# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 2 Methodology
### Vehicle Platoon Control
The formulation can have multiple feasible solutions if \(S_{g}\leq|C_{i}W_{i}|\leq S_{b}\), where \(S_{g}\) and \(S_{b}\) are the blue (right)-shaded and green (left)-shaded areas, respectively, in Figure 4. The blue-shaded area stands for the maximum distance a vehicle can travel within \(\Delta t_{des}\) (the vehicle accelerates at \(a_{max}\), all the way to \(v_{max}\), then maintains the speed until an instant after which it can keep decelerating at \(a_{min}\) to arrive at \(C_{i}\) with desired final speed \(v_{des}\)). Similarly, the green-shaded area is the lower bound of the distance with the given constraints (9b-d). One feasible strategy is to follow the speed trajectory indicated by the yellow (dash-dot)/red (solid) lines when \(|C_{i}W_{i}|\) is greater than or equal to/less than \(\frac{v_{max}^{2}}{2a_{max}}+v_{des}(\Delta t_{des}-\frac{v_{des}}{a_{max}})\). Specifically, the yellow dash-dot speed profile indicates that the vehicle accelerates at \(a_{max}\), all the way to a specific speed \(v^{\prime}\), then maintains the speed until an instant after which it can keep decelerating at \(a_{min}\) to arrive at \(C_{i}\) with the desired final speed \(v_{des}\). The red solid line shows that the vehicle stops at \(W_{i}\) until moment \(t^{\prime}\), then accelerates at \(a_{max}\) until the speed reaches \(v_{des}\), and maintains this speed up to the end \(\Delta t_{des}\). Both \(v^{\prime}\) and \(t^{\prime}\) can be derived from the constraint (9a). The merge-in procedure will be seamless if the transition of two stages is as precise as described by (9a-d). However, it is possible that when the matching slot arrives at the virtual critical position, the vehicle is not exactly at the critical position at the desired speed because a critical parameter \(\Delta t_{des}\) is estimated and possibly inaccurate when solving (9a-d). For example, in Figure 3, the matching slot of vehicle \(B_{2}\) arrives at the diverging point with the diverging vehicle \(G_{3}\) earlier than \(\Delta t_{des}\). Additionally, it is possible that, at \(\Delta t_{des}\), \(G_{3}\) still needs a little more time to get to the off-ramp. Then, \(B_{2}\) will have passed the critical point when \(G_{3}\) leaves the platoon. Note that the start time of the virtual platoon control stage is when the available slot arrives at the diverging point, not when the vehicle arrives at the critical point or the blue lines in Figure 3. Therefore, the virtual platoon control should ensure stability to mitigate the initial speed and position errors introduced by virtual vehicles when they enter the virtual platoon control stage. To ensure stability, the circular platoon in the roundabout is investigated, which consists of real vehicles, virtual vehicles projected from the vehicles on on-ramps at the second merge-in stage, and some spare vehicle slots. Note that here we assume a circular roundabout for simplicity; however, the model is not restricted only to ideal circular  
Figure 4: Feasible speed trajectories for vehicles in the self-adjusting stage. roundabouts. The platoon control model can be extended to non-circular roundabouts by mapping the roadways to circular roundabouts with the same road length. Suppose that when the roundabout serves at its maximum capacity, there are \(N\) homogeneous vehicles cruising in the roundabout with a desired angular speed \(\omega_{des}=v_{des}/R\) (where \(R\) is the radius of the roundabout). It implies that the roundabout has \(N\) vehicle slots labeled from \(1\) to \(N\). When there are spare vehicle slots, the circular platoon can be converted into one (if there is only one spare vehicle slot) or more linear vehicle platoons using the spare slots to separate them. Then, existing linear CACC platoon control strategies [17] can be applied. For a circular platoon with no spare vehicle slots, a symmetric bidirectional control architecture is developed (where predecessor and follower position errors influence the vehicle controller symmetrically).