\), at a specific point \(C\) as depicted in Figure 3[11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. To enhance path tracking performance, this paper introduces a preview function, as used in the previous works [24; 26; 30; 31; 32]. The preview distance, \(L_{p}\), is calculated using Equation (5), where \(k_{\varphi}\) is the velocity gain representing the preview interval. In Figure 3, the point C is located at the vehicle's center of gravity. The preview point, \(P\), and the point \(Q\) on the target path are determined with \(L_{p}\). At the point \(Q\), the time derivatives of \(e_{y}\) and \(e_{\varphi}\) are calculated as Equation (6), under the assumption that \(e_{\varphi}\) is limited to an angle less than 10 degrees and, consequently, can be approximated as \(\text{sine}_{\varphi}\approx e_{\varphi}\). \[L_{p}=k_{\varphi}\cdot v_{x} \tag{5}\]  
\[\left\{ \begin{array}{l}\dot{e}_{y}(t)=v_{x}\sin e_{\varphi}(t)-v_{x}\beta(t )-L_{p}\gamma(t)\approx v_{x}e_{\varphi}(t)-v_{x}\beta(t)-L_{p}\gamma(t)\\ \dot{e}_{\varphi}(t)=\dot{\varphi}_{d}(t)-\dot{\varphi}(t)=v_{x}\chi(t)- \gamma(t)\end{array} \right. \tag{6}\]  
The vectors \(\mathbf{x}\), \(\mathbf{w}\), and \(\mathbf{u}\) of the states, the disturbances, and control inputs are defined as Equation (7), respectively [11; 12; 14; 15; 16; 17; 19; 20; 21; 23; 28; 29; 31; 32]. With those vectors and Equations (4), (6) and (7), the state-space equation for PTC is derived as Equation (8), where the matrices \(\mathbf{A}\), \(\mathbf{B}_{1}\) and \(\mathbf{B}_{2}\) are defined as Equation (9) [32]. \[\left\{