# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### Vehicle Longitudinal Dynamics Models with Bidirectional Information Flow  
In CAV environment, the dissemination of real-time data is essential for ensuring the safety and stability of vehicles within a platoon. This critical process is supported by advanced sensor and communication technologies, which are essential for enabling automated vehicles to share information seamlessly. Key sensor devices include RADAR, cameras, and LIDAR, etc. Furthermore, advanced communication technologies, including Dedicated Short-Range Communications (DSRC) and 5G, are indispensable for facilitating Vehicle-to-Vehicle (V2V), Vehicle-to-Infrastructure (V2I), and Vehicle-to-Everything (V2X) communications [33]. Meanwhile, the effectiveness of information delivery within this communication network heavily relies on communication protocols. These protocols are designed to ensure that necessary information is disseminated to the intended vehicle in a reliable and low-latency manner [34]. The development and implementation of a variety of communication protocols have been proposed to significantly improve the safety and stability of vehicular platoons [3; 4; 35]. The integration of information technologies and protocols enables CAVs to exchange information seamlessly with each other. In a CAV platoon, the control system operates the vehicle using locally sensed information and information shared among vehicles [36]. A car-following model can be used to describe the vehicle's longitudinal dynamics. Note that, at present, our analysis is confined to a one-dimensional perspective. The car-following model, along with our safety and stability analysis, does not account for lateral dynamics, such as left and right turns. In the literature, continuous-time car-following models have a generalized form, which can be expressed below [37; 38]; we adapted this format to include additional information:  
\[ \begin{split}\dot{x}_{n}(t)=v_{n}(t)\\ \dot{v}_{n}(t)=f(s_{n}(t),\;v_{n}(t),\;\Delta v_{n}(t),\;K(t)) \end{split}  \tag{1}\]  
where \(\dot{v}_{n}(t)\) is the control variable, which represents the acceleration (or deceleration) of the \(n\)th vehicle at time \(t\). This variable is what the control system aims to adjust through the function. \(x_{n}\) and \(v_{n}\) represent the position and velocity of vehicle \(n\), \(s_{n}=x_{n-1}-x_{n}-l_{n-1}\) represents the net distance between vehicle \(n\) and vehicle \(n-1\), \(l_{n-1}\) indicates the length of vehicle \(n-1\), and \(\Delta v_{n}=v_{n}-v_{n-1}\) represents the velocity difference between vehicle \(n\) and vehicle \(n-1\). In this study, we apply a broader interpretation of net distance, referring to it as \"spacing.\" In Equation (1), \(K(t)\) symbolizes additional information, encompassing vehicle dynamic data from the following vehicle in bidirectional car-following control, like velocity or spacing. Moreover, it extends to represent various other data types, such as acceleration [13], traffic jerk [14; 15], visual angle [16], and electronic throttle opening angle [17; 18]. For platoon control with bidirectional information flow topology, the target vehicle reacts not only to the preceding vehicle but also the dynamics of the following vehicle to adjust its speed, as shown in Figure 1. Typically, bidirectional control utilizes two types of back-looking information from the following vehicle, i.e., spacing \(s_{n+1}\) and velocity difference \(\Delta v_{n+1}\) information. We denote as \(s_{n+1}=x_{n}-x_{n+1}-l_{n}\) the spacing between vehicle \(n+1\) and vehicle \(n\), and as \(\Delta v_{n+1}=v_{n+1}-v_{n}\) the velocity difference between vehicle \(n+1\) and vehicle \(n\). The car-following models with bidirectional communication topology have a generic form as in Equation (2). If only spacing information \(s_{n+1}\) is utilized, the resulting model is formulated in Equation (3) and referred to in this research as the spacing bidirectional control model. \[