# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### Assessment of Rear-End Collision Risk  
Rear-end collisions are prevalent on freeways, representing one of the most common types of accidents [44]. The advent of vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communications has made it possible to use individual vehicle information to assess collision risks and enhance traffic safety. Therefore, this section focuses on assessing the collision risk associated with the bidirectional control model. In this research, surrogate safety measures (SSMs) are utilized to evaluate collision risks. SSMs are safety performance indicators that estimate accident risks based on microscopic traffic parameters like speed, space headway, and time headway [44]. Several SSMs have been developed for estimating collision risks. Time to collision (TTC) is one of the most used SSMs. The concept of TTC, introduced in [45], refers to the time remaining until collision occurs between the leading and the following vehicle if velocity difference is maintained. This research adopts TTC as a metric to analyze the risk of rear-end collisions. Given that perturbation is introduced to the following vehicle \(n+1\) at time \(t\), it impacts the dynamics of vehicle \(n\) due to the bidirectional control mechanism. Our analysis specifically focuses on the collision risk between vehicle \(n+1\) and vehicle \(n\), as this pair presents a heightened risk of collision, particularly when the following vehicle \(n+1\) undergoes sudden acceleration. The TTC for vehicle \(n+1\) is calculated as follows:  
\[TTC_{n+1}(t)=\left\{ \begin{array}{l}\frac{x_{n}(t)-x_{n+1}(t)-l_{n}}{v_{n+1} (t)-v_{n}(t)}\;\textit{if }v_{n+1}(t)>v_{n}(t)\\ \infty,\;\textit{if }v_{n+1}(t)<v_{n}(t)\end{array} \right. \tag{8}\]  
where \(TTC_{n+1}(t)\) denotes the TTC value of vehicle \(n+1\) at time \(t\), \(x_{n}\) and \(x_{n+1}\) are the positions of vehicles \(n\) and \(n+1\), \(v_{n}\) and \(v_{n+1}\) are the velocities of vehicles \(n\) and \(n+1\), and \(l_{n}\) is the length of vehicle \(n\). A smaller TTC value indicates a higher risk of collision. As discussed earlier, the OP effect increases the risk of collisions, especially when the following vehicle experiences sudden acceleration. Thus, we consider the perturbation in Section 3.2 when \(\mu_{n+1}(t)>0\) (\(\varphi_{n+1}(t)<0\)). The TTC of vehicle \(n+1\) after time \(\Delta t\) for spacing and velocity difference bidirectional control can be represented using Equations (9) and (11), respectively. For the spacing bidirectional control model, \(TTC_{n+1}(t+\Delta t)\), denoted as \(TTC_{n+1}^{s}(t+\Delta t)\), is expressed as follows:  
\[ \begin{array}{l}TTC_{n+1}^{s}(t+\Delta t)=\frac{x_{n}(t+\Delta t)-x_{n+1}(t +\Delta t)-l_{n}}{v_{n+1}(t+\Delta t)-v_{n}(t+\Delta t)}\\ =\frac{\theta_{n+1}(t+\Delta t)-x_{n+1}(t)}{\mu_{n+1}(t+\Delta t)-f_{n+1}\theta _{n+1}(t)\Delta t}\end{array}  \tag{9}\]By taking partial derivation of Equation (9) with respect to \(f_{s_{n+1}}\), we obtain the following equation:  
\[\frac{\partial TTC_{n+1}^{s}(t+\Delta t)}{\partial f_{s_{n+1}}}\bigg{|}_{( \overline{v},\ \overline{s})}=\frac{(\varphi_{n+1}(t+\Delta t)+s_{e})\varphi_{n+1}(t) \Delta t}{\left(\mu\,_{n+1}(t+\Delta t)-f_{s_{n+1}}\varphi_{n+1}(t)\Delta t \right)^{2}} \tag{10}\]  
Given that perturbation \(\varphi_{n+1}(t)\) is small and does not result in an immediate collision, we can infer that \(\varphi_{n+1}(t+\Delta t)+s_{e}>0\). Thus, we can conclude that \(\frac{\partial TTC_{n+1}(t+\Delta t)}{\partial f_{s_{n+1}}}<0\). Since \(\frac{\partial TTC_{n+1}^{s}(t+\Delta t)}{\partial f_{s_{n+1}}}<0\), it can be observed that when \(f_{s_{n+1}}<0\) (IP), \(TTC_{n+1}(t+\Delta t)\) increases, resulting in lower collision risk and improved safety. On the contrary, when \(f_{s_{n+1}}>0\) (OP), \(TTC_{n+1}(t+\Delta t)\) decreases, leading to higher collision risk.