# Spacing Allocation Method for Vehicular Platoon: A Cooperative Game Theory Approach
## 4 Methodology Based on Cooperative Game Model for Platoon
### String Stability and Traffic Flow Stability  
In the research of the platoon, the string stability is required to ensure that the error will not propagate with the upstream of the platoon. Refer to the [33; 34; 35], the string stability can be described as Equation (26). \[\left|G(s)\right|=\left|\frac{e_{i}(s)}{e_{i-1}(s)}\right|=\left|\frac{V_{i} (s)}{V_{i-1}(s)}\right|\leq 1 \tag{26}\]  
where \(G(s)\) denotes the transfer function between the following errors. According to Equation (8) and Equation (10), we can get the equation of the space policy Equation (27). \[ \begin{array}{l}R=d+\frac{\omega_{x}}{2}\Big{(}\dot{\Delta}_{i+1}-\dot{ \Delta}_{i}\Big{)}+\frac{\omega_{x}}{2}(\Delta_{i+1}-\Delta_{i})\\ =d-\frac{\omega_{x}}{2}\dot{\Delta}_{i}-\frac{\omega_{x}}{2}\dot{\Delta}_{i}+ \frac{\omega_{x}}{2}\dot{\Delta}_{i+1}+\frac{\omega_{x}}{2}\dot{\Delta}_{i+1} \end{array}  \tag{27}\]  
Combing with the control law, we can get Equation (28). \[ \begin{array}{l}-\eta\Big{[}\dot{\Delta}_{i}-\left(-\frac{\omega_{x}}{2} \dot{\Delta}_{i}-\frac{\omega_{x}}{2}\ddot{\Delta}_{i}+\frac{\omega_{x}}{2} \dot{\Delta}_{i+1}+\frac{\omega_{x}}{2}\ddot{\Delta}_{i+1}\right)\Big{]}\\ =\ddot{\Delta}_{i}-\left(-\frac{\omega_{x}}{2}\ddot{\Delta}_{i}-\frac{\omega_{ x}}{2}\ddot{\Delta}_{i}+\frac{\omega_{x}}{2}\ddot{\Delta}_{i+1}+\frac{\omega_{x}}{2} \ddot{\Delta}_{i+1}\right)\end{array}  \tag{28}\]Through the Laplace transform and letting the \(L(\dot{\Delta_{i}})=\Delta V_{i}\), we may get Equation (29). \[ \begin{array}{c}-\eta\Big{[}\Delta V_{i}-\Big{(}-\frac{\omega_{c}}{2}\Delta V_{ i}-\frac{\omega_{c}}{2}s\Delta V_{i}+\frac{\omega_{c}}{2}\Delta V_{i+1}+\frac{ \omega_{c}}{2}s\Delta V_{i+1}\Big{)}\Big{]}\\ =s\Delta V_{i}-\Big{(}-\frac{\omega_{c}}{2}s\Delta V_{i}-\frac{\omega_{c}}{2}s^ {2}\Delta V_{i}+\frac{\omega_{c}}{2}s\Delta V_{i+1}+\frac{\omega_{c}}{2}s^{2} \Delta V_{i+1}\Big{)}\Leftrightarrow\\ \Big{[}-\eta\Big{(}1+\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}s\Big{)}-s- \frac{\omega_{c}}{2}s-\frac{\omega_{c}}{2}s^{2}\Big{]}\Delta V_{i}\\ =\Big{[}-\eta\Big{(}\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}s\Big{)}-\frac{ \omega_{c}}{2}s-\frac{\omega_{c}}{2}s^{2}\Big{]}\Delta V_{i+1}\end{array}  \tag{29}\]  
Finally, the transfer function \(G(s)\) is simplified in Equation (30). \[G(s)=\frac{\Delta V_{i+1}}{\Delta V_{i}}=\frac{-\eta-s-\eta\Big{(}\frac{\omega_ {c}}{2}+\frac{\omega_{c}}{2}s\Big{)}-\frac{\omega_{c}}{2}s-\frac{\omega_{c}}{2 }s^{2}}{-\eta\Big{(}\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}s\Big{)}-\frac{ \omega_{c}}{2}s^{2}} \tag{30}\]  
Substitute the \(s=jw\) into Equation (31), and the calculating progress is shown in Equation (30). \[ \begin{array}{c}\left\lfloor\frac{-\eta-\eta\frac{\omega_{c}}{2}+\frac{ \omega_{c}}{2}w^{2}-jw-\eta\big{(}\frac{\omega_{c}}{2}w\big{)}-\frac{\omega_ {c}}{2}w\big{)}}{-\eta\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}w^{2}-\eta\frac {\omega_{c}}{2}w-\frac{\omega_{c}}{2}w}\right\rfloor\Leftrightarrow\\ \frac{\big{(}-\eta-\eta\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}w^{2}\big{)}^{ 2}+\big{(}-1-\eta\frac{\omega_{c}}{2}-\frac{\omega_{c}}{2}\big{)}^{2}w^{2}}{ \left(-\eta\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}w^{2}\right)^{2}+\big{(}- \eta\frac{\omega_{c}}{2}-\frac{\omega_{c}}{2}\big{)}^{2}w^{2}}\leq 1\\ \eta^{2}-2\eta\Big{(}\eta\frac{\omega_{c}}{2}+\frac{\omega_{c}}{2}w^{2}\Big{)}+ w^{2}+2\big{(}\eta\frac{\omega_{c}}{2}-\frac{\omega_{c}}{2}\big{)}w^{2}\leq 0 \\ -\Big{(}\eta^{2}+\eta^{2}\omega_{c}\Big{)}+[\eta\omega_{v}-1-(\eta\omega_{v}+ \omega_{c})]w^{2}\leq 0\end{array}  \tag{31}\]  
To make sure the inequality always true, the Equation (32) should be satisfied. \[\left\{ \begin{array}{c}\Big{(}-\eta^{2}+\eta^{2}\omega_{c}\Big{)}\leq 0\\ -\eta\omega_{v}+1+(\eta\omega_{v}-\omega_{c})\leq 0\end{array} \right. \tag{32}\]  
Therefore, the sufficient condition of the string stability which is used to build the spacing policy is \(\omega_{c}\geq 1\). String stability should be considerate to avoid the amplification effect of the following errors and we will further verify this conclusion.