## 2 Simplified Vehicle Model  
A simplified vehicle model for the longitudinal motion scenario design can simulate vehicle motion by inputting the required acceleration (\(a_{x,des}\)), which is designed by the driver model proposed in Section 3. First, the simplified vehicle model assumes that the vehicle is traveling on flat ground, thus the relationship between the acceleration and velocity is shown in Equation (1) [43].  
\[\dot{v}_{x}=a_{x} \tag{1}\]  
\[ma_{x}=F_{x}-F_{r}-F_{a} \tag{2}\]  
\[w.r.t\left\{\begin{array}{l}F_{r}=N\big{(}C_{r,0}+C_{r,1}v_{x}+C_{r,2}v_{x} ^{2}\big{)}\\ F_{a}=0.5C_{d}\rho Av_{x}^{2}\end{array}\right. \tag{3}\]  
\[a_{x}=\left\{\begin{array}{l}a_{x},u\quad\quad if\ a_{x,des}>a_{x,U}\\ a_{x,L}\quad\quad else\ if\ a_{x,des}<a_{L}x\\ a_{x,des}\quad\quad\quad\quad else\end{array}\right. \tag{4}\]  
To consider the dynamic characteristics of the vehicle, a longitudinal dynamic vehicle model is described in Equation (2). The vehicle acceleration is deduced by the longitudinal  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Researcher (Year)** & **Model** \\ \hline \hline [PERSON] [24] (1953) & \(a_{F}(t+\tau)=C\dot{R}(t)\) \\ \hline [PERSON] [25] (1961) & \(a_{F}(t+\tau)=C\dot{R}(t)\frac{v_{T}(t)^{m}}{\dot{R}(t)^{m}}\) \\ \hline [PERSON] [26] (1961) & \(v_{F}(t+\tau)=V_{d}\cdot\Big{\{}1-\exp\Big{(}-C\frac{\dot{R}(t)-R_{min}}{V_{d }}\Big{)}\Big{\}}\) \\ \hline [PERSON] [27] (1964) & \(a_{F}(t+\tau)=C_{V}\cdot\dot{R}(t)+C_{S}\cdot(R(t)-C_{t}\cdot v_{F}(t))\) \\ \hline [PERSON] [28] (1981) & \(\min\left\{\begin{array}{l}v_{F}(t+\tau)=\\ v_{F}(t)+2.5\cdot A\cdot\tau\cdot\big{(}1-\frac{v_{F}(t)}{v_{x}}\big{)}\cdot \sqrt{0.025+\frac{v_{T}(t)}{V_{d}}}\\ B\cdot\tau+\sqrt{B^{2}\cdot\tau^{2}-B\cdot\Big{\{}2\cdot(R(t)-R_{min})-v_{F}(t) \cdot\tau-\frac{v_{L}(t)^{2}}{B}\Big{\}}}\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Deterministic classical car-following models.  
force (\(F_{x}\)), rolling resistance (\(F_{r}\)), and aerodynamic resistance (\(F_{a}\)) of the vehicle. The rolling resistance (\(F_{r}\)) is composed of the normal force of the vehicle (\(N\)), the constant coefficient (\(C_{r,0}\)), and velocity coefficients (\(C_{r,1},\ C_{r,2}\)). The aerodynamic resistance (\(F_{a}\)) is composed of the aerodynamic drag coefficient (\(C_{d}\)), the density of the air (\(\rho\)), and the area of the vehicle (\(A\)). The rolling resistance and the aerodynamic resistance are dependent on the vehicle velocity. The longitudinal force is constrained by the performance of the vehicle, which is related to the vehicle's powertrain and braking system. For this reason, the longitudinal acceleration of the vehicle was constrained by an upper bound (\(a_{x,U}\)) and lower bound (\(a_{x,L}\)). Figure 1a describes the upper constraint of the longitudinal force that varies with the transmission gear ratio. For simplification, a simplified acceleration constraint, as shown in Figure 1b, is designed to consider dynamic characteristics of the vehicle in the longitudinal motion scenario design. It constrains the desired acceleration (\(a_{x,des}\)) from the driver model to describe realistic vehicle motions. In other words, the actual vehicle is composed of complex dynamic systems, such as powertrain and hydraulic systems. However, in the case of this simplified vehicle model, it is used to simulate vehicle movement by considering acceleration and its related constraints. Acceleration and deceleration situations can occur depending on the sign of the longitudinal acceleration (\(a_{x}\)), and the simplified vehicle model moves by causing a change in vehicle speed.