## 3 Required Data Rate in V2V for Safe Overtaking
### Vehicle Movement
As explained, the clothoid is suitable to design the driving path in terms of linearly increasing curvature, so it is hard to handle analytically. Therefore, we used the sigmoid curve in our simulation. The characteristics of the sigmoid curve are that it is easier to configure and compute than the clothoid [25]. \[y(x)=\frac{B}{1+\mathrm{e}^{-\alpha x}} \tag{3}\]  
The function of the sigmoid curve and its parameters \(B,~{}a\) are shown in Equation (3). The way to construct the driving path with the sigmoid curve is shown in Figure 3. In other words, half of the driving path consists of two sigmoid curves that have a mirror symmetry. In order to configure the sigmoid curve, we needed to determine the \(B,~{}a,~{}x_{0}\) parameters. The parameter \(B\) depends on the road width. In this driving path, the ego vehicle was assumed to move from the center of the lane to the center of the neighbor lane, then return to the first lane. From this assumption, \(B\) is equal to the width of a single lane. The parameter \(a\) determines the curvature of the sigmoid curve. In order to determine \(a\), we considered a slip and constructed a sigmoid curve that does not cause slip at a minimum curvature radius shown in Figure 4. The judgment of the slip is performed by the following formula. \[\frac{mv^{2}}{R}\lessapprox\mu mg\left\{ \begin{array}{l}\text{ slip: }\frac{mv^{2}}{R}>\mu mg\\ \text{safe: otherwise}\end{array} \right. \tag{4}\]  
where \(m\) is the mass of the vehicle, \(v\) is the velocity of the vehicle, \(R\) is the curvature radius at the point where the vehicle places, \(\mu\) is the coefficient of static friction, and \(g\) is gravity acceleration. Since we assumed that the ego vehicle drives along the path at a constant velocity, the minimum curvature radius without slip is \(R_{\text{min}}=v^{2}/\mu g\) from Equation (4). Finally, the parameter \(x_{0}\) determines the length of the sigmoid curve. The length is determined by the duration to complete the overtaking, which was set in advance. Figure 5 shows the examples of the driving path at 20 km/h and 50 km/h. From the figure, it is shown that when the vehicle velocity becomes low, the slope of the driving path becomes steep. This can be explained by the definition of \(R_{\min}\). In other words, a vehicle driving at a low velocity can turn sharply without slipping. To compare with the distance required by the braking, firstly, we derived the distance for the overtaking driving path. Since the oncoming vehicle drives on the neighboring lane, the ego vehicle has to finish overtaking by the time the oncoming vehicle arrives at the collision point. Figure 4: The parameters for the sigmoid curve under \(a=5\) and the location of the minimum curvature radius. Figure 5: The examples of two driving paths where the duration to complete the overtaking is 5 s.  
Figure 6 shows both driving paths from the start point to the collision point, where \(v\) is the velocity of the vehicle, \(t_{\rm o}\) is the duration to complete the overtaking, and \(x_{0}\) is shown in Figure 4. The collision occurs when the ego vehicle moves to the center of the neighboring lane to overtake the blocking vehicle. Since the driving path is a mirror symmetry curve, when the ego vehicle arrives at the center of the neighbor lane, the oncoming vehicle moves for \(t_{\rm o}/2\). Therefore, the distance required for the overtaking driving path becomes \(vt_{\rm o}/2+2x_{0}\). From the above discussion, we combined the distance required for the driving path and the comfortable braking. Figure 7 shows the distance required for comfortable braking and for the three driving path cases. It is shown that the driving path required a larger distance than the comfortable braking distance at a low velocity, and this was reversed at a high velocity. Therefore, considering the driving path is important especially at a low velocity. Namely, the required distance \(d_{\rm req}\) can be formulated as follows.