## 2 Steering and Vehicle Systems Modeling
### Dynamic Model of Vehicle Tires  
To seamlessly integrate the vehicle dynamics framework with the steering system, a 2-DOF vehicle bicycle model is employed to elucidate the lateral dynamics during the path following process. The simplified structure is depicted in Figure 2. Assuming small angles for \(\delta\), specifically \(\cos\delta\approx 1\) and \(\sin\delta\approx\delta\), the longitudinal velocity \(v_{x}\) is considered constant, and the lateral acceleration \(\dot{v}_{y}\) and yaw acceleration \(\dot{\varphi}\) of the 2-DOF vehicle are expressed as follows:  
\[\dot{v}_{y} =\frac{F_{yf}\Big{(}\alpha_{f}\Big{)}}{m}+\frac{F_{yf}(\alpha_{r} )}{m}-v_{x}\dot{\varphi} \tag{5}\] \[\ddot{\varphi} =\frac{aF_{yf}\Big{(}\alpha_{f}\Big{)}}{I_{z}}-\frac{bF_{yf}( \alpha)}{I_{z}}\]  
where \(m\) is the mass of the vehicle, \(\beta\) is the sideslip angle of the center of mass of the vehicle, \(I_{z}\) is the moment of inertia of the vehicle, and \(v_{x}\) and \(v_{y}\) are the longitudinal velocity and lateral velocity of the vehicle, respectively. \(F_{yf}\) and \(F_{yr}\) are the lateral forces of the front and  
Figure 1: Vehicle steering system architecture. rear tires, respectively, and \(\alpha_{f}\) and \(\alpha_{r}\) are the lateral slip angles of the front and rear tires, respectively, expressed as  
\[\alpha_{f} =\frac{v_{y}+a\dot{\varphi}}{v_{x}}-\delta \tag{6}\] \[\alpha_{r} =\frac{v_{y}-b\dot{\varphi}}{v_{x}}\]  
where the distances from the center of the front and rear axes of the wheel to the center of the body are denoted as \(a\) and \(b\), respectively. In order to clarify the impact of tire models on vehicle dynamics, handling stability, braking safety, and ride comfort, this study uses the Fiala tire model to capture the nonlinear features of automotive tires [30]:  
\[F_{y}= \begin{cases}-\mu_{s}F_{z}\bigg{[}1-\Big{(}1-\frac{C_{\alpha}|\tan\alpha _{s}|}{3\mu_{s}F_{z}}\Big{)}^{3}\bigg{]}\operatorname{sgn}(\alpha_{s}),&| \alpha_{s}|<\alpha_{sl}\\ -\mu_{s}F_{z}\operatorname{sgn}(\alpha_{s}),&|\alpha_{s}|<\alpha_{sl}\end{cases}  \tag{7}\]  
where \(F_{y}\) represents the lateral force on the tire, \(\alpha_{s}\) is the tire slip angle, \(C_{\alpha}\) is the cornering stiffness, \(\alpha_{sl}=\arctan\Big{(}\frac{3\mu_{s}F_{z}}{C_{\alpha}}\Big{)},F_{z}\) represents the vehicle vertical force, and \(\mu_{s}\) is the road friction coefficient. As shown in Figure 3, to simplify the nonlinear characteristics of tires and reduce computational complexity, the nonlinear tire lateral force is approximated as a linearized function near the working point \(\alpha_{f}\) and \(\alpha_{r}\) of the Fiala tire model curve. Specifically, this approximation is expressed as follows:  
\[F_{yf}\Big{(}\alpha_{f}\Big{)}=F_{yf0}+C_{f0}\Big{(}\alpha_{f}- \alpha_{f0}\Big{)} \tag{8}\] \[F_{yr}(\alpha_{r})=F_{yr0}+C_{r0}(\alpha_{r}-\alpha_{r0})\]  
Figure 2: Vehicle dynamics model. where \(F_{yf/0}\) denotes the lateral force of the front wheel at the starting point and \(F_{yf/0}\) denotes the lateral force of the rear wheel at the starting point. Additionally, \(C_{f/0}\) and \(C_{r0}\) denote the slope of the Fiala tire model at \(\alpha_{f/0}\) and \(\alpha_{r0}\), respectively. To characterize the vehicle's real trajectory, the lateral position of the vehicle and the yaw angle of the vehicle, measured on a fixed ground axis, are combined into the state vector \(x_{2}=\left[Y,v_{y},\varphi,\phi\right]^{T}\), and the vehicle dynamics system can be represented as follows:  
\[ \begin{array}{l}\dot{x}_{2}=A_{2}x_{2}+B_{2}\delta+N_{2}{\xi_{s}}^{T}\\ y_{2}=C_{2}x_{2}\end{array}  \tag{9}\]  
where  
\[A_{2}=\left[ \begin{array}{cccc}0&1&v_{x}&0\\ 0&\frac{C_{f/0}+C_{r0}}{mv_{x}}&0&\frac{aC_{f/0}-C_{r0}}{mv_{x}}\\ 0&0&0&1\\ 0&\frac{aC_{f/0}-C_{r0}}{l_{x}v_{x}}&0&\frac{aC_{f/0}+b^{2}C_{r0}}{l_{x}v_{x} }\end{array} \right]\]  
\[B_{2}=\left[ \begin{array}{cccc}0&-\frac{C_{r0}}{m}&0&-\frac{aC_{f/0}}{l_{x}} \end{array} \right]^{T}\]  
\[N_{2}=\left[ \begin{array}{cccc}0&1&0&0\\ 0&0&0&1\end{array} \right]^{T}\]  
\[C_{2}=\left[