# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 5 Numerical Verification
### Numerical Investigation on Helly's Model with Bidirectional Information  
Numerical experiments using the spacing bidirectional Helly's model are first carried out. The values of model parameters are from [58] as summarized in Table 5.  
The simulation results are shown below. Figures 6-8 show vehicle speed profiles when \(\gamma_{x}=-0.4\), 0 and 0.4, respectively. Note that when \(\gamma_{x}=0\), the model degenerates to Helly's model. In each figure, Figures 6a, 7a and 8a show the speed profile of the perturbed vehicle and its preceding vehicles, demonstrating how back-looking information impacts downstream vehicle dynamics, especially in terms of safety. Figures 6b, 7b and 8b show the speed profile of the perturbed vehicle and its following vehicles, demonstrating how back-looking information impacts upstream vehicles dynamics, especially in terms of string stability. Figure 9 illustrates the minimum TTC value between the perturbed vehicle and its nearest preceding vehicle. When \(\gamma_{x}=-0.4\) and \(f_{n+1}^{s}<0\), based on the analysis in Table 2, spacing information results in the IP effect. As shown in Figure 6a, when the perturbed 10 th vehicle undergoes abrupt acceleration and deceleration, the nearest preceding 9 th vehicle also accelerates and decelerates. In Figure 9, a larger minimum TTC value is observed when \(\gamma_{x}=-0.4\) compared to the value when \(\gamma_{x}=0\), indicating lower collision risk. Figure 6b demonstrates that the following vehicles smoothly converge to steady speed without significant  
Figure 6: Vehicle speed profile under the bidirectional Helly’s model when \(\gamma_{x}=-0.4:\) (**a**) 1 st–10 th vehicles, (**b**) 10 th–20 th vehicles. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Initial Speed (m/s)** & \(\mathbf{\lambda_{x}}\) & \(\mathbf{\lambda_{v}}\) & \(\mathbf{\tau}\) \\ \hline
15 & 1 & 1 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters in Helly’s model. oscillations. These results validate that incorporating spacing information with the IP effect can reduce collision risk and improve platoon stability. When \(\gamma_{x}=0.4\) and \(f_{n+1}^{s}>0\), spacing information results in the OP effect. As shown in Figure 8a, when the perturbed 10 th vehicle undergoes abrupt acceleration and deceleration, the nearest preceding 9 th vehicle reacts by decelerating and then accelerating adversely. Figure 8: Vehicle speed profile under the bidirectional Helly’s model when \(\gamma_{x}=0.4:\) (**a**) 1 st–10 th vehicles, (**b**) 10 th–20 th vehicles. Figure 7: Vehicle speed profile under the bidirectional Helly’s model when \(\gamma_{x}=0:\) (**a**) 1 st–10 th vehicles, (**b**) 10 th–20 th vehicles. Figure 9: Minimum TTC variation with respect to \(\gamma_{x}\) in the bidirectional Helly’s model. The speed variation of the ninth vehicle also affects the eighth vehicle, resulting in adverse behavior with respect to the ninth vehicle. In Figure 9, a smaller minimum TTC value is observed when \(\gamma_{x}=0.4\), indicating higher collision risk. Figure 8b shows that the speed fluctuations are enlarged upstream in the platoon due to string instability. These results validate that incorporating spacing information with the OP effect can increase collision risk and worsen platoon stability. Figure 9 demonstrates that an increase in \(\gamma_{x}\) leads to a smaller minimum TTC value, indicating higher collision risk. When \(\gamma_{x}\) approaches 0.6, the perturbed 10 th vehicle collides with the 9 th vehicle due to the OP effect. Numerical experiments using the velocity difference bidirectional Helly's model are then carried out. Figures 10-12 show the vehicle speed profiles when \(\gamma_{v}\)= 0.4, 0 and \(-0.4\), respectively. Figure 13 illustrates the minimum TTC value between the perturbed vehicle and its nearest preceding vehicle. When \(\gamma_{v}=0.4\) and \(f_{n+1}^{\Delta v}>0\), the velocity difference information results in the IP effect.