# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### String Stability Analysis
\begin{split}\big{(}f_{s}\big{)}^{2}-\frac{\big{(}f_{v_{n}}\big{)} ^{2}}{2}\big{(}f_{s_{n}}-f_{s_{n+1}}\big{)}-f_{s}f_{v_{n}}f_{\Delta v}<0\end{split}  \tag{21}\]  
where \(f_{s}=f_{s_{n}}+f_{s_{n+1}}f_{\Delta v}=f_{\Delta v_{n}}+f_{\Delta v_{n+1}}\)  
If \(f_{s_{n+1}}=0\) and \(f_{\Delta v_{n+1}}=0\), the equation then simplifies to \(f_{s_{n}}-\frac{\big{(}f_{s}^{\pi}\big{)}^{2}}{2}-f_{v_{n}}f_{\Delta v_{n}}<0\), which matches the string stability condition of the traditional predecessor following scheme [40; 49; 50]. This section first derives the stability condition of the spacing bidirectional control model based on Equation (21) when \(f_{\Delta v_{n+1}}=0\) and \(f_{s_{n+1}}\
eq 0\). The platoon is stable if \(\eta^{s}<0\), which is formulated in Equation (22),  
\[\eta^{s}=\ \ \big{(}f_{s}\big{)}^{2}-\frac{\big{(}f_{v_{n}}\big{)}^{2}}{2} \big{(}f_{s_{n}}-f_{s_{n+1}}\big{)}-f_{s}f_{v_{n}}f_{\Delta v_{n}} \tag{22}\]  
It should be noted that in the spacing bidirectional control model, spacing information \(s_{n+1}\) from the following vehicle is usually coupled with spacing information \(s_{n}\) from the preceding vehicle as a stimulative term, \(w(s_{n+1})-w(s_{n})\). Therefore, \(f_{n}^{s}\) is composed of two parts, denoted as \(f_{n}^{\mathrm{s1}}\) and \(f_{n}^{\mathrm{s2}}\), where \(f_{n}^{\mathrm{s1}}\) represents the original part in Equation (1) and \(f_{n}^{\mathrm{s2}}=-f_{n+1}^{\mathrm{s}}\) represent the part in a stimulative term as shown in the equation below:  
\[f_{s_{n}}=f_{s_{n}}^{\mathrm{1}}+f_{s_{n}}^{\mathrm{2}} \tag{23}\]  
where \(f_{s_{n}}^{\mathrm{2}}=-f_{s_{n+1}}\). Then, the stability condition of the spacing bidirectional control model can be rewritten as shown in Equation (24),  
\[\eta^{\mathrm{s}}=\left(f_{n}^{\mathrm{s1}}\right)^{2}-\frac{\left(f_{v_{n}} \right)^{2}}{2}\Big{(}f_{s_{n}}^{\mathrm{1}}-2f_{s_{n+1}}\Big{)}-f_{s_{n}}^{ \mathrm{1}}f_{v_{n}}f_{\Delta v_{n}} \tag{24}\]  
Taking the partial derivative of Equation (24) with respect to \(f_{n+1}^{\mathrm{s}}\) yields Equation (25) as follows:  
\[\frac{\partial\eta^{\mathrm{s}}}{\partial f_{s_{n+1}}}=\left(f_{v_{n}}\right) ^{2} \tag{25}\]  
Since \(\frac{\partial\eta^{\mathrm{s}}}{\partial f_{s_{n+1}}}>0\), it can be observed that when \(f_{s_{n+1}}<0\) (IP), \(\eta^{\mathrm{s}}\) decreases, indicating an improvement in string stability. On the contrary, when \(f_{s_{n+1}}>0\) (OP), \(\eta^{\mathrm{s}}\) increases, leading to worse stability. In this section, we derive the stability condition of the velocity difference bidirectional control model based on Equation (21) when \(f_{\Delta v_{n+1}}\
eq 0\) and \(f_{s_{n+1}}=0\). The platoon is stable if \(\eta^{\Delta v}<0\), which is formulated in Equation (26),  
\[\eta^{\Delta v}=\left(f_{s_{n}}\right)^{2}-\frac{\left(f_{v_{n}}\right)^{2}}{ 2}f_{s_{n}}-f_{s_{n}}f_{v_{n}}f_{\Delta v} \tag{26}\]  
Taking the partial derivative of Equation (26) with respect to \(f_{\Delta v_{n+1}}\) yields Equation (27),  
\[\frac{\partial\eta^{\Delta v}}{\partial f_{\Delta v_{n+1}}}=-f_{s_{n}}f_{v_{n}} \tag{27}\]  
where \(f_{s_{n}}>0\) and \(f_{v_{n}}<0\). Since \(\frac{\partial\eta^{\Delta v}}{\partial f_{\Delta v_{n+1}}}>0\), it can be observed that when \(f_{\Delta v_{n+1}}<0\) (OP), \(\eta^{\Delta v}\) decreases, resulting in improved stability. On the contrary, when \(f_{\Delta v_{n+1}}>0\) (IP), \(\eta^{\Delta v}\) increases, leading to worse stability.