# Reservation-Based 3D Intersection Traffic Control System for Autonomous Unmanned Aerial Vehicles
## 4 UAV's Behavior in the Approaching Area
### Avoiding Collision with a UAV Ahead
Now we will explain how to determine \(r_{B,j+1}\). At \(t_{j}\), the available parameters are \(s_{A,j}\), \(s_{B,j}\), \(d_{A,B,j}\), \(r_{A,j}\), \(r_{B,j}\). From those values, we can get \(s_{A,j+1}\), \(s_{B,j+1}\), and \(d_{A,B,j+1}\). As an example, consider the values of \(r_{A,j}\), \(r_{B,j}\), and \(d_{A,B,j+1}\) shown in Figure 2 and set \(d_{min}\) as 1 m. The blue dashed lines represent the distance travelled by \(UAV_{A}\) from \(t_{j+1}\) assuming it will slow down using \(dr_{max}\) until it stops. A dashed line in the figure means the UAV decelerates at \(dr_{max}\). Since we assume that \(UAV_{A}\) will slow down at \(dr_{max}\) until it stops, \(UAV_{B}\) should also slow down until it stops to avoid collision with \(UAV_{A}\). One approach is to make \(UAV_{B}\) slow down using \(dr_{max}\) from \(t_{j+1}\) until it stops, as shown by the yellow dashed lines in the figure. While this makes the distance between them always at least \(d_{min}\), we do not want \(UAV_{B}\) to slow down at rate \(dr_{max}\) when it can use a smaller deceleration rate that will still maintain the minimum distance from \(UAV_{A}\). In this approach, as can be seen from the figure, the shortest distance between them, which is when both UAVs have stopped, is twice the value of \(d_{min}\). This means \(UAV_{B}\) can use a smaller deceleration rate than \(dr_{max}\) and still maintain \(d_{min}\). Moreover, the distance between them will be greater than the value in the example when \(UAV_{A}\) does not actually slow down using \(dr_{max}\) from \(t_{j+1}\). Hence, to avoid this scenario, we assume that \(UAV_{B}\) will slow down using \(dr_{max}\) at \(t_{j+2}\), as shown by the red dashed lines. Then, we get \(r_{B,j+1}^{d_{min}}\), the value of \(r_{B,j+1}\) such that the distance when both have stopped, \(d_{A,B}^{stop}\), is equal to \(d_{min}\). Figure 2: Distance between \(UAV_{A}\) and \(UAV_{B}\) on different acceleration rates. The lines represent the acceleration rates of \(UAV_{A}\) and \(UAV_{B}\) at different time steps. A dashed line in the figure means the UAV decelerates at \(dr_{max}\). The blue lines represent the acceleration rates of \(UAV_{A}\), slowing down from \(t_{j+1}\) until it stops. The yellow lines represent the acceleration rates of \(UAV_{B}\), slowing down from \(t_{j+1}\) until it stops. The red lines represent the acceleration rates of \(UAV_{B}\), slowing down starting at \(t_{j+2}\) until it stops. The value of \(d_{A,B}^{stop}\) is equal to \(d_{min}\) when \(UAV_{A}\) and \(UAV_{B}\) follow the acceleration rates represented by the blue and yellow lines respectively. The green lines represent the acceleration rate of \(UAV_{B}\) when \(r_{B,j+1}\) is equal to \(r_{max}\) and then it decelerates using \(dr_{max}\) at \(t_{j+2}\). In this case, \(UAV_{B}\) collides with \(UAV_{A}\) before \(t_{j+3}\). We also show another scenario, as shown by the green solid line, wherein the distance travelled by \(UAV_{B}\) when \(r_{B,j+1}\) is equal to \(r_{max}\) to show that there can be collision even when \(d_{A,B,j+2}\) is greater than \(d_{min}\). In this scenario, even though \(d_{A,B,j+2}\) is greater than \(d_{min}\), \(UAV_{B}\) will collide with \(UAV_{A}\) before \(t_{j+3}\) even when \(UAV_{B}\) will decelerate using \(dr_{max}\) at \(t_{j+2}\). This happens when \(s_{B,j+2}\) is greater than \(s_{A,j+2}\) because this makes \(d_{B,j+2}\) greater than \(d_{A,j+2}\). Hence, we must make sure that distance between them is always at least \(d_{min}\) until they both have stopped. Therefore, \(UAV_{B}\) should use a rate less than or equal to \(r_{B,j+1}^{d_{min}}\) for the distance between them to be always at least \(d_{min}\). However, \(UAV_{B}\) cannot just use any rate less than or equal to \(r_{B,j+1}^{d_{min}}\). The speed of \(UAV_{B}\) at \(t_{j+2}\) should not exceed \(s_{max}\) and the rate must not exceed \(r_{max}\). Considering this, let \(r_{B,j+1}^{s_{max}}\) be the rate that will make \(s_{B,j+2}\) equal to \(s_{max}\).