# Spacing Allocation Method for Vehicular Platoon: A Cooperative Game Theory Approach
## 3 Flock's Model for Platoon  
The connected automated vehicles (CAVs) in platoon are always considered as agents in multiple-agent systems which can use bionic concepts to describe the group behaviors. In this section, a novel flock's model which is suitable for cooperative control of platoon is introduced according to the consistency control principles of the MAS. The principles can be expressed as follows when applied in platoon control. Collision Avoidance: avoid collisions with neighboring vehicles. Velocity matching: attempt to match velocity with leading vehicle or vehicle in front. Center gathering: attempt to stay close to the center of the neighboring vehicles. Collision avoidance serves to establish the minimum required separation distance while velocity matching tends to maintain it. Center gathering is used to ensure the traffic flow stability. The relationship of the CAVs in platoon has obvious differences with normal self-driving vehicle. For examples, the CAVs in platoon are more like to cooperate than competition. With reasonable rules, they can achieve consistent motion pattern and avoid harmful effects caused by irrational competition. Therefore, the cooperative game theory is suitable for solving the cooperative control of the CAVs in platoon. In this section, a spacing policy and the performance function are proposed referring to the aforementioned distributed model based on flocks. In addition to several general spacing policies, the relationship between car follower model (CFM) and traffic oscillations has been extensively studied [27; 28]. The traditional model considers that the traffic flow is in a steady state, and the desired spacing between vehicles is related to the velocity. However, the later cellular automaton (CA) model believed that the randomization enables the models to depict the spontaneous formation of jams [29]. In actual traffic flow, driving behavior is closely related to time difference, random behavior, etc. [30]. Therefore, a reasonable modeling model should be able to consider the relationship between distance and speed, and have resistance to the impact of random interference fluctuations on traffic flow. With reference to the three principles in the previously mentioned content, the desired spacing will be calculated with positions \(X_{i}\), and velocities \(\dot{X}\) of the vehicles. \(\omega_{v}\), \(\omega_{c}\) are denote the weight coefficients of velocity match principle and center gathering principle respectively. Avoiding-collision profit \(J_{a}\) is a constant space to ensure a safe zone between two vehicles which is shown in Equation (4) and Figure 1.  
\[J_{a}=d \tag{4}\]  
where \(d\) denotes the constant distance in stationary state which is related to \(L_{i}\), and the \(L_{i}\) demonstrates the length of the vehicle \(i\). Velocity-match profit \(J_{v}\), the velocity of the host vehicle \(\dot{X}_{i}\) should match with the average velocity of the nearest vehicles, as shown in Equation (5) and Figure 2. When the time gradually increases and the platoon enters a steady state, the velocities of all vehicles will tend to be equal. \[J_{v}=\omega_{v}\bigg{(}\dot{X}_{i}-\frac{\dot{X}_{i-1}+\dot{X}_{i+1}}{2} \bigg{)} \tag{5}\]  
Figure 1: Avoid collision principle. Considering that the last vehicle in platoon does not have two reference nearest vehicles, the equation should be modified as Equation (6). \[J_{v}=\omega_{v}\Big{(}\dot{X}_{i}-\dot{X}_{i-1}\Big{)} \tag{6}\]  
Center-gathering profit \(J_{c}\), the vehicles should be located in the middle of its previous vehicle and following vehicle as shown in Equation (7). As shown in Figure 3, the vehicle \(i\) should decelerate to reach the center point of vehicle \(i-1\) and \(i+1\). \[J_{c}=\omega_{c}\bigg{(}X_{i}-\frac{X_{i-1}+X_{i+1}}{2}\bigg{)} \tag{7}\]  
Obviously, the host vehicle has not center point while \(i=n\) which means that \(J_{c}=~{}0\). Combining Equations (4), (5) and (7), we may obtain the whole desired spacing from Equation (8). \[