# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 2 Methodology
### Vehicle Platoon Control  
Once vehicles choose to merge into the roundabout based on the probabilistic decision control model, they will start the merge-in procedure and eventually cruise with the circular platoon until they diverge to the approaches they are heading to. Note that this subsection focuses on longitudinal control therefore does not consider non-holonomic vehicle dynamics or low-level vehicle control such as throttle/steering control. The merge-in procedure consists of a self-adjusting stage and a virtual platoon control stage. Figure 3 illustrates an example of the two stages using the merge-in procedure of vehicle \(B_{1}\). First, \(B_{1}\) waits at position \(W_{2}\) (marked by a red line) when it is at the head of the queue on approach 2. When the decision control model allows it to merge into a spare slot (labeled the matching slot of \(B_{1}\)) in the platoon, \(B_{1}\) enters the self-adjusting stage. The objective of this stage is to have vehicle \(B_{1}\) drive through a critical position \(C_{2}\) (marked by a blue line on approach 2) to enter the virtual platoon control stage at the desired speed \(v_{des}\) after some desired time \(\Delta t_{des}\). The desired speed is also the desired cruising speed of the platoon. The desired time \(\Delta t_{des}\) is estimated as the time it takes for the matching slot to travel to a virtual critical position \(V_{2}\) (marked by the yellow line on the right part of the roundabout in Figure 3) from the beginning of the self-adjusting stage. The virtual critical position \(V_{2}\) is a projection of \(C_{2}\) on the roundabout circle, that is \(|M_{2}V_{2}|=|M_{2}C_{2}|\), where \(M_{2}\) is a merging point of the roundabout and the on-ramp from approach 2. When vehicle \(B_{1}\) travels from \(C_{2}\) to \(M_{2}\), a virtual vehicle \(Y_{1}\), which is the projection of \(B_{1}\), also travels from \(V_{2}\) to \(M_{2}\). At the same time, \(B_{1}\) communicates with vehicles \(G_{1}\) and \(G_{2}\), making \(Y_{1}\) behave like a vehicle communicating with its surrounding vehicles in the circular platoon. This stage is labeled as the virtual platoon control stage. Note that when determining the length of \(|M_{2}C_{2}|\), \(|M_{2}D_{2}|\geq|M_{2}C_{2}|=|M_{2}V_{2}|\) should be ensured, where \(D_{2}\) is the diverging point of the off-ramp to approach 2. It is assumed that once a diverging vehicle passes the diverging point and drives to the off-ramp, it is no longer a part of the circular platoon (i.e., its movement will not affect surrounding vehicles in the platoon anymore). This guarantees that during the virtual platoon control stage, there will be no real diverging vehicle at the slot where the virtual vehicle projected from a real merge-in vehicle is located. Otherwise, they may have a conflicting influence on the surrounding vehicles since they are at the same slot in the platoon. The control of a vehicle from approach \(i\) in the self-adjusting stage can be formulated as  
\[\int\limits_{0}^{\Delta t_{des}}v(t)dt =|C_{i}W_{i}| \tag{9a}\] \[v(\Delta t_{des}) =v_{des} \tag{9b}\]  
Figure 3: Illustration of self-adjusting stage and virtual platoon control stage in vehicles’ merge-in procedure. \[0\leq v(t)\leq v_{max},\ \forall t\in[0,\ \Delta t_{des}] \tag{9c}\] \[a_{min}\leq\frac{dv(t)}{dt}\leq a_{max},\ \forall t\in[0,\ \Delta t_{des}] \tag{9d}\]  
where \(v_{max},\ a_{min},\ a_{max}\) are the maximum speed, maximum deceleration, and maximum acceleration, respectively. As shown in Figure 4, in a speed-time graph, the formulation seeks for trajectories starting from \((0,0)\) ending at \((\Delta t_{des},\ v_{des})\), whose time integration equals to \(|C_{i}W_{i}|\).