# Virtual Belt Algorithm for the Management of Isolated Autonomous Intersection
## 5 Initial Idea of Virtual Belt Algorithm
### Demonstration of the Initial Idea  
A simplified intersection with three through lanes is shown in Figure 13. There are two approaching CAVs for each lane. According to the time of getting entry to the intersection, the orders of these CAVs are labeled from 1 to 6 as shown in this figure. The length and width of each lane are 120 m and 4 m, respectively. The length and width of these CAVs are 4 m and 2 m, respectively. It is preferred for these CAVs to pass the intersection with the speed of 2 m/s and least time delay. The collisions among these CAVs are prohibited. It can be found that there are potential collisions among these CAVs. Thus, these CAVs should be coordinated to pass the intersection by the IMU. The designed VPs based on the simplified intersection are shown in this figure as well. These VPs are the same as these shown in Figure 12. Thus, the VBs in Figure 12 are used to demonstrate the initial idea. To show the initial ideal in a simple way, the dynamic limits of these CAVs are ignored. Thus, when a CAV is assigned to a VG, the CAV can reach the position of the VG with neglectable time. It is assumed that a CAV can keep a safe clearance with the front CAV in the same lane. These simplifications are made to show the initial idea of VB algorithm clearly. The VB algorithm considering the dynamic limits of CAVs will be designed next section. Several definitions are made for VG to simplify the discussion:  
* A VG that approaches to the intersection is called a _closing VG_. * A VG that has been assigned with a CAV is called an _occupied VG_. * A VG without CAV is called a _free VG_. A occupied VG becomes free at the time of born. * A VG that may transfer a CAV cross the intersection with collision is called an _unsafe VG_. Naturally, a collision avoidance policy can be proposed to prevent collisions among these CAVs: only the VGs before collision zone can be used for the transfer of CAVs. When a VG is occupied, the CVGs of the VG should be marked to avoid assignment of CAVs. Note that the mark should be cleared in certain conditions. Or a marked VG will not be utilized for CAVs forever, even all of its CVGs become free. To prevent such a condition, the mark will be cleared when the VG is dead. When a VG is born, it should be checked if there are potential conflict VGs. If a conflict VG is found, the VG should be marked to avoid usage. An unmarked VG before collision zones is called a _candidate VG_. In order to avoid collision among CAVs, a policy is used to coordinate approaching CAVs: only a candidate VG can be used for the transfer of CAVs. The candidate VGs are ordered according to their distances to the intersection. The closer the distance, the higher order the VG. To transfer a CAV with the least time, it is preferred to assign a VG with the candidate VG with higher order. Figure 13: Simplified intersection. The steps to control CAVs to pass the intersection are shown below:  
Step 1: At the time of 0 s, as shown in Figure 14, find the VG (\(G_{v9}^{B1}\)) with the highest order in \(B_{v1}\) for the first waiting CAV. Then, assign the first waiting CAV to \(G_{v9}^{B1}\). Finally, mark the CVGs of \(G_{v9}^{B1}\) and fill them with gray. When CAV 1 is assigned to \(G_{v9}^{B1}\), CAV 2 becomes the first waiting CAV. According to the assumption at the beginning of the section, the dynamic limits of CAVs are ignored in this section. It is assumed that a CAV can catch up with the assigned VG in no time. Similar to step 1, the other CAVs can be assigned to the optimal VGs with steps 2-6 as shown in Figure 15.  
When all CAVs have been assigned with VGs, these CAVs can get through the intersection without collision as demonstrated in Figure 16.  
Figure 14: Step 1, (**a**) find the optimal VG, (**b**) assign the first waiting CAV, (**c**) mark the CVGs. Figure 15: Steps 2–6, (**a**) step 2, (**b**) step 3, (**c**) step 4, (**d**) step 5, (**e**) step 6. It can be seen from the above discussion that, the designed VBs can be used for the management of intersection.