# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### Phase Shift Effects in Bidirectional Communication  
The present research employs the proposed framework centered around phase shift to characterize the interconnected behavior of the following vehicle and the target vehicle in a platoon under bidirectional communication topology. In this scenario, the following vehicle serves as the source vehicle, transmitting stimulus information, while the target vehicle receives this stimulus information and responds accordingly. As an initial step to analyze safety impacts, this research narrows the focus to two specific and extreme cases of phase shift without considering communication latency: the in-phase (IP) effect and the opposite-phase (OP) effect. The in-phase (IP) effect indicates a synchronized manner between the following vehicle and the target vehicle. For instance, if the following vehicle accelerates abruptly, the target vehicle also accelerates, as depicted in Figure 3a. In contrast, the opposite-phase (OP) effect reflects an adverse manner. For instance, if the following  
Figure 2: Demonstration of phase shift. (**a**) Phase shift. (**b**) In Phase. (**c**) Opposite Phase. vehicle accelerates abruptly, the target vehicle decelerates to adjust its speed, as shown in Figure 3b. The rest of this section outlines the conditions for the IP and OP effects in the spacing bidirectional control model and the velocity difference bidirectional control model. Considering that a platoon of CAVs runs on a single lane with all vehicles in an equilibrium state, i.e., maintaining the same spacing and velocity, perturbation is introduced to the following vehicle \(n\)+1 at time \(t\), resulting in a deviation denoted as velocity deviation \(\mu_{n+1}(t)=v_{n+1}-v_{n}\) and spacing deviation \(\varphi_{n+1}(t)=s_{n+1}(t)-s_{e}\), where \(s_{e}\) represents spacing at the equilibrium state. We have  
\[ \begin{array}{l}\mu_{n+1}(t)=-\dot{\varphi}_{n+1}(t)\\ \varphi_{n+1}(t)*\mu_{n+1}(t)<0\end{array}  \tag{5}\]  
In a bidirectional control model, the velocity deviation of the following vehicle \(n\)+1 prompts the target vehicle \(n\) to adjust its speed. By taking the first-order Taylor expansion of Equation (3) of the spacing bidirectional control model, we can express the acceleration of target vehicle n as shown in Equation (6). \[\dot{v}_{n}(t)=f_{s_{n+1}}\varphi_{n+1}(t) \tag{6}\]  
By taking the first-order Taylor expansion of Equation (4) of the velocity difference bidirectional control model, the acceleration of the target vehicle \(n\) can be expressed as in Equation (7). \[\dot{v}_{n}(t)=f_{\Delta v_{n+1}}\mu_{n+1}(t) \tag{7}\]  
where \(f_{s_{n+1}}=\left.\frac{\partial f_{s}}{\partial s_{n+1}}\right|_{\left( \overline{\mathcal{D}},\,\overline{\mathcal{D}}\right)}\) represents velocity differential with respect to spacing change, and \(f_{\Delta v_{n+1}}=\left.\frac{\partial f_{s}}{\partial\Delta v_{n+1}}\right|_ {\left(\overline{\mathcal{D}},\,\overline{\mathcal{D}}\right)}\) represents velocity differential with respect to velocity difference change. Let us first concentrate on the IP effect. When \(\varphi_{n+1}(t)\,*\,\dot{v}_{n}(t)<0\) or \(\mu_{n+1}(t)\,*\,\dot{v}_{n}(t)>0\), it implies that if vehicle \(n+1\) accelerates or decelerates, target vehicle \(n\) will also accelerate or decelerate, as demonstrated in Figure 3a. The IP effect is determined by the positivity  
Figure 3: Illustration of the (**a**) in-phase effect (**b**) opposite-phase effect. or negativity of \(f_{s_{n+1}}\) and \(f_{\Delta v_{n+1}}\). If \(f_{s_{n+1}}<0\), then \(\varphi_{n+1}(t)\;*\;\dot{v}_{n}(t)<0\). If \(f_{\Delta v_{n+1}}>0\), then \(\mu_{n+1}(t)\;*\;\dot{v}_{n}(t)>0\). Correspondingly, when \(\varphi_{n+1}(t)\;*\;\dot{v}_{n}(t)>0\) or \(\mu_{n+1}(t)\;*\;\dot{v}_{n}(t)<0\), it means that if vehicle \(n+1\) accelerates or decelerates, the target vehicle \(n\) will respond oppositely by decelerating or accelerating; this phenomenon is denoted as the OP effect, as illustrated in Figure 3b.