# Evaluating the Impact of Drone Signaling in Crosswalk Scenario
## 3 Materials and Methods  
### Problem Statement  
We consider a crosswalk involving a car and a pedestrian. The car approaches the crosswalk (at a distance \(d_{c}\), with a speed \(v_{c}\) and an acceleration \(a_{c}\)). The pedestrian intends to cross the road at the same moment. This scenario is represented in Figure 1.  
To avoid the collision, the car must stop and let the pedestrian safely cross the intersection (final speed \(v_{c,f}=0\)). When the pedestrian becomes in the field of view of the car driver, the latter brakes to stop before the crosswalk. The traveled distance during this process is the stopping distance \(d_{c,s}\), defined by Equation (1), where \(d_{c,p-r}\) is the perception-reaction distance (this corresponds to the traveled distance during the perception-reaction time \(t_{p-r}\) of the driver), and \(d_{c,b}\) is the braking distance (this corresponds to the distance while the speed is decreasing up to the vehicle stops). In addition, we use Equation (2) to compute \(d_{c,p-r}\). \[d_{c,s}=d_{c,p-r}+d_{c,b}, \tag{1}\]  
\[d_{c,p-r}=t_{p-r}\cdot v_{c}. \tag{2}\]  
To compute the braking distance, we consider the model of kinetic energy dissipation. The theoretical braking distance can be computed by determining the work required to dissipate the vehicle's kinetic energy. The model is detailed in Reference [44]. The braking distance \(d_{c,b}\) satisfies Equation (3), where \(m\) is the mass of the car, and \(L\) represents the longitudinal frictional forces (see Equation (4)). \[\int_{0}^{d_{c,b}}Ldl=\frac{1}{2}\cdot m\cdot v_{c}^{2}, \tag{3}\]  
\[L=f_{l}\cdot N=f_{l}\cdot m\cdot g, \tag{4}\]  
with \(N\) being the normal tire-pavement reaction, \(f_{l}\) the coefficient of friction between the road surface and the tires, and \(g\) the gravity of Earth. If we consider an average coefficient of friction over the braking distance (not depending on the speed), by applying [PERSON]'s second law and equation of motion, the braking distance is determined by equation  
\[d_{c,b}=\frac{v_{c}^{2}}{2\cdot f_{l}\cdot g}. \tag{5}\]  
By combining Equations (2) and (5), it yields:  
\[d_{c,s}=t_{p-r}\cdot v_{c}+\frac{v_{c}^{2}}{2\cdot f_{l}\cdot g}. \tag{6}\]  
To avoid collisions, \(d_{c,s}\) must be lower than \(d_{c}\), at the instant the car driver sees the pedestrian. This condition is likely to be satisfied if \(v_{c}\) is below a threshold \(v_{0}\) (usually the authorized speed limit). However, if \(v_{c}\) is too high, the stopping distance would be too large, and there would be a collision between the car and the pedestrian at the limit condition of braking. In fact, a too high value of \(v_{c}\) can be due to several factors, essentially related to both drivers and pedestrians behaviors:  
* visibility limits when driving by night or by curved road,
* lack of attention from the driver,
* lack of attention from the pedestrian, and
* infrastructure characteristics. Figure 1: Classical crosswalk: one pedestrian and one car coming through a straight line. Moreover, during braking, to evaluate exactly the evolution of the speed \(v_{c,\textit{braking}}\) as a function of the distance (\(d_{c,\textit{braking}}\)), we consider that \(f_{l}\) depends on the speed (\(v_{c}\)), and, in this case, we consider an average deceleration (\(a_{c,\textit{braking}}\)), obtained empirically [45]. \[v_{c,\textit{braking}}=\sqrt{v_{c}^{2}-2\cdot a_{c,\textit{braking}}\cdot d_{c, \textit{braking}}} \tag{7}\]  
In this work, we focused on adding a drone on the top of the scene, just over the crosswalk, named \"controlled by drone\". The drone observes the vehicle and sends alerts to its driver and/or to the infrastructure in case of harmful situation. The authors in Reference [29] mentioned that there are alerts for both pedestrians and vehicle drivers for signalized street crossing scenario. The alert for pedestrians is based on Mobile Accessible Pedestrian Signal System. Message exchanges are suggested as starting before the driver sees the pedestrian.