# Investigating and Modeling of Cooperative Vehicle-to-Vehicle Safety Stopping Distance
## 4 Communication-Assisted Stopping Distance
### Stopping Distance  
This section shows results on how the testbed could be used in collision avoidance, such as rear end collisions or emergency braking. This is based upon the latency finding of our experiments between sending and receiving nodes. Latency or information freshness is critical for CAVs as this is the speed at which the nodes are communicating information and is required to be 100 ms for standard BSM [39; 40]. The latency that we monitored can be seen in Figure 1, and this experiment was conducted by measuring processing time and calculating the propagation time. The equation that is used to calculate End to End processing time (\(PT_{EEE}\)) is shown in Equation (1), and it shows the Time sent (\(T_{s}\)) and Time received (\(T_{R}\)) of a packet when transmitted through direct connection via SubMiniature version A (SMA) cable, which eliminates the propagation time. We then subtract the times from one another, leaving us with the processing time of both systems, which we assume is even on both sides. Then, we calculated the propagation time (\(T_{P}\)) for each distance, and, because we are only using distances up to 100 m, the propagation is at most 33.4 micro seconds. We then add the two times to find total End to End Latency (\(T_{EEE}\)). This latency, however, ignores the logical processing inside of the receiving device for reacting to the received packet, which would be negligible. \[PT_{EEE}=\frac{(T_{R}-T_{S})}{2}\] \[T_{EEE}=PT_{EEE}+T_{P}. \tag{1}\]  
Figure 1: End to End latency distribution. The results for latency can be seen in Figure 1 and are the product of over 2000 samples. We found that the latency ranged between 2.1 and 2.9 ms with a mean of 2.48 ms, showing the latency is lower than expected for information freshness of 50 ms required for pre-crash sensing [41]. To calculate the stopping distance (\(D_{\text{S}}\)), we chose to use two different methods available and used in the literature. The first equation shown in Equation (2) is the most commonly used formula used in physics to calculate braking distance (\(D_{\text{B}}\)); however, it is not based upon vehicle mass and assumes a rate of deceleration from 0 to max deceleration (maximum braking force) [35; 36; 37; 38], where g is acceleration due to gravity (9.81 m/s\({}^{2}\)), V is velocity of vehicle, and u is the coefficient of friction. This equation is then used with an equation for thinking distance to calculate total stopping distance. Thinking distance (\(D_{\text{R}}\)) is calculated with the velocity (\(V\)) and reaction time (\(T_{R}\)). \(T_{R}\) can also be represented in our case with \(T_{EEE}\) when using the testbed. \[ \begin{split} D_{\text{S}}(m)=D_{\text{R}}+D_{\text{B}}\\ D_{\text{R}}(m)=V*T_{\text{R}}\\ D_{\text{B}}(m)=\frac{V^{2}}{2g\mu}.\end{split}  \tag{2}\]  
The coefficient of friction is a variable to describe the friction between the rubber tires of the vehicle and the road. Table 1 shows some road examples that are based on weather and ground type, with the average coefficient values associated [42]. In our experiments, we chose to use 0.8 as the coefficient of friction as this is the standard for normal dry road conditions for both asphalt and concrete. The second equation shown in Equation (3) is used by many researchers, like in Reference [42; 43; 44; 45], as this equation considers more in-depth variables concerning the type of car, road conditions, air conditions, and other parameters. Table 2 details each component of Equation (3), along with the value we chose to use and the range of values that are typically used. This is then added to \(D_{\text{R}}\), which is calculated in the same way as previously and leads to the total \(D_{\text{S}}\). \[ \begin{split} D_{\text{B}}(m)=\frac{W}{2 gC_{\text{ae}}}ln(1+ \frac{C_{\text{ae}}V^{2}}{\eta_{\text{b}}tW+f_{r}cos\theta+Wsin\theta})\\ whereC_{\text{ae}}=(p*Af*Cd)/2^{2}\\ D_{\text{S}}(m)=D_{\text{B}}+D_{\text{R}}\end{split}  \tag{3}\]