We propose a real-time quantification model called “the Virtual Confidence (VC)” for simulating a driver’s driving confidence during the AV operating process. The VC can emulate the real-time acceptance pattern of human drivers about the status of the vehicle by considering three main components: (1) Quantification of the integrated threat to the AV (2) Quantification of the AV’s driving status and violations (3) Quantification of the complexity of the environment in which the AV is exposed. In this section, the design of the specific components and the preliminary algorithm scheme is described and analyzed in detail.
3.1. Model Framework
The calculation of the confidence value can be expressed by the following equation. The operating logic of the VC model can be understood by combining the content of
Figure 2 with Equation (
1).
In the equation, (
,
,
,
,
) refers to (Virtual confidence value, Outside influencing value, Self influencing value, Recovery value, Maximum virtual confidence value). Where
t and
indicate two adjacent moments in terms of frequency.
is the specific confidence value at the corresponding moment. In the operation process, this confidence value is controlled to be no more than a maximum value of
and no less than 0. The confidence value is specified to be the maximum value at
when the vehicle system starts. We specify the
as 100 when the surrounding complexity is in the most stable state.
for the vehicle’s out factors indicates that the vehicle at the time of
t is affected by the external potential threat. This value is the sum of the influence value of the diverse disturbance that the vehicle is subjected to, including the influence of other vehicles’ lane changing, the influence of vehicles’ trajectory overlapping, etc.
is the self-factor of the ego vehicle, which indicates the sum value of the abnormal behavior’s impact or unusual state of the vehicle at the moment
t, including continuous violation of regulations or interference with other traffic participants.
is the amount of confidence value’s recovery. The value of
is obtained by calculating the complexity of the driving environment around the ego and converting it using the entropy method. The calculation details of the factors in Equation (
1) will be fully described in the following sections.
Section 3.2 is for
,
Section 3.3 is for
and
Section 3.4 is for
and
.
The output values of the VC model are presented in the form shown in
Figure 3.
It should be mentioned that the dynamic influence on a human driver’s confidence usually exists within the observable range of the driver. In our confidence model, we have set a fixed computational constraint range for all calculations of influence factors, which means only the factors within the range of the ego vehicle will be considered.
3.2. 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:
where
v represents the instantaneous speed of the vehicle.
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.
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):
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
, solve for
t is the required time. When
, the required time is
. This calculation is aiming to figure out the timestamp when the interacting vehicle is expected to drive into and out of the interaction area. By using Equation (
4), we can obtain four time points, the entry time
and the exit time
of the ego vehicle and the entry time
and the exit time
of the corresponding vehicle:
The intersection of the two time periods
is calculated.
i denotes the vehicle serial number that has an intersection with the reachable area of the ego, and if this intersection is empty, it means that under the assumption of this paper, there is no possibility of a future collision between the two vehicles. If the intersection set
) exists, then calculate the collision possibility of ego using Equation (
5):
where
denotes the temporal crossover rate between the ego and the vehicle with serial number
i during the interactions.
In addition to the degree of temporal crossover of the two vehicles, the location of the interaction region relative to the ego vehicle can lead to different degrees of influence on driving confidence. We use the minimum distance
from the front of the ego vehicle to each interaction area, combined with the temporal crossover rate above, to calculate the final influence of interaction on confidence. Using Equation (
6), the value of the interference during interaction can be figured out.
where
k is a human-set parameter, which indicates the different impact coefficients of different types of vehicles. For example, the
k for large delivery vehicles is higher than that for sedans.
is a human-set parameter, which determines the standard value of interaction loss at the reference distance.
In addition to the interaction interference, the confidence of human drivers will also be affected by the abnormal actions of the surrounding traffic participants. The sudden braking operation of the surrounding vehicles may represent a sudden dangerous situation. In the calculation, we only focus on the vehicle with the maximum acceleration around the ego. The influence of the vehicle on the confidence value can be expressed as Equation (
7):
where
k is the human-set parameter.
a is the acceleration value of the maximum accelerating vehicle.
is the constant value for abnormal acceleration, which can be regarded as the response threshold and can always be set to be 7 m/s
.
m is the human-set constant used to determine the minimum amount of affection in case of abnormal acceleration. As the abnormal acceleration’s value rises, the confidence impact will present an exponential expansion.
By combining the interaction and the abnormal behavior confidence interference, the combined effect of two major external interference factors on the ego can be described as Equation (
8):
where
t in the equation represents the time of the calculation.
and
indicate the proportional relationship between the two main influences.
3.3. The Influence of Self Factors
We define the relationship related to the time of the autonomous driving system publication as the baseline coefficients for the self-factor value before calculating the confidence risk associated with the abnormal behavior and state of the ego vehicle, as the following Equation (
9):
where
y is the length of time the system has been in use.
is a set parameter, used to adjust the relationship between driving confidence and the length of time the system has been released.
is also a set parameter, used to determine the initial value for the new autonomous driving system.
From the kinetic aspect to detect the abnormal state of the ego, we believe that as long as the ego vehicle is in a state of especially high acceleration or deceleration, as well as a high wheel steering rate can lead to the loss of confidence. We adopt the same equation as the abnormal acceleration and deceleration of the other car in Equation (
7) with the different threshold settings to guarantee comfort. We use the rate of change of the ego vehicle’s heading angle to represent the wheel steering which is shown in Equation (
10):
where
f is the operating frequency of the VC model.
is the value of yaw change at the moment
t.
n is the baseline changing rate.
k is the human-set parameter and
m is the constant that is used to determine the minimum amount of loss.
Combining with the abnormal state, the final value of confidence loss due to the unusual driving state of the ego can be expressed as Equation (
11):
where
is the value of abnormal confidence loss for acceleration and deceleration,
is the value of confidence loss for abnormal steering rate.
t represents the moment
t as mentioned above.
and
are set parameters that can adjust the numerical proportional relationship between the two affections.
For the possible violations of autonomous vehicles, we take into account three common but relatively not completely unacceptable violations that can bring direct danger to the vehicle: (1) Driving over the speed limit (2) Driving against traffic (3) Frequent Lane changes.
For speeding, the traffic law allows vehicles to exceed a certain percentage of the road limit for a short period of time during overtaking. However, when the vehicle is seriously speeding, the greater margin of speeding the greater the negative impact on drivers’ confidence. At the same time, the instantaneous speeding penalty should be capped. We use the following Equation (
12) to quantify the impact of the ego vehicle’s speeding:
where
is the speed limit of the road section.
is the set over speeding amplitude.
k is a set parameter, which determines the maximum value of the impact of each time period because of the speeding reason.
In the case of going against traffic, although overtaking on a two-lane road will require temporally driving on the opposite lane, the duration of driving on the opposite lane should not be too long. There will be an amount of impact on the driver’s confidence when traveling in the reverse lane. The VC model will check the vehicle’s heading and the road heading. While the two heading value is significantly different, the confidence loss calculation will be carried out according to Equation (
13).
The loss constant
in Equation (
13) can be set artificially to describe the driver’s tolerance of avoiding or penalizing the vehicles against traffic time.
The behavior of continuous fast lane change is relatively easy to define, it only needs to set a fixed loss of confidence for each lane change, which can generate a large number of penalty values in a short period of frequent lane change scenarios. Once the ego vehicle touches the road edge, it can be considered a lane change.
in Equation (
14) is the loss constant that can be set artificially. The value of
should not be set too large to avoid excessive confidence impact caused by the usual lane change.
The equation for the confidence loss of the ego vehicle’s law-breaking actions is as follows:
The overall self-factors influence on confidence value can be synthesized as Equation (
16).
where
and
are the set parameters.
U for the publication time according to Equation (
9).
3.4. The Recover Rate
The value of recovery in the driving confidence model is negatively correlated with the complexity of the dynamic and static surrounding environment in which the vehicle is driven [
21,
22]. This paper specifies that
in Equation (
1) is always a value not less than 0.
We refer to the design framework of safety entropy [
23] for the solution of environmental complexity and determine the influencing factors in safety entropy by investigating the environmental factors. The factors that affect the environmental complexity are divided into three parts: (1) The traffic participant’s properties and status, (2) The static environment of the current road where the ego vehicle is traveling, and (3) The real-time weather condition. The specific details of the influencing factors can be seen in
Table 1.
As can be seen from
Table 1, each of the three major factors affecting environmental complexity is subdivided into four complexity-related subfactors.
We normalize the values of the subfactors in
Table 1 and unifies them as the greater the value the higher the impact. In addition, all the values are positive, and the normalization equation is as shown:
where
x represents the value of the influence subfactor. The constant 1.001 is set to avoid a normalized value of 0.
We analyzed the survey report and combined the results with real-world driving experience to figure out that human drivers will drive relatively more cautiously in scenarios where the average speed of the surrounding active traffic participants is higher [
24]. In the road traffic participants’ part, higher average speeds increase the danger level in case of an accident. Meanwhile, the larger the maximum speed difference between the surrounding active state participants, the greater the diversity in their behavior, making cautious driving necessary in such scenarios. Greater numbers of surrounding traffic participants and large vehicles such as trucks and vans require a higher level of caution. The closer the road is to the border, the greater the influence on driver confidence.
We consider the type of road construction, such as viaducts or underground passages, when assessing the influence on driver confidence. We assign a fixed value to this factor to deal with the non-quantifiable problem, for example, 0 on a common road and 1 underground. For the number of lanes factor, the more the number of lanes in the same direction of the current road, the more operational options for vehicles, and the higher the complexity.
For the weather condition factors, we adopt some simple quantification methods for factors: Rainfall is quantified as mm/h. The visibility parameter is quantified as m, and the visibility should take inverse preprocessing. Furthermore, the wind speed parameter’s unit m/s. The travel time factor is non-quantifiable which includes day and night labels.
The overall calculation of complexity is as follows. Equation (
18) for each main part to solve the occupancy rate of its indicators.
The information entropy
for each major type was obtained using Equation (
19). Furthermore, the weighting relationship between each major category of influencing factors was obtained using Equation (
20).
Finally, we use the specific normalized values of each subfactor to calculate the entropy value of each major factor category by using Equation (
21). Combining the weights in Equation (
20), we can obtain the overall complexity value of the AVs’ driving environment named
.
In Equation (
23),
is the numerical amplification and
is the standard environmental norm. The cooperation of the two quantities serves to amplify and obtain a more pronounced change in the entropy value. By setting reasonable parameters, the distribution of environmental complexity values obtained by our experimental calibration and their corresponding complexity levels are listed in
Table 2:
Based on the value of the environmental complexity, the amount of recovery of real-time driving confidence can be set. A simulation program with a clock interval of 0.05 s is used in this paper, and according to the size of this interval, we first divide five different environmental complexity levels. Where each level will correspond to a number to facilitate the quantification of the recovery amount. The recovery value can be calculated as Equation (
24).
in the equation corresponds to the value of the complexity level. is the current vehicle’s confidence value. We assume that the recovery of confidence value should be relatively difficult when confidence is already in a lower value state. This results in the higher the complexity of the environment, the longer time it takes to re-establish confidence. k is the base recovery amount constant.
The maximum value of confidence can also be set according to the . However, in this paper, we specify as 100 in order to simplify the verification process.