# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 2 Methodology  
### Roundabout Flow Control  
Consider a typical roundabout shown in Figure 2, which has four approaches indexed by \(i=1,\ 2,\ 3,\ 4\). The incoming flows on the four approaches are denoted by \(Q_{i}\), assumed constant during each time period \(\tau\), and known at the beginning. In real world, \(Q_{i}\) can be  
Figure 1: Conceptual structure of the roundabout control strategy. predicted using real-time loop detector data, while the length of \(\tau\) can be adjusted according to the changing rate of \(Q_{i}\). The queue length on approach \(i\), \(l_{i}(t)\), is a state variable of \(t\). In each time period, \(t\) starts from 0 and ends at \(\Delta\tau\) (the time period length). Merge-in flows, \(q_{i}(t)\), are control inputs, which represent the flows merging into the circular flow within the roundabout from approach \(i\). The queue length dynamics can be described as  
\[\frac{dl_{i}(t)}{dt}=Q_{i}-q_{i}(t),\ \forall i=1,\ 2,\ 3,\ 4. \tag{1}\]  
Apart from the non-negative constraint, it is also reasonable to have an upper bound \(L_{i,max}\) for queue length \(l_{i}(t)\). In applications, \(L_{i,max}\) can be decided by the critical queue length that may cause spillback on approach \(i\). Therefore, the state constraints are  
\[0\leq l_{i}(t)\leq L_{i,max},\ \forall i=1,\ 2,\ 3,\ 4. \tag{2}\]  
To model the constraints of control inputs \(q_{i}\), the roundabout segments are introduced and described. As seen in Figure 2, the circular roundabout is separated by four approaches into four arc-shaped segments, \(S_{1},\ S_{2},\ S_{3}\), and \(S_{4}\). Since these segments are likely to have similar road design as they belong to the same roundabout, it is assumed that all segments have the same service capacity \(\mathcal{Q}_{S,max}\) (i.e., the upper bound of segment flow \(\Pi_{S_{k}},\ k=1,\ 2,\ 3,\ 4\)). Note that \(\Pi_{S_{k}}\) may consist of vehicles from every approach. For example, vehicles entering from approach 1 heading to approach 3 will drive through \(S_{1}\) and \(S_{2}\), and thereby contribute to both \(\Pi_{S_{1}}\) and \(\Pi_{S_{2}}\). Similarly, vehicles entering from approach 2 heading to approach 1 will also drive through \(S_{2}\), thereby contributing to \(\Pi_{S_{2}}\). Denote flows from approach \(i\) heading to approach \(j\) as \(q_{ij}\) (\(i,j=1,\ 2,\ 3,\ 4\)), which can be represented as \(q_{ij}=\eta_{ij}q_{i}\), where \(\eta_{ij}\) is the proportion of the flow heading to approach \(j\) in the merge-in flow from approach \(i\). To describe the relationships between \(S_{k}\) and \(q_{ij}\), define the service tables of four segments as follows (note that this study assumes that vehicles will drive through each segment no more than once when passing through the roundabout). The element in row \(i\) and column \(j\) of the service table for \(S_{k}\), \(a_{ij}^{k}\), is an indicator variable of whether \(q_{ij}\) will pass through \(S_{k}\). For instance, \(a_{22}^{4}=1\), indicating the U-turn flow from approach 2 will pass through \(S_{4}\) in the roundabout; while \(a_{32}^{2}=0\) means vehicles coming from approach 3 heading to approach 2 will not drive through \(S_{2}\). Therefore, the flow on \(S_{k}\) is  
\[\Pi_{S_{k}}(t)=\sum_{i=1}^{4}\sum_{j=1}^{4}a_{ij}^{k}\eta_{ij}q_{i}(t),\ \forall k=1,\ 2,\ 3,\ 4 \tag{3}\]  
Then, the control input constraints can be represented as  
\[\Pi_{S_{k}}(t)\leq\mathcal{Q}_{S,max},\ \forall k=1,\ 2,\ 3,\ 4 \tag{4}\]  
Figure 2: Four-approach roundabout. \[q_{i}(t)>0,\ \forall i=1,\ 2,\ 3,\ 4 \tag{5}\]  
The control objective is  
\[\min_{q_{1},q_{2},q_{3},q_{4}}\int\limits_{0}^{\Delta\tau}\left(\sum_{i=1}^{4} \frac{Q_{i}l_{i}(t)}{q_{i}(t)}\right)dt \tag{6}\]  
where \(l_{i}(t)/q_{i}(t)\) is the estimated waiting time for vehicles at the end of the queue on approach \(i\), and \(Q_{i}dt\) is the number of incoming vehicles or vehicles joining the queue at the end on approach \(i\). Therefore, the integration can be interpreted as the total estimated waiting time of vehicles arriving from \(t=0\) to \(t=\Delta\tau\).