# Modeling Driver's Real-Time Confidence in Autonomous Vehicles
## 3 Modeling Confidence
### The Influence of Out Factors  
In the process of driving a vehicle, the interference of human drivers comes mainly from the surrounding traffic participants, especially moving vehicles [17]. Therefore, we focus on direct external influences on nearby vehicles that may intersect spatio-temporally with the ego vehicle. Human drivers can subconsciously predict the movement of surrounding objects and even the possible thoughts and behaviors of surrounding drivers. Once the predicted results present a certain possibility of danger, driving confidence decreases, and response actions are made synchronously. At the same time, humans are able to predict not only the occurrence of a collision but the approximate location of the potential accident, including the location of the road where the collision may occur and the damaged position of the vehicle. To simplify the calculation cost, we adopt the assumption of the \"Safety Force Field\" system proposed by NVIDIA, which assumes that traffic participants on the road are rational humans who will take brake response when danger is predicted in any situation [18]. Based on this assumption we set up a simple Spatio-temporal intersection prediction. The prediction can determine the magnitude of the disturbance to the ego vehicle based on the interval. Assuming the emergency response measure performed by the rational person is simple linear braking, then the expected distance traveled by each vehicle can be described as the following equation:  
\[d=vs.t_{0}+\frac{v^{2}}{2a} \tag{2}\]  
where \(v\) represents the instantaneous speed of the vehicle. \(t_{0}\) represents the average braking reaction time. \(a\) is the deceleration of emergency braking. For autonomous vehicles, precise calculations of the vehicle's speed, acceleration, and other related information can be obtained through onboard devices such as GPS, IMU, and cameras [19; 20]. For human drivers, the prediction of the moving target will not only focus on the position of the vehicle's gravity center but also on the moving position of the whole physical entity of the vehicle in space. We assume that the vehicle reachable area is a rectangle extending the length of expected travel distance \(d\) from the left and right corners of the vehicle to its forward direction based on the brake assumption. Each active vehicle, on the road around the ego, generates a predicted reachable area, and the intersection of two or more reachable areas can generate different kinds of polygonal intersection areas. The regions intersecting with the ego vehicle are quadrilateral in most cases and triangular or pentagonal in a few, as shown in Figure 4.  
Figure 3: The form of the VC output. In Figure 4, (a) and (c) are the general interacting scenarios among two vehicles where the interaction regions are all quadrilateral, and (b) and (d) are where the regions are triangular or pentagonal appear in the specific situation. In the calculation process, we use the two-dimensional coordinates of the vertices of each vehicle's reachable regions to calculate the intersection set and obtain the coordinates of the interaction polygon. The properties of interactions are regular under the simple definition, so the coordinates of the vertices can relatively easily be figured out. After obtaining the vertex coordinates, the positions of the closest and farthest points to the ego and the target vehicle can be obtained by comparing the distances to each vertex, respectively. The relationship between distance and time to possible collision is expressed as Equation (3):  
\[d_{i}= \begin{cases}vt,&\text{if }d_{i}\leq vs.t_{0}\\ vt_{0}+\frac{at^{2}}{2},&\text{otherwise}.\end{cases}  \tag{3}\]  
We can use the equation to calculate the time required for the front of two vehicles to the nearest point of the interaction area and the two vehicles' rear to the farthest interaction point. When \(d_{i}\leq vs.t_{0}\), solve for \(t\) is the required time. When \(d_{i}>vs.t_{0}\), the required time is \(t+t_{0}\).