# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 4 Analytical Verification on Specific Car-Following Models
### Non-Linear Car-following Model  
The research further employs the Intelligent Driver Model (IDM) car-following model, proposed by [PERSON] et al. [12], to describe the non-linear following behavior of CAVs due to several of its advantages. First, the IDM is a multi-regime model, which captures the dynamics of different traffic congestion levels more realistically than other models [33]. Second, it provides collision-free behavior and smooth traffic flow [56]. Third, it is well-accepted to model connected automated vehicles' longitudinal dynamics [57]. In the IDM model, the acceleration of vehicle \(n\) at time \(t\) is determined by its current velocity \(v_{n}\), headway \(s_{n}\), and velocity difference \(\Delta v_{n}\) to the preceding vehicle, which can be expressed as  
\[ \begin{split}&\dot{v_{n}}(t)=a\bigg{(}1-\Big{(}\frac{v_{n}(t)}{v_{0}} \Big{)}^{\delta}-\Big{(}\frac{s^{*}(v_{n}(t),\ \Delta v_{n}(t))}{s_{n}(t)}\Big{)}^{2}\bigg{)}\\ & s^{*}(v_{n}(t),\ \Delta v_{n}(t))=s_{0}+v_{n}(t)T+\frac{v_{n}(t) \cdot\Delta v_{n}(t)}{2\sqrt{ab}}\end{split}  \tag{33}\]  
where \(\dot{v_{n}}(t)\) and \(v_{n}\) denote the acceleration and speed of vehicle \(n\) at time \(t\); \(a\) and \(b\) denote the maximum acceleration and deceleration of the vehicle, respectively; \(v_{0}\) denotes the free-flow speed; \(\delta\) is the acceleration exponent parameter; \(s_{0}\) represents the minimum bumper-to-bumper gap in traffic jam states; \(T\) is the desired time gap; \(s_{n}\) denotes the net distance, \(s_{n}=x_{n}-x_{n-1}-l\), between vehicle \(n\) and its preceding vehicle \(n-1\), where \(l\) is vehicle length and \(x_{n}\) denotes the position of vehicle \(n\) at time \(t\). \(\Delta v_{n}\) denotes the velocity difference between vehicle \(n\) and its preceding vehicle \(n-1\). The research adapts the bidirectional distance balanced model (BDBM) proposed in [26] and further modifies it by incorporating velocity difference information from the nearest following vehicle. The structure of the new bidirectional IDM model is expressed as follows:  
\[ \begin{split}&\dot{v_{n}}(t)=a\bigg{(}1-\Big{(}\frac{v_{n}(t)}{v_{0}} \Big{)}^{\delta}-\Big{(}\frac{s^{*}(v_{n}(t),\ \Delta v_{n}(t))}{s_{n}(t)}\Big{)}^{2}\bigg{)}\\ & s^{*}(v_{n}(t),\ \Delta v_{n}(t))=s_{0}+v_{n}(t)T+\frac{v_{n}(t) \cdot\Delta v_{n}(t)}{2\sqrt{ab}}\\ &\qquad+\gamma_{x}(s_{n+1,t}(t)-s_{n,t}(t))+\gamma_{v}\Delta v _{n+1}(t)\end{split}  \tag{34}\]  
where \(s_{n+1}\) and \(\Delta v_{n+1}\) denote the spacing and velocity difference information received from the following vehicle, respectively. \(\gamma_{x}\) represents the sensitivity to spacing and \(\gamma_{v}\) represents the sensitivity to velocity difference. By setting \(\gamma_{v}=0\) and \(\gamma_{x}=0\), the spacing and the velocity difference bidirectional IDM model can be obtained, respectively. We can obtain the analytic expressions of the partial differential equations for the bidirectional IDM model, as shown in the equations below:  
\[ \begin{split}& f_{s_{n}}^{1}=\frac{2\pi s^{*2}}{s^{3}},\ f_{s_{n}}^{2}=\frac{2\pi s^{*}\gamma_{x}}{s^{2}},\ f_{v_{n}}=-\frac{4\pi v_{v}^{3}}{v_{0}^{2}}- \frac{2\pi Ts^{*}}{s^{2}},\\ & f_{\Delta v_{n}}=-\frac{\sqrt{a}\pi s^{*}}{s^{2}\sqrt{b}},\ f_{s_{n+1}}=-\frac{2\pi s^{*}\gamma_{x}}{s^{2}},\ f_{\Delta v_{n+1}}=-\frac{2\pi s^{*}\gamma_{x}}{s^{2}} \end{split}  \tag{35}\]  
For the spacing bidirectional IDM, by applying the partial differential equations in Equation (35) to Equation (10), we obtain Equation (36). \[\frac{\partial TTC_{n+1}^{s}(t+\Delta t)}{\partial f_{s_{n+1}}}\bigg{|}_{( \mathfrak{P},\mathfrak{P})}=\frac{(\varphi_{n+1}(t+\Delta t)+s_{e})\varphi_{n+ 1}(t)\Delta t}{\left(\mu_{n+1}(t+\Delta t)+\frac{2\pi s^{*}\gamma_{x}}{s^{2}} \varphi_{n+1}(t)\Delta t\right)^{2}} \tag{36}\]Since \(\frac{\partial TTC_{n+1}^{s_{n+1}(t+\Delta t)}}{\partial f_{n+1}}<0\), it can be observed that when \(\gamma_{x}>0\) and \(f_{s_{n+1}}<0\) (IP), \(TTC_{n+1}(t+\Delta t)\) increases, resulting in lower collision risk.