Study and Simulation Analysis of Vehicle Rear-End Collision Model considering Driver Types

. The reasonable distance between adjacent cars is very crucial for roadway traﬃc safety. For diﬀerent types of drivers or diﬀerent driving environments, the required safety distance is diﬀerent. However, most of the existing rear-end collision models do not fully consider the subjective factor such as the driver. Firstly, the factors aﬀecting driving drivers’ characteristics, such as driver age, gender, and driving experience are analyzed. Then, on the basis of this, drivers are classiﬁed according to reaction time. Secondly, three main factors aﬀecting driving safety are analyzed by using fuzzy theory, and the new calculation method of the reaction time is obtained. Finally, the improved car-following safety model is established based on diﬀerent reaction time. The experimental results have shown that our proposed model obtained more accurate vehicle safety distance with varied traﬃc kinematic conditions (i.e., diﬀerent traﬃc states, varied driver types, etc.). The ﬁndings can help traﬃc regulation departments issue early warnings to avoid potential traﬃc accidents on roads.


Introduction
Private car has become affordable for public with the quick development of economic and promotion of manufacture technique.An adverse effect of is increasing traffic accidents on roadways which greatly imperil traffic safety and efficiency.
e previous studies have shown that drivers are assumed to the take the major responsibility for the traffic accidents (i.e., a few accidents are triggered by vehicle defects).Indeed, over 80% traffic accidents on roads can be ascribed to driver misconduct (e.g., answering cell phone, smoking, and taking naps) [1,2].It is observed that varied driver characteristics (e.g., age, gender, experience, and speed of response) can impose different impact on individual vehicle maneuver procedure.
e driver response time for different driver types for taking actions against dangerous driving situations is varied, which has attracted a lot of research attentions [3][4][5].More specifically, the aggressive drivers require smaller displacement between neighboring vehicles when traveling on road, while the mild drivers are likely to keep sufficient vehicle headway for the purpose of avoiding potential traffic accidents.Note that quantifying such vehicle rear end displacement considering driver types is not easy, which is indeed a hot topic in the transportation safety research community.
Several studies have been conducted to establish minimum safety following distance model with vehicle kinematic data (i.e., driver types, vehicle acceleration/decelerations distributions, etc.).Zhang and Hao deeply analyzed the resistance influence on the minimum safety distance, which was involved with air resistance, road resistance, vehicle wheel rolling resistance, etc. [6].Xu et al. proposed a minimum safety car-following model under different driver states by considering the vehicle acceleration/deceleration in vehicle braking process [7].Spyropoulou proposed a novel vehicle safety distance model which considered the constraints of vehicle speed variation, minimal safety distance, etc. [8].Hu et al. firstly studied driver behavior differences and empirical judgment ratio distributions in an abnormal traffic scenario and then proposed a car-following model for estimating the minimum safety distance for the emergency evacuation vehicle [9].Similar research studies can be found in [10][11][12][13][14].
Many studies have been conducted to analyze the relationship between traffic safety and drivers characteristic, including driver personality, physical fitness, driver distraction, etc. [15][16][17][18].Tang and Xia implemented a series of experiments to measure the reaction time of four different driver types and further studied its impact on the minimal safety distance of preventing rear-end collision [17].Tang and Xia analyzed a variety of factors that affect the driver reaction time by using fuzzy mathematics theory and then proposed a novel model to estimate driver reaction time [17].By considering influence of driver individual differences, vehicle braking performance, and driving states, Zhang et al. established a carfollowing model with minimum safety distance for three typical traffic states [19].Similarly, Xue et al. proposed a rearend collision behavior model considering the individual differences of drivers [20].Some scholars analyzed the relationship between other influencing factors and car safety, such as the speed relation among adjacent vehicles [21,22].
Fuzzy relevant models have been proposed to simulate real-world traffic situations considering varied driver types and thus provide optimal traffic control strategies for the purpose of ensuring traffic safety and efficiency (i.e., with minimal waiting time, short queue length, etc.).Chai et al. proposed a simulation-based approach to measure driver cognitive failures, which was implemented with fuzzy logic and cellular automata model [23].Azimirad et al. proposed a novel fuzzy traffic controller to formulate and optimize traffic control at isolated signalized intersection [24].Li proposed a dynamic fuzzy neural networks traffic flow prediction model to accurately obtain traffic flow prediction under chaos traffic state [25].Bocklisch et al. proposed an adaptive fuzzy pattern classification model for simulating nonlinear, multidimensional transition processes, which aims to identify the lane change intentions for varied driver types [26].
e previous studies focusing on vehicle rear-end collision analysis mainly employed the kinematic data to estimate minimal neighboring vehicle distance with the quantitative indicators (i.e., speed, displacement, etc.), which did not consider the qualitative indicators (e.g., driving behavior differences influence) and mainly led to biased results.e advantages of fuzzy theory can help us quantify the physiological and psychological characteristics of vehicle drivers and thus provide us a more holistic view of the car-following model.e paper is organized as follows.Driver features were deeply analyzed in Section 2, and the proposed safety model was illustrated in Section 3. e experimental results were described in Section 4, and Section 5 concluded the paper.

Driving Characteristics Analysis.
e vehicle controlling procedure consists of environment perception, potential risk judgment and traveling decision making, and vehicle maneuver, which can be found in Figure 1.In the perception stage, the driver's control of car speed and perception of surrounding environment have a close bearing on the driver's reaction speed to the unexpected situation.In the judgment and decisionmaking stage, the driver makes control decisions based on driving experience to ensure driving safety.In the operation stage, the driver controls the car accelerating, decelerating, turning, and braking according to the decision.Note that over 80% traffic accidents are triggered due to the driver errors [16,27,28].To avoid the potential traffic accidents, drivers are supposed to take early and correct maneuver activities (i.e., identify risky behaviors, decide the suitable travel behavior, take maneuver activities, etc.) for each of vehicle maneuver steps.e driver's individual characteristics can be analyzed from physiological and psychological factors, for example, the driver's age, gender and driving age, fatigue and emotions while driving, the driver's own driving style and safety attitude, traffic regulations awareness level, and so on.ese research studies show that [29][30][31] (1) the reaction speed to the emergency situation will be significantly reduced with age; (2) in terms of emergency response capacity and the ability to cope with the complex external environment, the female drivers are worse than the male drivers; (3) the experienced driver has the faster reaction in the emergency situation; and (4) the adventure driver is more likely to be in danger of rapid acceleration, deceleration, and changing lanes than the cautious driver.Based on the above analysis, it is necessary to quantify the driving characteristics into quantifiable parameters, such as reaction time, which can be introduced into the modeling process of the improved car-following model.

Relationship Analysis between Driver Characteristics and Reaction Time.
e driver age, driving experience duration, and fatigue level are considered as crucial elements for quantifying analysis on the relationship between driver characteristics and reaction time [32,33].Previous studies have evaluated the reaction time variation under different traffic flow density constraints by building three kinds of traffic flow density scenarios, which suggested that the driver reaction time tends to be shorter in higher traffic flow density.Factually, the driver in a conservative driving style (intends to obtain much larger distance) has the longest reaction time regardless of the vehicle braking manner.e driver age has a significant interference to driver reaction time which is obviously observed in the emergency traffic situations.
e statistical distributions of empirical traffic data suggested that the average and variance of collision avoidance reaction time of 18 to 30 years old drivers are the smallest and the drivers over 51 years old have the longest collision avoidance reaction time.Previous studies suggested that aging drivers need more reaction time because they require longer time to think carefully about various potential travel strategies [18,34,35]: where t r is the driver's reaction time, C is the response degree of different types of drivers under different conditions, and Q is vehicle braking type.

Journal of Advanced Transportation
According to the formula, the reaction time of several representative driver types is obtained which is shown in Table 1 (for details, see [9]).

Analysis and Improvement of Safety
Distance Model e reaction time of various types of drivers is different when they face the same situation, so the required safety distance is also different.e improvement of safety distance model mainly researches on driver's reaction time and confirms a more reasonable safety distance.When establishing the model, the driver's reaction time varies from different influencing factors.Some researchers discovered that these influencing factors are not absolutely independent, but have a certain intersection, and they have characteristics of fuzziness [36,37].erefore, this study uses the fuzzy mathematics principle to calculate reaction time of different types of drivers.

Reaction Time Analysis.
ere are many factors affecting the driver's driving behavior, and the impact of these factors can be measured by the reaction time of drivers.rough indepth analysis and comparative research, three main factors affecting response time were selected in this manuscript: age, driving age, and fatigue.Other factors will be considered in future research.
In the fuzzy inference system, there are three variables including input variables, fuzzy rules, and fuzzy output, as shown in Figure 2.After determining the influencing factors, the fuzzy distribution method is used to determine membership function, and the Gaussian or semi-Gaussian distributions are selected which include small, large, and intermediate types that correspond exactly to the conservative, adventurous, and conventional types of driver classification [38,39].
( e inference system finally derives the driver's reaction time according to fuzzy theory.Based on a large number of survey results, it is generally believed that the driver's response time interval is [0.2, 3.0], which can be roughly divided into short, medium, and long.erefore, the Gaussian-type function is used for the membership function.
After inputting the driver's age, driving age, and fatigue level, the fuzzification process can be performed to obtain the approximate value of the driver's reaction time, and then the center of gravity defuzzification method is used for antifuzzification, and the precise value of the reaction time can be obtained.Its expression is where  is the integral of all subsets in the continuous domain y and y 0 means that the area of left and right sides is the same.

Improved Safety Distance Model
Establishment. e reaction time is used to distinguish different types of drivers and is used as a parameter in the process of model building.In order to analyze different driving conditions comprehensively, three states of static, uniform, and deceleration of the preceding car were analyzed in order to establish the safety distance model.
(1) Static state of the front car: When the front vehicle is stationary, the positional relationship between the two vehicles is shown in Figure 3.

Journal of Advanced Transportation
At position 1, the rear car begins to realize the danger.After the reaction time t, the rear car will arrive at position 2. e rear car maintains a constant speed v 1 during the whole process, and traveling distance in reaction time t is At position 2, the rear car starts to brake at the maximum deceleration, and its speed drops to 0 at position 3.At this time, the distance between the front and rear cars is d 0 , and the distance traveled by rear car during this time is To avoid collision of two cars, the minimum safety distance needed to maintain is where a 1 is the maximum braking deceleration of the rear car and d 0 is the minimum safety distance required for the two cars.(2) Uniform state of the front car: When two cars are driving at a constant speed, a collision occurs only when the speed of the following vehicle is faster than the speed of the preceding vehicle.e positional relationship is shown in Figure 4.
Assume that the rear car is in position 1 and the front car is in position 3. e rear car will reach position 2 after reaction time t, and then it starts to brake.During the reaction time, the rear car maintains a constant speed, and the distance traveled is At position 2, assume that the rear car begins to decelerate until its speed is the same with the front car; meanwhile, the front car arrives at position 5 and the rear car arrives at position 4.In this process, the distance traveled by the rear car and the front car is, respectively, To avoid collisions of two cars, the minimum safety distance needed to maintain is (3) Deceleration state of the front car: When the speed of the rear car is faster than that of the front car, the two cars have the possibility of collision.In order to prevent this collision, the speed of the two cars is equal or the speed is zero when the distance between two cars is d 0 .e position relationship is shown in Figure 5.
With the previous analysis, the rear car travels from position 1 to position 2 within the reaction time t. e distance traveled is At position 2, the rear car begins to brake.When the speed of two cars is equal (set as v), the rear and the front car  e travel distances are as follows: To avoid collisions, the minimum safety distance to be maintained is where a 2 is the braking deceleration of the front car and the remaining parameters have the same meaning as mentioned above.  2 shows the driver reaction time obtained by substituting three groups of factors into the fuzzy system.From Table 2, some conclusions can be obtained as follows.(1) When the driver's age and driving age are constant, the driver's reaction time is directly proportional to the fatigue degree.(2) When the driver's age and fatigue degree are constant, the driver's reaction time is directly proportional to the driving age.(3) When the driver's driving age and fatigue value are constant, the driver's reaction time is directly proportional to the age.And, between the ages of 30 and 45, the driver's reaction time grows slowly, and after the age of 45, the growth rate accelerated.

Comparative Analysis of
(b) e Honda model: (c) Research institutes' model: (d) Universities' models: In the formulas, d, D, S 0 , d w , and D x represent the safety distance, v 1 and v 2 represent the speed of the leading and following car, a 1 and a 2 represent the acceleration of the leading and following car, v r is the relative speed, d 0 and d represent the minimum safety spacing, l is the length of car, τ is the response time of driver, and the remaining parameters are the same as before.

Model Comparison and Analysis.
e reaction time deduced by the fuzzy theory is substituted into different safety distance models.Under different reaction times, the safety distances calculated by different models are different.Four groups of data are randomly selected as comparisons.
(1) Static state of the front car: D 1 represents the safety distance of modified model when the leader car is in the anchoring state.In such traffic state, the parameters are set as follows: the initial speed of the rear car is 60 km/h and the deceleration is 4 m/s 2 .e safety distance between the two cars is 50 m.More specific parameter settings are shown in Table 3.
(a) We employ MATLAB to implement experiment when the driver reaction time is set to 1.21 s, and the corresponding results are shown in Figure 6. e comparison result of safety distances for different models is shown in Figure 7.
From Figure 6, we can see that the two curves intersect at 50 m, which proves that setting the safety distance to 50 m is not enough and a collision will occur.Under this condition, the safety distance of rear-end collision model is 54.89 m by calculation.From Figure 7, it can be found that in all the models, the improved model D 1 is the closest to the minimum safety distance.
(b) We employ MATLAB to implement experiment when the driver reaction time is set to 1.58 s, and the corresponding results are shown in Figure 8. e comparison chart of safety distances for different models is shown in Figure 9.
From Figure 8, the safety distance is 61.06 m.From Table 3, it can be seen that the models D 1 and D are close to the minimum safety distance.However, due to the need of keeping a certain distance between two cars when they are still, the model D 1 is more appropriate.
(2) Uniform state of the front car: D 2 represents the safety distance of the modified model when the front car is in the uniform state.In such traffic state, the parameters are set as follows: the speed and deceleration of the rear car are 80 km/h and 4 m/s 2 and the speed of the front car is 40 km/h.e safety distance between the two cars is 50 m.More specific traffic parameters for obtaining safety distance in the traffic state are shown in Table 4.
(a) We employ MATLAB to implement experiment when the driver reaction time is set to 1.39 s, and the corresponding results are shown in Figure 10.e comparison chart of safety distances for different models is shown in Figure 11.
It can be seen from Figure 10 that since there are no intersections between the two curves, it can be seen that the safety distance of 50 m is sufficient, and the calculated safety distance to prevent rear end is 31.37 m. Figure 11 shows that the safe distance of model D 2 is closest to 31.37 m, so the model D 2 is most suitable.
(b) We employ MATLAB to implement experiment when the driver reaction time is set to 1.73 s, and the corresponding results are shown in Figure 12. e comparison of the safety distances of different models is shown in Figure 13.
It can be seen from Figure 12 that since there are no intersections between the two curves, it can be seen that the 50 m safety distance is sufficient.According to MATLAB calculations, the safety distance is 34.78 m if there is no collision.From Figure 13, it is found that the safe distance of model D 2 is closest to 31.37 m, so the model D 2 is most suitable.
(3) Deceleration state of the front car: D 3 represents the safety distance of the modified model when the front car is in the deceleration state.In this driving condition, the parameters are set as follows: the speed and deceleration of the rear car are 100 km/h and 5 m/s 2 and the speed and deceleration of the front car are 60 km/h and 3 m/ s 2 .e safety distance between the two cars is 50 m.e safety distance of different models (in the deceleration state) is shown in Table 5.

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Journal of Advanced Transportation (1) When the driver's reaction time is 1.21 s, use MATLAB simulation to obtain results as shown in Figure 14. e comparison of the safety distance of different models is shown in Figure 15.
From Figure 14, there are intersections between the two curves, which proves that it is not enough to set the safety distance to 50 m, and there will be a collision.After the simulation calculation, the    safety distance for preventing rear-end collision is 64.5 m.It can be seen from Figure 15 that the safety distance of model D 3 is the closest to 64.5 m. us, the model D 3 is most suitable.
(2) We employ MATLAB to implement experiment when the driver reaction time is set to 1.73 s, and the corresponding results are shown in Figure 16.From Figure 16, it can be seen that there are intersections between the two curves, which proves that setting the safety distance to 50 m is not enough and a collision will occur.After simulation calculation, the safety distance of preventing rear-end collision is 78.67 m.As can be seen from Figure 17, the safety distance of model D 3 is the closest to 78.67 m, so the model D 3 is most suitable.
We have evaluated varied minimal car-following distances at different speed variations.For instance, the minimal vehicle headways are 32.04 m and 63.86 m when the    following car moves at 60 km/h and 90 km/h with the leading vehicle in the uniform state.e minimal vehicle distance can be shorter when the leading vehicle is in the deceleration state and the following car has same speed constraints (i.e., 60 km/h and 90 km/h).Vehicle speed influence on the simulation model can be summarized as twofolds: (1) the required minimum following distance for the following car is different when the leading car is in different traffic states (e.g., different speeds) and ( 2) the traveling speed of the following car is in positive relationship to that of the minimal car-following distance (i.e., higher speeds requires larger minimal distance).

Conclusions
We have analyzed different driver types' influence on roadway safety and thus quantified the influence by establishing a novel car rear-end collision model.We employed the fuzzy theory to build the reasoning model with the input of typical traffic safety factors (i.e., driver age, driving age, and fatigue degree) and output the driver reaction time.With the aim of obtaining minimum safety following distance, we have deeply analyzed the safety distance between two neighboring vehicles with leading vehicle at varied states.We have estimated minimal safety distances for the two vehicles with the leading car in varied kinematic states (i.e., static, deceleration, and constant speed), which are indeed commonly encountered on the roadway traffic scenarios.Suppose two of the traffic factors are in constant states, driver reaction time is proportional to the variation of the remaining factor (either in negative or positive).
Compared with the traditional rear-end model, the proposed safety distance model established can be adapted to different types of drivers, which can better benefit traffic management and control.

Figure 1 :
Figure 1: Control flow for vehicle maneuver procedure.

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Journal of Advanced Transportation will arrive the position 4 and 5, respectively.

Figure 5 :
Figure 5: Positional relationship (in the deceleration state of the front car).

Figure 6 :
Figure 6: Distance change (in the static state on front car).

Figure 7 :Figure 8 :
Figure 7: Safety distance comparison (in the static state on front car).

Figure 9 :
Figure 9: Safety distance comparison (in the static state of front car).

Figure 10 :
Figure 10: Distance change (in the uniform state of front car).

Figure 11 :
Figure 11: Safety distance comparison (in the uniform state of front car).

Figure 12 :
Figure 12: Distance change (in the uniform state of front car).

Figure 13 :
Figure 13: Safety distance comparison (in the uniform state of front car).

Figure 14 :
Figure 14: Distance change (in the deceleration state of front car).

Figure 15 :Figure 16 :
Figure 15: Safety distance comparison (in the deceleration state of front car).

Figure 17 :
Figure 17: Safety distance comparison (in the deceleration state of front car).

Table 1 :
Reaction time of different types of drivers (s).
Different Models 4.2.1.Traditional Safety Distance Model.So far, there are many classic car safety distance models, such as, the Mazda model, the Honda model, and the models proposed by the domestic research institutes and various universities [14], as follows: (a) e Mazda model:Figure 4: Positional relationship (in the uniform state of the front car).

Table 2 :
Reaction time in different combinations.

Table 3 :
Safety distance of different models (in the static state).

Table 4 :
Safety distance of different models (in the uniform state).

Table 5 :
Safety distance of different models (in the deceleration state).