# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### Phase Shift Effects  
In a CAV system, the dynamics of vehicles are impacted by several factors, including the vehicle's control mechanisms, types of exchanged information, communication delays, packet losses, and the design of information flow topology. For instance, in CACC systems that utilize bidirectional information flow topology, vehicles take into account information from the following vehicle, which affects the dynamics of the preceding vehicles, potentially giving rise to substantial safety concerns. However, there has not yet been a generalized model to comprehensively assess such impacts on vehicle dynamics. Inspired by our recent discovery that a perturbed vehicular platoon exhibits periodic oscillatory dynamics characterized by inherent frequency [39; 40; 41], this research proposed a generalized analysis model based on the concept of phase shift to capture the interactions among connected vehicles' dynamics. Our research demonstrated that the oscillation dynamics of a perturbed vehicle platoon can be described by a second-order non-homogeneous ordinary differential equation (ODE), resulting in periodic oscillations  
Figure 1: Illustration of the platooned vehicles using bidirectional communication. propagating within the platoon. In the context of a CAV platoon employing bidirectional communication, the effects of perturbations can be transmitted both forward and backward simultaneously, leading to an overlapping effect of two periodic oscillations. This overlapping effect can be analyzed using the concept of phase shift. The concept of phase shift is commonly used in physics, for example, in acoustics and optics [42; 43], to describe the differences, \(\varphi(t)=\phi_{C}(t)-\phi_{F}(t)\), between the phases of two periodic signals \(F\) and \(G\), as depicted in Figure 2a. Specifically, when the difference is zero, the two signals are in phase (IP), as shown in Figure 2b, indicating perfect synchronization. On the contrary, when the difference is not zero, implying \(\varphi(t)\
eq 0\), the two signals are termed out-of-phase, as illustrated in Figure 2a. For sinusoidal signals, when difference \(\varphi(t)\) is 180 deg, the two phases are opposite, defined as opposite phase (OP), as illustrated in Figure 2c. The concept of phase shift provides a valuable framework for understanding the interactions among vehicles through information exchange. In this framework, we distinguish between different vehicles involved in the information exchange process by categorizing them as either the \"source vehicle\" or the \"target vehicle\". The \"source vehicle,\" represented as signal \(F\) in Figure 2, refers to the vehicle that transmits stimulus information. On the other hand, the \"target vehicle,\" denoted as signal \(G\) in Figure 2, refers to the vehicle that receives this stimulus information and responds accordingly. Phase shift plays a crucial role in depicting target vehicle response to stimulus information, encompassing factors such as latency and response patterns. To represent latency, let us consider \(\dot{v}_{n}(t)\), which represents vehicle dynamics with delay \(t_{d}\) indicating the latency in the target vehicle's response to perturbation from the source vehicle. In this context, phase shift \(\varphi(t)\) can serve as a representation of latency \(t_{d}\). The relationship between phase shift \(\varphi(t)\) and latency \(t_{d}\) is associated with the wavelength of the resulting platoon oscillation, which can be analytically derived using the corresponding ODE [41]. Furthermore, phase shift can also be utilized in depicting response patterns. When phase shift \(\varphi(t)=0\) and there is no latency (\(t_{d}=0\)), the dynamics of the two adjacent vehicles synchronize their responses to perturbation. If the target vehicle responds to the source vehicle in a synchronized manner, this can be denoted as \"in-phase.\" On the contrary, if the response is unsynchronized with phase shift \(\varphi(t)\
eq 0\), it is denoted as being \"out of phase\".