# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 4 Analytical Verification on Specific Car-Following Models
### Linear Car-following Model  
Car-following models have been developed for more than half a century, and numerous models have been proposed to model the longitudinal behaviors of vehicles. Among them, [PERSON]'s linear car-following model has been widely applied to describe CAV behavior [54] due to its simple and intuitive feature [40]. This research adopts [PERSON]'s car-following model to model the linear following behavior of CAVs and bidirectional information flow topology effects. The formulation of the [PERSON]'s model is denoted as below [55]:  
\[\dot{v}_{n}(t)=\lambda_{x}(s_{n}(t)-\tau v_{n}(t)-s_{0})-\lambda_{v}\Delta v_{ n}(t) \tag{28}\]  
where \(\lambda_{v}\) represents the sensitivity to velocity difference between target vehicle and preceding vehicle, \(\lambda_{x}\) represents the sensitivity to spacing between target vehicle and preceding vehicle, \(s_{0}\) is the minimum distance allowed as a safety gap, and \(\tau\) represents reaction time. Note that \(\tau v_{n}(t)+s_{0}\) indicates the desired space gap. In this model, the acceleration of a vehicle presents a linear relationship with deviation from spacing and velocity difference between two successive vehicles. Incorporating the information from the nearest following vehicle, the bidirectional [PERSON]'s model can be expressed as:  
\[ \begin{split}\dot{v}_{n}(t)=\lambda_{x}(s_{n}(t)-\tau v_{n}(t)-s _{0})-\lambda_{v}\Delta v_{n}(t)\\ +\gamma_{x}(s_{n+1}(t)-s_{n}(t))+\gamma_{v}\Delta v_{n+1}(t)\end{split}  \tag{29}\]  
where \(s_{n+1}\), \(\Delta v_{n+1}\) represent the spacing and velocity difference information from the following vehicle, respectively. Difference \(s_{n+1}(t)-s_{n}(t)\) represents the spacing stimulus. \(\gamma_{x}\) represents sensitivity to spacing, and \(\gamma_{v}\) represents sensitivity to velocity difference. By \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Model** & **Spacing Information** & **Rear-End Collision Risk** & **String Stability** \\ \hline In-Phase Effect & \(f_{s_{n+1}}<0\) & Decreased & Improved \\ \hline Opposite-Phase Effect & \(f_{s_{n+1}}>0\) & Increased & Worsened \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Impact of spacing information on platoon safety and stability. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Model** & **Velocity Difference** & **Rear-End Collision** & **String Stability** \\  & **Information** & **Risk** & **Riring Stability** \\ \hline In-Phase Effect & \(f_{\Delta v_{n+1}}>0\) & Decreased & Worsened \\ \hline Opposite-Phase Effect & \(f_{\Delta v_{n+1}}<0\) & Increased & Improved \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Impact of velocity difference information on platoon safety and stability. setting \(\gamma_{v}=0\) and \(\gamma_{x}=0\), the spacing and the velocity difference bidirectional Helly's model can be obtained, respectively. We can obtain the analytic expressions of the partial differential equations for the bidirectional Helly's model, as shown in the equations below:  
\[ \begin{array}{cc}f_{s_{n}}^{1}=\lambda_{x},&f_{s_{n}}^{2}=-\gamma_{x},&f_{ \bar{\
u}_{x}}=-\lambda_{x}\tau,&f_{\Delta\varphi_{n}}=-\lambda_{v},\\ &f_{s_{n+1}}=\gamma_{x},&f_{\Delta\varphi_{n+1}}=\gamma_{v}\end{array}  \tag{30}\]  
For the spacing bidirectional Helly's model, by applying the partial differential equations to Equation (10), we obtain Equation (31). \[\frac{\partial TTC_{n+1}^{s}(t+\Delta t)}{\partial f_{s_{n+1}}}\bigg{|}_{( \overline{v},\,\overline{s})}=\frac{(\varphi_{n+1}(t+\Delta t)+s_{e})\varphi_ {n+1}(t)\Delta t}{\left(\mu_{n+1}(t+\Delta t)-\gamma_{x}\varphi_{n+1}(t) \Delta t\right)^{2}} \tag{31}\]  
Since \(\frac{\partial TTC_{n+1}^{s}(t+\Delta t)}{\partial f_{s_{n+1}}}<0\), it can be observed that when \(f_{s_{n+1}}<0\) (IP), \(TTC_{n+1}(t+\Delta t)\) increases, resulting in lower collision risk. On the contrary, when \(f_{s_{n+1}}>0\) (OP), \(TTC_{n+1}(t+\Delta t)\) decreases, leading to higher collision risk.