## 2 Longitudinal Dynamics of Vehicles  
In this study, a longitudinal dynamics model was developed to describe the longitudinal behavior of an HDV that takes into account the vehicle's forward motion and the external forces acting on it. According to [33], the model was formulated based on simplified equations of motion that capture the relationship between vehicle mass and external forces, allowing for effective use in controller design and performance analysis. The longitudinal dynamics of a vehicle is shown in Figure 1. This longitudinal model has been widely used as a fundamental model to analyze the longitudinal be havior of HDVs, and many research groups have applied it to the design and validation of different controllers [34; 35]. The equation of motion of a vehicle moving on an inclined load along the vehicle's longitudinal axis is represented based on [PERSON]'s second law:  
\[ma_{x}=F_{T}-F_{R}, \tag{1}\]  
where the parameter \(m\) denotes the vehicle mass, \(a_{x}\) represents the vehicle's longitudinal acceleration, \(F_{T}\) is the total longitudinal traction force, and \(F_{R}\) signifies the resistance force, which can be further expressed as  
\[F_{R}=F_{g}+F_{r}+F_{a}, \tag{2}\]  
\[where\left\{ \begin{array}{c}F_{g}=mgssin\theta\\ F_{r}=\frac{C_{r}mgscon\theta}{r_{eff}}\\ F_{a}=0.5\rho C_{d}A_{f}(V_{x}-V_{air})^{2}\end{array} ,\right. \tag{3}\]  
Here, \(F_{g}\) comprises the gravitational force, \(F_{r}\) is the rolling resistance, and \(F_{a}\) denotes the aerodynamic resistance. The parameter \(g\) represents the gravitational acceleration, \(\theta\) is the road slope, \(C_{r}\) indicates the rolling resistance coefficient, \(r_{eff}\) represents the effective tire radius, \(\rho\) denotes the air density, \(C_{d}\) indicates the aerodynamic drag coefficient, \(A_{f}\) denotes the frontal area of the vehicle, \(V_{x}\) is the absolute vehicle speed, and \(V_{air}\) is the absolute speed of the wind. According to [36], the acceleration performance of a fully loaded HDV has a maximum acceleration of 0.05 m/s\({}^{2}\) on a flat road when the vehicle speed exceeds 25 mph. Because HDVs have relatively low driving forces, performance can be compromised even under loading conditions that are not a problem for passenger cars. In addition, variations in road gradient and payload may exceed the vehicle's overall operational limits. Control commands that do not consider these physical limitations are likely to result in engine power saturation, which not only reduces the acceleration performance of the vehicle but also decreases the accuracy of speed tracking. Furthermore, actuator saturation can negatively impact vehicle stability, particularly in platooning scenarios where multiple vehicles follow each other in a coordinated formation, often using automated systems to maintain safe distances, where precise control is essential. To address these challenges, the authors of [11] created an artificial saturation point to impose an acceleration limit that prevents the vehicle's actuator from reaching saturation:  
\[a_{x,act}(t)=\min\big{[}a_{x}(t),a_{x,Lim}\big{(}m(t),F_{g}(t)\big{)}\big{]}, \tag{4}\]  
Here, \(a_{x,act}\) denotes the actual acceleration and \(a_{x,Lim}\) denotes the limitation of acceleration. From Equation (4), it can be seen that the vehicle mass and the road grade both signifi  
Figure 1: Longitudinal dynamics of a vehicle on an inclined road. cantly influence the longitudinal dynamics of the vehicle. These factors are time-varying and must be considered in the control design to ensure stable and efficient operation. Furthermore, the actuation pathway from the control command to the actual vehicle acceleration introduces system lag. This delay arises from the throttle response, transmission dynamics, and engine inertia.