# Modeling Analysis of Improved Minimum Safe Following Distance under Internet of Vehicles

Authors: Qiang Luo; Meining Ling; Xiaodong Zang; Cong Zhai; Liming Shao; Junheng Yang
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

The development of the Internet of vehicles technology can improve the communication between vehicles, thereby changing the driving behavior of drivers. Therefore, the traditional safe-following model cannot accurately describe the driving behavior and needs to be improved accordingly. First, two key parameters (i.e., drivers’ reaction sensitivity and road friction coefficient) are obtained through a comprehensive comparative analysis of influencing factors on the Internet of vehicles environment. And the calculation methods of these two parameters are proposed by using the multilevel comprehensive weighted evaluation method and the BP neural network. Then, these two key parameters are used to improve the traditional minimum safety distance model for adapting to driving behavior under the Internet of vehicles environment. Finally, through setting up simulation experiments and comparative analysis, the relationship between different influencing factors and the minimum safe following distance is obtained, and the influence degree of different influencing factors is sorted. The most important factor affecting car-following safety is the drivers’ characteristics. It can provide strong theoretical support for the safe driving assistance system of vehicles.


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## Body

## 1. Introduction

The investigation of lots of traffic accidents indicates that the rear-end accident takes up a relatively high proportion and is one of the main traffic accidents. The studies suggested that the rear-end accident is mainly affected by drivers’ characteristics, the acceleration/deceleration performance of vehicles, road condition, and weather situation [1–4]. The influence degree of the four types of factors is different.Jiang and Yu established a highway antitailing model based on a radial basis function neural network by using traditional algorithms [5]. Nevertheless, the model is mainly applied to highways, and the influencing factors considered are relatively single. Based on the drivers’ characteristics (e.g., driving age and gender), Xia established a fuzzy inference system to determine different drivers’ response times and proposed an improved safe car-following model [3]. Peng derived the modified Korteweg-de Vries (mKdV) equation to describe the traffic behavior near the critical point by using linear stability theory and proposed an improved car-following (MCF) model based on the full velocity difference (FVD) model and considering multiple information inputs from preceding vehicles [6, 7]. Wilson and Ward suggested that all vehicle car-following models should be tested for stability to ensure improved performance [8]. Tan and Huang [9] proposed a cooperative collision warning system for vehicle-to-vehicle based on DGPS. However, the safety distance algorithm does not consider the drivers’ driving characteristics and actual road conditions; thus, the practical application of the warning system is low. Luo et al. [10, 11] proposed an improved vehicle rear-end collision model by considering the driver types and multiple factors on the traditional driving environment.With the emergence and development of the Internet of vehicles technology, the driver can obtain more information around in time, and the accuracy of the information is high. It has a greater auxiliary effect on driving behavior and can effectively reduce the occurrence of traffic accidents, especially vehicle rear-end accidents. Sun et al. [12], Wang et al. [13], Hua et al. [14], and Monteil et al. [15] comprehensively consider the drivers’ comprehensive grasp of driving data of the vehicle ahead in the vehicle-to-vehicle interconnection environment, and some improved models have been established. Wang et al. [13] not only considers the influence of the vehicle ahead on the car-following safety but also considers the influence of the secondary leading car and lane-changing behavior, and then, a combined car-following and lane-changing model considering dual leading vehicles is proposed. Ge et al. [16] consider that the rear neighboring vehicles have a certain urging effect on the target vehicle, and the target vehicle will also pay attention to the distance between the rear neighboring vehicle and itself in order to avoid collisions between vehicles. Through analysis and verification, an extended OV model based on backward observation is proposed. The research results show that considering the rearview effect of the rear vehicle can significantly enhance the stability of traffic flow. Gu [17] corresponded to the research objectives in the development stage of the Internet of vehicles with adaptive cruise technology (ACC) and cooperative adaptive cruise technology (CACC) and proposed the multimode, multiobjective, and multilevel car-following models of different development stages.The driver can comprehensively obtain the road and other surrounding vehicles’ information in the Internet of vehicles environment. Thus, the influence of external factors on driving safety becomes weaker. However, some subjective factors will still have an important impact on driving safety. Through comprehensive analysis and comparison, we obtained that the drivers’ reaction sensitivity and the road surface have a greater impact on car-following safety. Thus, this paper sets up the estimation method of these two parameters and then improves the classical minimum car-following distance model by using them. The improved car-following model can be more in line with the driving behavior in the Internet of vehicles environment and can better provide assistance for car-following driving. At the same time, the influence degree of different influencing factors obtained by simulation comparison can provide strong support for vehicle-assisted driving decision-making.


## 2. Analysis of Factors Affecting the Car-Following Safety

From the analysis of the traffic operation system, the factors affecting driving safety can be divided into driver factors, vehicle factors, road factors, and traffic environment factors. There have been many relevant research reports on these factors. For example, the classic safe following distance model mainly focuses on the research of the minimum safety distance between the target vehicle and the leading vehicle during the following process. When the target vehicle brakes after a series of reactions and operational delays, a certain safety distance can be maintained with the front vehicle, and no vehicle rear-end collisions will occur. Under the environment of the vehicle network, the drivers can perceive the information of surrounding vehicles and the traffic environment in time. Thus, the influence of many objective factors has been reduced. By analyzing the existing research literature, two key parameters that affect the minimum safety distance can be obtained as follows: drivers’ reaction sensitivity and road friction coefficient. The reaction sensitivity of the driver can be measured by the reaction time. And the factors affecting the two key parameters will be analyzed as follows.


### 2.1. Drivers’ Reaction Sensitivity

#### 2.1.1. Analysis of Influencing Factors

The factors affecting the drivers’ driving reaction sensitivity can be divided into drivers’ characteristics, traffic conditions, and environmental factors.(1)
Driver’s characteristics: the investigation of many accidents shows that more than 80% of traffic accidents are caused directly or indirectly due to drivers [18]. Thus, the driver’s characteristic is the main factor affecting the driving safety and can be divided into gender, age, driving experience, and physiological status. According to the survey, male and female drivers have different abilities to identify and judge traffic conditions and respond to sudden accidents. In general, the drivers’ reaction sensitivity slows down with age [19]. The more experienced in driving, and the more mature in the treatment of various sudden traffic conditions. When the driver has a good mental state, the perception of road traffic information is accurate [20].(2)
Traffic condition: traffic condition is also an important external factor affecting car-following safety. It can be subdivided into the road feature and the car-following status, which has a great influence on driving behavior. First, the reasonable design of road alignment from three-dimensional (3D) angles (i.e., horizontal, vertical, and cross section) can provide suitable driving conditions and improve the safety of driving. On the contrary, the unreasonable design will cause the vehicle to run unsteadily or even lead to traffic accidents. Second, the car-following status affects the drivers’ driving psychology and increases driving difficulty, thereby affecting the safety of driving.(3)
Environmental factors: for the environmental factors, the factors affecting the drivers’ reaction sensitivity are driving period and weather characteristics. In accordance with traffic accident statistics, the accident-prone time periods are concentrated at noon and at early morning. During the two time periods, the physiological activities should be in a sleepy state. Under the limited lighting conditions at night, the drivers’ visual perception range is reduced, and the field of view is narrowed. In addition, the bad weather affects the drivers’ reaction sensitivity and driving operation, thereby adversely affecting driving safety.To sum up, various factors affecting the driving reaction sensitivity are analyzed. We use the reaction time to quantify the drivers’ reaction sensitivity, and the calculation method of reaction time will be introduced as follows.


#### 2.1.2. Calculation Method of Reaction Time

(1) Hierarchical Classification of Influencing Factors. According to the above analysis of influencing factors, a three-level evaluation system of drivers’ reaction sensitivity (expressed by reaction time) was established. The goal of each layer is one of the three factors analyzed in the previous section. The factor set of the first-level layer contains three factors: driver characteristics, traffic conditions, and environmental factors. The first factor set of the second-level layer contains four factors: gender, age, driving experience, and physical state. The second factor set of the second-level layer contains two factors: road feature and car-following status. The third factor set of the second-level layer contains two factors: driving period and weather.The evaluation and classification of different influencing factors on the second-level factor set are shown in Table1.Table 1
Evaluation and classification of different influencing factors.
FactorsClassification and evaluationGender (I)MaleFemale(80–100)(70–90)Age (I)18–3030–5050–70(80–100)(70–80)(60–70)Driving experience (I)>10 years6–10 years1–5 years(80–100)(70–80)(60–70)Physical state (I)NormalFatigueIllness(80–100)(70–80)(60–70)Road feature (II)Normal roadMonotonous roadSharp-tend road(80–100)(70–80)(60–70)Car-following status (II)Free flowNormal followingEmergency following(80–100)(70–80)(60–70)Driving period (III)Others12 a.m.–14 p.m.22 p.m.–6 a.m.(80–100)(70–80)(60–70)Weather (III)SunnyRainFog(80–100)(70–80)(60–70)(2) Weight Determination of Indicators. The weight set of the first-level factor is set to S, which contains three members: S1, S2, and S3. The weight set of the second-level factor subset is set to Si=si1,si2,……,sit, i = 1, 2, 3.(1)
Evaluation score of the hierarchical factors: the influence degree of the various factors on the second level is scored, as shown in Table2.(2)
Calculation of the weight coefficient: the weight coefficient of each influencing factor is expressed asw1,w2,…wi, and their quantity relationship is ∑i=1nwi=1.MatrixD can be obtained by comparing the above weight coefficients [21].(1)S=w1w1w1w2…w1wnw2w1⋮w2w2⋮…w2wn⋱⋮wnw1wnw2…wnwn.The weight vectorw=β1,β2,β3,……,βnT is used to right multiplying matrix D, and the characteristic equation of matrix D can be expressed as(2)Dw=λw,wherew is a vector combined with a weight factor βi for each influencing factor and λ is the characteristic value of matrix D. The various influencing factors are compared in pairs. The elements dij in matrix D are used as an indicator to measure the relative importance of influencing factors Yi. Thus, the relative importance value can be obtained according to the value of dij. The values of dij are 1, 2, 3, 4, 5, 6, 7, 8, and 9. The more the value of dij, the more important that di has than dj.We tested the consistency of the paired comparison matrix composed of the above impact indicators and found that the consistency ratio (CR) is less than 0.1, which shows that the judgment matrix passes the consistency test [21, 22]. Then, the weight values of the influencing factors at all levels are obtained, as detailed in Table 2.(3)
Calculation of drivers’ reaction sensitivity: the evaluation score of each index at the first level can be obtained in accordance with the influence evaluation score of the corresponding factors at the second level. The calculation expression is(3)Di=∑i=1nYiwi.(4)
Score determination of the evaluation system: the evaluation scoreR of the driver reaction evaluation system can be obtained according to the evaluation score at the first level. The calculation expression is(4)R=∑i=13Diwi.Table 2
Weight values of influencing factors.
1st levelDriver characteristicsTraffic conditionEnvironment factorsWeight0.400.290.312nd levelGenderAgeDriving experiencePhysiological stateRoad featureFollowing statusDriving timeWeatherWeight0.190.260.280.270.450.550.400.60


### 2.2. Road Friction Coefficient

#### 2.2.1. Analysis of Influencing Factors

The existing research literature shows that the factors affecting the road friction coefficient are mainly road characteristics, tire type, and force.(1) Road Characteristics. Roads with different characteristics have different road friction coefficients, and the driving comfort of vehicles driving on them is also different. The influencing factors mainly include the road roughness and wet/dry degree. According to the literature [22–24], the relationship between the road friction coefficient and road type is shown in Table 3.Table 3
Friction coefficients of four types of roads.
No.Road typeRange of variationSet value1Dry asphalt(0.80, 0.95]0.912Wet asphalt(0.61, 0.75]0.713Snow pavement(0.20, 0.27]0.244Ice surface(0, 0.10]0.10(2) Tire Type and Force. In addition to supporting the body weight of the car, the tire can transmit road condition information to the body. Existing researchers have proposed a tire model to describe the relationship between tire force and tire motion parameters, that is, the relationship between tire input and output during vehicle driving under different road conditions. A commonly used tire model consists of a drive force module, a slip angle module, a slip rate module, and a tire force module. The tire force module consists of lateral force and lateral velocity, longitudinal force and longitudinal velocity, return torque, and yaw velocity [25].The longitudinal force, lateral force, and vertical force generated in the process of tire contact with the ground have an important effect on the stability of the vehicle. If these three forces can be coordinated and controlled during driving, the driving smoothness and safety can be improved.


#### 2.2.2. Estimation Method

The road friction coefficient is a key factor affecting the car-following safety and is mainly influenced by road surface condition, vehicle speed, and vehicle tire type. In accordance with our previous research results, the neural network can be used to calculate the road friction coefficient. This method is used to establish a network structure with the input for road characteristics, tire pressure, vehicle speed, and the output for friction coefficient, as shown in Figure1. A detailed description can be found in the literature [10].Figure 1
Neural network structure.In general, the texture of tires can be divided into three categories, ordinary pattern, cross-country pattern, and mixed pattern. Thus, the input has three parameters in this network structure which are road characteristics, tire pressure, and vehicle speed. And the output is the friction coefficients of each tire type. The BP neural network structure adopts a three-layer structure, that is, an input layer, two hidden layers, and an output layer. The input layer contains three layers of input. The first and second layers have four neurons and three neurons, respectively, and the output layer has a single output, as shown in Figure1. The training data come from 120 experimental samples under two road surfaces (asphalt and cement concrete), different weather, and different driving speed.The log-sig, tan-sig, and purelin functions are chosen for the input transfer function of each layer, respectively, as follows [25–28]:(5)O1x=11+e−x,O2x=21+e−2x−1,O3x=x.The relationship between input and output function is(6)a3=O3ω3O2ω2O1ω1I+d1+d2+d3,where Ii≥0i=1,2,3 is an input parameter, Oi is the output result of the ith layer, and ωj,dj denote the weight matrix and an offset value vector at jj=1,2,3 layer. When j=3, the layer represents an output layer and f3 is the output parameter.


## 2.1. Drivers’ Reaction Sensitivity

### 2.1.1. Analysis of Influencing Factors

The factors affecting the drivers’ driving reaction sensitivity can be divided into drivers’ characteristics, traffic conditions, and environmental factors.(1)
Driver’s characteristics: the investigation of many accidents shows that more than 80% of traffic accidents are caused directly or indirectly due to drivers [18]. Thus, the driver’s characteristic is the main factor affecting the driving safety and can be divided into gender, age, driving experience, and physiological status. According to the survey, male and female drivers have different abilities to identify and judge traffic conditions and respond to sudden accidents. In general, the drivers’ reaction sensitivity slows down with age [19]. The more experienced in driving, and the more mature in the treatment of various sudden traffic conditions. When the driver has a good mental state, the perception of road traffic information is accurate [20].(2)
Traffic condition: traffic condition is also an important external factor affecting car-following safety. It can be subdivided into the road feature and the car-following status, which has a great influence on driving behavior. First, the reasonable design of road alignment from three-dimensional (3D) angles (i.e., horizontal, vertical, and cross section) can provide suitable driving conditions and improve the safety of driving. On the contrary, the unreasonable design will cause the vehicle to run unsteadily or even lead to traffic accidents. Second, the car-following status affects the drivers’ driving psychology and increases driving difficulty, thereby affecting the safety of driving.(3)
Environmental factors: for the environmental factors, the factors affecting the drivers’ reaction sensitivity are driving period and weather characteristics. In accordance with traffic accident statistics, the accident-prone time periods are concentrated at noon and at early morning. During the two time periods, the physiological activities should be in a sleepy state. Under the limited lighting conditions at night, the drivers’ visual perception range is reduced, and the field of view is narrowed. In addition, the bad weather affects the drivers’ reaction sensitivity and driving operation, thereby adversely affecting driving safety.To sum up, various factors affecting the driving reaction sensitivity are analyzed. We use the reaction time to quantify the drivers’ reaction sensitivity, and the calculation method of reaction time will be introduced as follows.


### 2.1.2. Calculation Method of Reaction Time

(1) Hierarchical Classification of Influencing Factors. According to the above analysis of influencing factors, a three-level evaluation system of drivers’ reaction sensitivity (expressed by reaction time) was established. The goal of each layer is one of the three factors analyzed in the previous section. The factor set of the first-level layer contains three factors: driver characteristics, traffic conditions, and environmental factors. The first factor set of the second-level layer contains four factors: gender, age, driving experience, and physical state. The second factor set of the second-level layer contains two factors: road feature and car-following status. The third factor set of the second-level layer contains two factors: driving period and weather.The evaluation and classification of different influencing factors on the second-level factor set are shown in Table1.Table 1
Evaluation and classification of different influencing factors.
FactorsClassification and evaluationGender (I)MaleFemale(80–100)(70–90)Age (I)18–3030–5050–70(80–100)(70–80)(60–70)Driving experience (I)>10 years6–10 years1–5 years(80–100)(70–80)(60–70)Physical state (I)NormalFatigueIllness(80–100)(70–80)(60–70)Road feature (II)Normal roadMonotonous roadSharp-tend road(80–100)(70–80)(60–70)Car-following status (II)Free flowNormal followingEmergency following(80–100)(70–80)(60–70)Driving period (III)Others12 a.m.–14 p.m.22 p.m.–6 a.m.(80–100)(70–80)(60–70)Weather (III)SunnyRainFog(80–100)(70–80)(60–70)(2) Weight Determination of Indicators. The weight set of the first-level factor is set to S, which contains three members: S1, S2, and S3. The weight set of the second-level factor subset is set to Si=si1,si2,……,sit, i = 1, 2, 3.(1)
Evaluation score of the hierarchical factors: the influence degree of the various factors on the second level is scored, as shown in Table2.(2)
Calculation of the weight coefficient: the weight coefficient of each influencing factor is expressed asw1,w2,…wi, and their quantity relationship is ∑i=1nwi=1.MatrixD can be obtained by comparing the above weight coefficients [21].(1)S=w1w1w1w2…w1wnw2w1⋮w2w2⋮…w2wn⋱⋮wnw1wnw2…wnwn.The weight vectorw=β1,β2,β3,……,βnT is used to right multiplying matrix D, and the characteristic equation of matrix D can be expressed as(2)Dw=λw,wherew is a vector combined with a weight factor βi for each influencing factor and λ is the characteristic value of matrix D. The various influencing factors are compared in pairs. The elements dij in matrix D are used as an indicator to measure the relative importance of influencing factors Yi. Thus, the relative importance value can be obtained according to the value of dij. The values of dij are 1, 2, 3, 4, 5, 6, 7, 8, and 9. The more the value of dij, the more important that di has than dj.We tested the consistency of the paired comparison matrix composed of the above impact indicators and found that the consistency ratio (CR) is less than 0.1, which shows that the judgment matrix passes the consistency test [21, 22]. Then, the weight values of the influencing factors at all levels are obtained, as detailed in Table 2.(3)
Calculation of drivers’ reaction sensitivity: the evaluation score of each index at the first level can be obtained in accordance with the influence evaluation score of the corresponding factors at the second level. The calculation expression is(3)Di=∑i=1nYiwi.(4)
Score determination of the evaluation system: the evaluation scoreR of the driver reaction evaluation system can be obtained according to the evaluation score at the first level. The calculation expression is(4)R=∑i=13Diwi.Table 2
Weight values of influencing factors.
1st levelDriver characteristicsTraffic conditionEnvironment factorsWeight0.400.290.312nd levelGenderAgeDriving experiencePhysiological stateRoad featureFollowing statusDriving timeWeatherWeight0.190.260.280.270.450.550.400.60


## 2.1.1. Analysis of Influencing Factors

The factors affecting the drivers’ driving reaction sensitivity can be divided into drivers’ characteristics, traffic conditions, and environmental factors.(1)
Driver’s characteristics: the investigation of many accidents shows that more than 80% of traffic accidents are caused directly or indirectly due to drivers [18]. Thus, the driver’s characteristic is the main factor affecting the driving safety and can be divided into gender, age, driving experience, and physiological status. According to the survey, male and female drivers have different abilities to identify and judge traffic conditions and respond to sudden accidents. In general, the drivers’ reaction sensitivity slows down with age [19]. The more experienced in driving, and the more mature in the treatment of various sudden traffic conditions. When the driver has a good mental state, the perception of road traffic information is accurate [20].(2)
Traffic condition: traffic condition is also an important external factor affecting car-following safety. It can be subdivided into the road feature and the car-following status, which has a great influence on driving behavior. First, the reasonable design of road alignment from three-dimensional (3D) angles (i.e., horizontal, vertical, and cross section) can provide suitable driving conditions and improve the safety of driving. On the contrary, the unreasonable design will cause the vehicle to run unsteadily or even lead to traffic accidents. Second, the car-following status affects the drivers’ driving psychology and increases driving difficulty, thereby affecting the safety of driving.(3)
Environmental factors: for the environmental factors, the factors affecting the drivers’ reaction sensitivity are driving period and weather characteristics. In accordance with traffic accident statistics, the accident-prone time periods are concentrated at noon and at early morning. During the two time periods, the physiological activities should be in a sleepy state. Under the limited lighting conditions at night, the drivers’ visual perception range is reduced, and the field of view is narrowed. In addition, the bad weather affects the drivers’ reaction sensitivity and driving operation, thereby adversely affecting driving safety.To sum up, various factors affecting the driving reaction sensitivity are analyzed. We use the reaction time to quantify the drivers’ reaction sensitivity, and the calculation method of reaction time will be introduced as follows.


## 2.1.2. Calculation Method of Reaction Time

(1) Hierarchical Classification of Influencing Factors. According to the above analysis of influencing factors, a three-level evaluation system of drivers’ reaction sensitivity (expressed by reaction time) was established. The goal of each layer is one of the three factors analyzed in the previous section. The factor set of the first-level layer contains three factors: driver characteristics, traffic conditions, and environmental factors. The first factor set of the second-level layer contains four factors: gender, age, driving experience, and physical state. The second factor set of the second-level layer contains two factors: road feature and car-following status. The third factor set of the second-level layer contains two factors: driving period and weather.The evaluation and classification of different influencing factors on the second-level factor set are shown in Table1.Table 1
Evaluation and classification of different influencing factors.
FactorsClassification and evaluationGender (I)MaleFemale(80–100)(70–90)Age (I)18–3030–5050–70(80–100)(70–80)(60–70)Driving experience (I)>10 years6–10 years1–5 years(80–100)(70–80)(60–70)Physical state (I)NormalFatigueIllness(80–100)(70–80)(60–70)Road feature (II)Normal roadMonotonous roadSharp-tend road(80–100)(70–80)(60–70)Car-following status (II)Free flowNormal followingEmergency following(80–100)(70–80)(60–70)Driving period (III)Others12 a.m.–14 p.m.22 p.m.–6 a.m.(80–100)(70–80)(60–70)Weather (III)SunnyRainFog(80–100)(70–80)(60–70)(2) Weight Determination of Indicators. The weight set of the first-level factor is set to S, which contains three members: S1, S2, and S3. The weight set of the second-level factor subset is set to Si=si1,si2,……,sit, i = 1, 2, 3.(1)
Evaluation score of the hierarchical factors: the influence degree of the various factors on the second level is scored, as shown in Table2.(2)
Calculation of the weight coefficient: the weight coefficient of each influencing factor is expressed asw1,w2,…wi, and their quantity relationship is ∑i=1nwi=1.MatrixD can be obtained by comparing the above weight coefficients [21].(1)S=w1w1w1w2…w1wnw2w1⋮w2w2⋮…w2wn⋱⋮wnw1wnw2…wnwn.The weight vectorw=β1,β2,β3,……,βnT is used to right multiplying matrix D, and the characteristic equation of matrix D can be expressed as(2)Dw=λw,wherew is a vector combined with a weight factor βi for each influencing factor and λ is the characteristic value of matrix D. The various influencing factors are compared in pairs. The elements dij in matrix D are used as an indicator to measure the relative importance of influencing factors Yi. Thus, the relative importance value can be obtained according to the value of dij. The values of dij are 1, 2, 3, 4, 5, 6, 7, 8, and 9. The more the value of dij, the more important that di has than dj.We tested the consistency of the paired comparison matrix composed of the above impact indicators and found that the consistency ratio (CR) is less than 0.1, which shows that the judgment matrix passes the consistency test [21, 22]. Then, the weight values of the influencing factors at all levels are obtained, as detailed in Table 2.(3)
Calculation of drivers’ reaction sensitivity: the evaluation score of each index at the first level can be obtained in accordance with the influence evaluation score of the corresponding factors at the second level. The calculation expression is(3)Di=∑i=1nYiwi.(4)
Score determination of the evaluation system: the evaluation scoreR of the driver reaction evaluation system can be obtained according to the evaluation score at the first level. The calculation expression is(4)R=∑i=13Diwi.Table 2
Weight values of influencing factors.
1st levelDriver characteristicsTraffic conditionEnvironment factorsWeight0.400.290.312nd levelGenderAgeDriving experiencePhysiological stateRoad featureFollowing statusDriving timeWeatherWeight0.190.260.280.270.450.550.400.60


## 2.2. Road Friction Coefficient

### 2.2.1. Analysis of Influencing Factors

The existing research literature shows that the factors affecting the road friction coefficient are mainly road characteristics, tire type, and force.(1) Road Characteristics. Roads with different characteristics have different road friction coefficients, and the driving comfort of vehicles driving on them is also different. The influencing factors mainly include the road roughness and wet/dry degree. According to the literature [22–24], the relationship between the road friction coefficient and road type is shown in Table 3.Table 3
Friction coefficients of four types of roads.
No.Road typeRange of variationSet value1Dry asphalt(0.80, 0.95]0.912Wet asphalt(0.61, 0.75]0.713Snow pavement(0.20, 0.27]0.244Ice surface(0, 0.10]0.10(2) Tire Type and Force. In addition to supporting the body weight of the car, the tire can transmit road condition information to the body. Existing researchers have proposed a tire model to describe the relationship between tire force and tire motion parameters, that is, the relationship between tire input and output during vehicle driving under different road conditions. A commonly used tire model consists of a drive force module, a slip angle module, a slip rate module, and a tire force module. The tire force module consists of lateral force and lateral velocity, longitudinal force and longitudinal velocity, return torque, and yaw velocity [25].The longitudinal force, lateral force, and vertical force generated in the process of tire contact with the ground have an important effect on the stability of the vehicle. If these three forces can be coordinated and controlled during driving, the driving smoothness and safety can be improved.


### 2.2.2. Estimation Method

The road friction coefficient is a key factor affecting the car-following safety and is mainly influenced by road surface condition, vehicle speed, and vehicle tire type. In accordance with our previous research results, the neural network can be used to calculate the road friction coefficient. This method is used to establish a network structure with the input for road characteristics, tire pressure, vehicle speed, and the output for friction coefficient, as shown in Figure1. A detailed description can be found in the literature [10].Figure 1
Neural network structure.In general, the texture of tires can be divided into three categories, ordinary pattern, cross-country pattern, and mixed pattern. Thus, the input has three parameters in this network structure which are road characteristics, tire pressure, and vehicle speed. And the output is the friction coefficients of each tire type. The BP neural network structure adopts a three-layer structure, that is, an input layer, two hidden layers, and an output layer. The input layer contains three layers of input. The first and second layers have four neurons and three neurons, respectively, and the output layer has a single output, as shown in Figure1. The training data come from 120 experimental samples under two road surfaces (asphalt and cement concrete), different weather, and different driving speed.The log-sig, tan-sig, and purelin functions are chosen for the input transfer function of each layer, respectively, as follows [25–28]:(5)O1x=11+e−x,O2x=21+e−2x−1,O3x=x.The relationship between input and output function is(6)a3=O3ω3O2ω2O1ω1I+d1+d2+d3,where Ii≥0i=1,2,3 is an input parameter, Oi is the output result of the ith layer, and ωj,dj denote the weight matrix and an offset value vector at jj=1,2,3 layer. When j=3, the layer represents an output layer and f3 is the output parameter.


## 2.2.1. Analysis of Influencing Factors

The existing research literature shows that the factors affecting the road friction coefficient are mainly road characteristics, tire type, and force.(1) Road Characteristics. Roads with different characteristics have different road friction coefficients, and the driving comfort of vehicles driving on them is also different. The influencing factors mainly include the road roughness and wet/dry degree. According to the literature [22–24], the relationship between the road friction coefficient and road type is shown in Table 3.Table 3
Friction coefficients of four types of roads.
No.Road typeRange of variationSet value1Dry asphalt(0.80, 0.95]0.912Wet asphalt(0.61, 0.75]0.713Snow pavement(0.20, 0.27]0.244Ice surface(0, 0.10]0.10(2) Tire Type and Force. In addition to supporting the body weight of the car, the tire can transmit road condition information to the body. Existing researchers have proposed a tire model to describe the relationship between tire force and tire motion parameters, that is, the relationship between tire input and output during vehicle driving under different road conditions. A commonly used tire model consists of a drive force module, a slip angle module, a slip rate module, and a tire force module. The tire force module consists of lateral force and lateral velocity, longitudinal force and longitudinal velocity, return torque, and yaw velocity [25].The longitudinal force, lateral force, and vertical force generated in the process of tire contact with the ground have an important effect on the stability of the vehicle. If these three forces can be coordinated and controlled during driving, the driving smoothness and safety can be improved.


## 2.2.2. Estimation Method

The road friction coefficient is a key factor affecting the car-following safety and is mainly influenced by road surface condition, vehicle speed, and vehicle tire type. In accordance with our previous research results, the neural network can be used to calculate the road friction coefficient. This method is used to establish a network structure with the input for road characteristics, tire pressure, vehicle speed, and the output for friction coefficient, as shown in Figure1. A detailed description can be found in the literature [10].Figure 1
Neural network structure.In general, the texture of tires can be divided into three categories, ordinary pattern, cross-country pattern, and mixed pattern. Thus, the input has three parameters in this network structure which are road characteristics, tire pressure, and vehicle speed. And the output is the friction coefficients of each tire type. The BP neural network structure adopts a three-layer structure, that is, an input layer, two hidden layers, and an output layer. The input layer contains three layers of input. The first and second layers have four neurons and three neurons, respectively, and the output layer has a single output, as shown in Figure1. The training data come from 120 experimental samples under two road surfaces (asphalt and cement concrete), different weather, and different driving speed.The log-sig, tan-sig, and purelin functions are chosen for the input transfer function of each layer, respectively, as follows [25–28]:(5)O1x=11+e−x,O2x=21+e−2x−1,O3x=x.The relationship between input and output function is(6)a3=O3ω3O2ω2O1ω1I+d1+d2+d3,where Ii≥0i=1,2,3 is an input parameter, Oi is the output result of the ith layer, and ωj,dj denote the weight matrix and an offset value vector at jj=1,2,3 layer. When j=3, the layer represents an output layer and f3 is the output parameter.


## 3. Safe Following Distance Model Establishment

### 3.1. Existing Car-Following Distance Model

While the vehicle is running, the rear vehicle (vehicle B) should maintain a proper following distance from the front vehicle (vehicle A) to ensure the following safety and to prevent the occurrence of rear-end collisions during driving. The following relationship between the front and rear vehicles can be divided into three situations through the experimental investigation and analysis, and the specific models are described as follows [10, 11]. The car-following process is shown in Figure 2.Figure 2
Car-following process.(1) Stationary State of the Front Vehicle. When the front vehicle is in a stationary state, the minimum following distance ST can be expressed as(7)ST=SB+d=vBtr+ts2+vB22aB+d,where ST is the minimum safe following distance, SB are the distances traveled by vehicle B, respectively, vB and aB are the driving speed and braking deceleration of vehicle B, tr is the sum of drivers’ reaction time and brake coordination time, ts is braking deceleration increase time, and d is the safety distance (2–5 m).(2) Constant Speed State of the Front Vehicle. When the relative speed of the two vehicles is small or large, the required minimum distance is different; thus, the safe following distance ST can be expressed as(8)ST=v′tr+ts−aBts26+dIv′tr+ts2+v′2aB+dII,where v′ is the relative speed of two vehicles, case I denotes that the value of v′ is small, case II denotes that the value of v′ is large.(3) Uniform deceleration state of the front vehicle. There are three situations: the speed of the front vehicle is less than that of the rear vehicle, the speed of the front vehicle equals that of the rear vehicle, and the speed of the front vehicle is larger than that of the rear vehicle. For three different driving situations, the minimum safety distance models are different.(9)ST=vBtr+v′ts/2+vB2/2aB−vA2/2aA+dIvBtr+vB2/2aB−vA2/2aA+dIIvBtr+ts−2vBv′+vA2/2aA+vB2/2aB−vAts/2+dIII,where vA is the driving speed of the front vehicle, aA is the braking deceleration of the front vehicle. Case I denotes that the speed of the front vehicle is less than that of the rear vehicle, Case II denotes that the speed of the front vehicle equals that of the rear vehicle, and Case I denotes that the speed of the front vehicle is larger than that of the rear vehicle.


### 3.2. Improved Minimum Safe Following Distance Model

According to the motion characteristics of the vehicle, the brake acceleration of vehicles has the following relationship with the road friction coefficient.(10)amax≤μg,where aA=aB=amax=μg. The value of μA is the same as that of μB, can be denoted as μ, which is used to correct the acceleration. Then, the calculation expression of the correction factor of drivers’ reaction time is calculated as 75/R.According to the literature [28, 29], the value of drivers’ braking reaction time is 1.25 s, and under normal driving conditions, the value of the minimum safety distance is 2.5 m. Therefore, the corrected reaction time is 1.25P, and the minimum safety distance is 2.5P. The above-corrected parameters are substituted into the traditional safe vehicle following the distance model, and the improved models can be obtained as follows.(1)
Stationary state of the front vehicle:(11)ST=vB1.25P+ts2+vB22μBg+2.5P=vB1.25P+ts2+vB22μg+2.5P.(2)
Constant speed state of the front vehicle:(a).
The large relative speed:(12)ST=v′1.25P+ts−μBgts26+2.5P=v′1.25P+ts−μgts26+2.5P.(b)
The small relative speed:(13)ST=v′1.25P+ts2+v′2μBg+2.5P=v′1.25P+ts2+v′2μg+2.5P.(3)
Uniform deceleration state of the front vehicle:(a)
The speed of the front vehicle is less than that of the rear vehicle:(14)ST=1.25vBP+v′ts2+vB22μBg−vA22μAg+2.5P=1.25vBP+v′ts2+vB2−vA22μg+2.5P.(b)
The speed of the front vehicle equals that of the rear vehicle:(15)ST=1.25vBP+vB22μBg−vA22μAg+2.5P=1.25vBP+vB2−vA22μg+2.5P.(c)
The speed of the front vehicle is larger than that of the rear vehicle:(16)ST=vB1.25P+ts−2vBv′+vA2−vB22μg−vAts2+2.5P.


## 3.1. Existing Car-Following Distance Model

While the vehicle is running, the rear vehicle (vehicle B) should maintain a proper following distance from the front vehicle (vehicle A) to ensure the following safety and to prevent the occurrence of rear-end collisions during driving. The following relationship between the front and rear vehicles can be divided into three situations through the experimental investigation and analysis, and the specific models are described as follows [10, 11]. The car-following process is shown in Figure 2.Figure 2
Car-following process.(1) Stationary State of the Front Vehicle. When the front vehicle is in a stationary state, the minimum following distance ST can be expressed as(7)ST=SB+d=vBtr+ts2+vB22aB+d,where ST is the minimum safe following distance, SB are the distances traveled by vehicle B, respectively, vB and aB are the driving speed and braking deceleration of vehicle B, tr is the sum of drivers’ reaction time and brake coordination time, ts is braking deceleration increase time, and d is the safety distance (2–5 m).(2) Constant Speed State of the Front Vehicle. When the relative speed of the two vehicles is small or large, the required minimum distance is different; thus, the safe following distance ST can be expressed as(8)ST=v′tr+ts−aBts26+dIv′tr+ts2+v′2aB+dII,where v′ is the relative speed of two vehicles, case I denotes that the value of v′ is small, case II denotes that the value of v′ is large.(3) Uniform deceleration state of the front vehicle. There are three situations: the speed of the front vehicle is less than that of the rear vehicle, the speed of the front vehicle equals that of the rear vehicle, and the speed of the front vehicle is larger than that of the rear vehicle. For three different driving situations, the minimum safety distance models are different.(9)ST=vBtr+v′ts/2+vB2/2aB−vA2/2aA+dIvBtr+vB2/2aB−vA2/2aA+dIIvBtr+ts−2vBv′+vA2/2aA+vB2/2aB−vAts/2+dIII,where vA is the driving speed of the front vehicle, aA is the braking deceleration of the front vehicle. Case I denotes that the speed of the front vehicle is less than that of the rear vehicle, Case II denotes that the speed of the front vehicle equals that of the rear vehicle, and Case I denotes that the speed of the front vehicle is larger than that of the rear vehicle.


## 3.2. Improved Minimum Safe Following Distance Model

According to the motion characteristics of the vehicle, the brake acceleration of vehicles has the following relationship with the road friction coefficient.(10)amax≤μg,where aA=aB=amax=μg. The value of μA is the same as that of μB, can be denoted as μ, which is used to correct the acceleration. Then, the calculation expression of the correction factor of drivers’ reaction time is calculated as 75/R.According to the literature [28, 29], the value of drivers’ braking reaction time is 1.25 s, and under normal driving conditions, the value of the minimum safety distance is 2.5 m. Therefore, the corrected reaction time is 1.25P, and the minimum safety distance is 2.5P. The above-corrected parameters are substituted into the traditional safe vehicle following the distance model, and the improved models can be obtained as follows.(1)
Stationary state of the front vehicle:(11)ST=vB1.25P+ts2+vB22μBg+2.5P=vB1.25P+ts2+vB22μg+2.5P.(2)
Constant speed state of the front vehicle:(a).
The large relative speed:(12)ST=v′1.25P+ts−μBgts26+2.5P=v′1.25P+ts−μgts26+2.5P.(b)
The small relative speed:(13)ST=v′1.25P+ts2+v′2μBg+2.5P=v′1.25P+ts2+v′2μg+2.5P.(3)
Uniform deceleration state of the front vehicle:(a)
The speed of the front vehicle is less than that of the rear vehicle:(14)ST=1.25vBP+v′ts2+vB22μBg−vA22μAg+2.5P=1.25vBP+v′ts2+vB2−vA22μg+2.5P.(b)
The speed of the front vehicle equals that of the rear vehicle:(15)ST=1.25vBP+vB22μBg−vA22μAg+2.5P=1.25vBP+vB2−vA22μg+2.5P.(c)
The speed of the front vehicle is larger than that of the rear vehicle:(16)ST=vB1.25P+ts−2vBv′+vA2−vB22μg−vAts2+2.5P.


## 4. Simulation and Analysis of the Improved Models

In this study, three different driving states of the front vehicle (i.e., stationary state, uniform state, and uniform deceleration state) are defined according to the actual driving situation. Five different driving conditions (see Table4) are set according to the previous analysis of influencing factors. 100 experienced experts are selected to score according to the scoring rule of each factor set in Table 1 and then take the average value. The values in brackets in Table 4 are the evaluation scores of factors.Table 4
Simulation parameters of different scenarios.
Influencing factorsScenario 1Scenario 2Scenario 3Scenario 4Scenario 5Driver characteristicsGender (Y1)Male (80)Female (60)Male (80)Male (80)Female (60)Age (Y2)18–30 years old (85)50–70 years old (60)18–30 years old (85)18–30 years old (85)50–70 years old (60)Driving years (Y3)1–5 years (60)>15 years (85)1–5 years (60)1–5 years (60)>15 years (85)Psychology (Y4)Normal (85)Disease (60)Normal (85)Normal (85)Disease (60)Traffic environmentRoad feature (Y5)Normal road (85)Normal road (85)Sharp bend (60)Normal road (85)Sharp bend (60)Traffic density (Y6)Free flow (85)Free flow (85)Emergency follow (60)Free flow (85)Emergency follow (60)Environment factorsDriving period (Y7)12 : 00–14 : 00 (75)12 : 00–14 : 00 (75)12 : 00–14 : 00 (75)23 : 00–6:00 (60)23 : 00–6:00 (60)Weather (Y8)Sunny (85)Sunny (85)Sunny (85)High fog (60)High fog (60)Assuming that the front vehicle and the rear vehicle are vehicle A and vehicle B, respectively (the same as in the following), and the two vehicles are driving in the same direction in the same lane. The values of some parameters are set as follows:ts = 0.12 s, μ = 0.70, g = 9.8 m/s2, tr = 1.25P, dmin = 2.5P.The simulation of five representative situations is conducted to observe the effect of the drivers’ brake reaction time on the safe car-following in five conditions. The total scoreR of the brake reaction time is calculated by using formulae (3) and (4). The weight coefficients of the various factors affecting the drivers’ brake reaction time can be obtained from Table 4. After a comprehensive weighting evaluation, the total score value R and correction factor P of the drivers’ reaction braking time are obtained. The calculation results can be found in Table 5.Table 5
Weighted evaluations of different cases.
ParametersCase 1Case 2Case 3Case 4Case 5D166.2577.8077.8066.2577.80D260.0085.0060.0085.0085.00D360.0060.0081.2081.2081.20R62.50074.37073.69276.32280.942P1.0954451.0042271.0088360.9913010.962595


### 4.1. Stationary State of the Front Vehicle

The first driving state is that the state of the front vehicle A is in the stationary, the speed of vehicle B is set within the range of 60–62 km/h, and the minimum safety distance of five cases can be obtained in accordance with the formula (11), as shown in Figure 3.Figure 3
Minimum safety distance under the stationary state of the front vehicle.As shown in Figure3, the minimum safety distance has a linear increase in relation to the speed of vehicle B. When the speed of vehicle B ranges from 60 to 62 km/h, the variation range of the minimum safety distance is 340–375 m. It indicates that the value of the minimum distance increases by at least 35 m when the speed of vehicle B increases by 2 km/h. By comparing the safe following distance under five different driving conditions, the influence degree of various factors on the following distance can be obtained, the driver characteristics have the greatest impact, and the environmental condition and road characteristics have mostly the same influence degree.


### 4.2. Constant Speed State of the Front Vehicle

When the front vehicle drives at a constant speed, there are two cases according to the speed relations between the front vehicle and the rear vehicle. The changing analysis of the minimum safety distance under five different driving conditions is conducted when the speed of the front vehicle drives by 50 km/h.


#### 4.2.1. Relative Speed of Two Vehicles Is Large

When the speed of the rear vehicle is 80–85 km/h, then the relative speed of the two vehicles is 30–35 km/h, it belongs to the case where the relative speed between two vehicles is large. The changing of the minimum safety distance in this state can be obtained in accordance with formula (12), as shown in Figure 4.Figure 4
Minimum safety distances under constant speed state of the front vehicle (v′ is large).


#### 4.2.2. Relative Speed of Two Vehicles Is Small

When the speed of vehicle B is 55–65 km/h, then the relative speed of the two vehicles is 5–10 km/h, it belongs to the case where the relative speed between two vehicles is small. The changing of the minimum safety distance in this state can be obtained in accordance with formula (13), as shown in Figure 5.Figure 5
Minimum safety distances under constant speed state of the front vehicle (v′ is small).As can be seen from Figures4 and 5, the minimum safety distance of case 1 is the maximum, and that of case 5 is the minimum. The changing of the minimum safety distance in the five cases is almost the same. Compare Figures 4 and 5 when the driving condition is not good, the minimum safety distance is less than 15 m at a relative speed of 8 km/h. However, the minimum safety distance is almost triple (about 50 m) at a relative speed of 32 km/h. Thus, the required safety distance is big enough to ensure car-following safety. When the relative speed of two vehicles is relatively small, then the three factors (drivers’ own condition, environment condition, and road characteristics) have mostly the same degree of influence. And the drivers’ own conditions have a slightly great degree of influence. When the relative speed of two vehicles is large, the order of influence degree of three factors on safe following distance from small to large is drivers’ own condition, environmental condition, and road characteristics.


### 4.3. Uniform Deceleration State of the Front Vehicle

When the front vehicle drives at a uniform deceleration, there are three cases according to the speed relations between vehicle A and vehicle B. The changing analysis of the minimum safety distance under five different driving conditions is conducted.


#### 4.3.1. The Speed of the Front Vehicle Is Less Than That of the Rear Vehicle

When the speed of the front vehicle is 40–42 km/h, and the speed of the rear vehicle is 60–62 km/h. The minimum safety distance of five cases can be obtained in accordance with formula (14), as shown in Figure 6.Figure 6
Minimum safety distance under uniform deceleration state of the front vehicle (VA < VB).


#### 4.3.2. The Speed of the Front Vehicle Equals That of the Rear Vehicle

When the speed of both the front vehicle and the rear vehicle is 50–52 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (15), as shown in Figure 7.Figure 7
Minimum safety distance under uniform deceleration state of the front vehicle (VA = VB).


#### 4.3.3. The Speed of Front Vehicle Is Larger Than That of the Rear Vehicle

In the third following situation of vehicle A in a uniform deceleration state, when the speed of vehicle A is 60–62 km/h, and the speed of vehicle B is 40–42 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (16), as shown in Figure 8.Figure 8
Minimum safety distance under uniform deceleration state of the front vehicle (VA > VB).As can be seen from Figures6 to 8, the minimum safety distance increases as the speed of vehicle A (or vehicle B) increases. By comparing the changing of the minimum safety distance under uniform deceleration of vehicle A, some conclusions can be get: (1) the minimum safety distance is the maximum when the speed of vehicle A is less than that of vehicle B and is the minimum when the speed of vehicle A speed equals that of vehicle B. (2) The drivers’ characteristics are the most important factor affecting the braking reaction time. The road characteristics and environmental conditions have similar and weak influence. When the speed of vehicle A equals that of vehicle B, the influence of environmental condition is slightly greater than that of road characteristics. (3) In summary, when vehicle A is in a uniform deceleration state, the influence degree of the factors can be sorted: drivers’ characteristics, environmental conditions, and road characteristics. The influence degree of the environmental condition is slightly greater than that of road characteristics under certain special conditions.


## 4.1. Stationary State of the Front Vehicle

The first driving state is that the state of the front vehicle A is in the stationary, the speed of vehicle B is set within the range of 60–62 km/h, and the minimum safety distance of five cases can be obtained in accordance with the formula (11), as shown in Figure 3.Figure 3
Minimum safety distance under the stationary state of the front vehicle.As shown in Figure3, the minimum safety distance has a linear increase in relation to the speed of vehicle B. When the speed of vehicle B ranges from 60 to 62 km/h, the variation range of the minimum safety distance is 340–375 m. It indicates that the value of the minimum distance increases by at least 35 m when the speed of vehicle B increases by 2 km/h. By comparing the safe following distance under five different driving conditions, the influence degree of various factors on the following distance can be obtained, the driver characteristics have the greatest impact, and the environmental condition and road characteristics have mostly the same influence degree.


## 4.2. Constant Speed State of the Front Vehicle

When the front vehicle drives at a constant speed, there are two cases according to the speed relations between the front vehicle and the rear vehicle. The changing analysis of the minimum safety distance under five different driving conditions is conducted when the speed of the front vehicle drives by 50 km/h.


### 4.2.1. Relative Speed of Two Vehicles Is Large

When the speed of the rear vehicle is 80–85 km/h, then the relative speed of the two vehicles is 30–35 km/h, it belongs to the case where the relative speed between two vehicles is large. The changing of the minimum safety distance in this state can be obtained in accordance with formula (12), as shown in Figure 4.Figure 4
Minimum safety distances under constant speed state of the front vehicle (v′ is large).


### 4.2.2. Relative Speed of Two Vehicles Is Small

When the speed of vehicle B is 55–65 km/h, then the relative speed of the two vehicles is 5–10 km/h, it belongs to the case where the relative speed between two vehicles is small. The changing of the minimum safety distance in this state can be obtained in accordance with formula (13), as shown in Figure 5.Figure 5
Minimum safety distances under constant speed state of the front vehicle (v′ is small).As can be seen from Figures4 and 5, the minimum safety distance of case 1 is the maximum, and that of case 5 is the minimum. The changing of the minimum safety distance in the five cases is almost the same. Compare Figures 4 and 5 when the driving condition is not good, the minimum safety distance is less than 15 m at a relative speed of 8 km/h. However, the minimum safety distance is almost triple (about 50 m) at a relative speed of 32 km/h. Thus, the required safety distance is big enough to ensure car-following safety. When the relative speed of two vehicles is relatively small, then the three factors (drivers’ own condition, environment condition, and road characteristics) have mostly the same degree of influence. And the drivers’ own conditions have a slightly great degree of influence. When the relative speed of two vehicles is large, the order of influence degree of three factors on safe following distance from small to large is drivers’ own condition, environmental condition, and road characteristics.


## 4.2.1. Relative Speed of Two Vehicles Is Large

When the speed of the rear vehicle is 80–85 km/h, then the relative speed of the two vehicles is 30–35 km/h, it belongs to the case where the relative speed between two vehicles is large. The changing of the minimum safety distance in this state can be obtained in accordance with formula (12), as shown in Figure 4.Figure 4
Minimum safety distances under constant speed state of the front vehicle (v′ is large).


## 4.2.2. Relative Speed of Two Vehicles Is Small

When the speed of vehicle B is 55–65 km/h, then the relative speed of the two vehicles is 5–10 km/h, it belongs to the case where the relative speed between two vehicles is small. The changing of the minimum safety distance in this state can be obtained in accordance with formula (13), as shown in Figure 5.Figure 5
Minimum safety distances under constant speed state of the front vehicle (v′ is small).As can be seen from Figures4 and 5, the minimum safety distance of case 1 is the maximum, and that of case 5 is the minimum. The changing of the minimum safety distance in the five cases is almost the same. Compare Figures 4 and 5 when the driving condition is not good, the minimum safety distance is less than 15 m at a relative speed of 8 km/h. However, the minimum safety distance is almost triple (about 50 m) at a relative speed of 32 km/h. Thus, the required safety distance is big enough to ensure car-following safety. When the relative speed of two vehicles is relatively small, then the three factors (drivers’ own condition, environment condition, and road characteristics) have mostly the same degree of influence. And the drivers’ own conditions have a slightly great degree of influence. When the relative speed of two vehicles is large, the order of influence degree of three factors on safe following distance from small to large is drivers’ own condition, environmental condition, and road characteristics.


## 4.3. Uniform Deceleration State of the Front Vehicle

When the front vehicle drives at a uniform deceleration, there are three cases according to the speed relations between vehicle A and vehicle B. The changing analysis of the minimum safety distance under five different driving conditions is conducted.


### 4.3.1. The Speed of the Front Vehicle Is Less Than That of the Rear Vehicle

When the speed of the front vehicle is 40–42 km/h, and the speed of the rear vehicle is 60–62 km/h. The minimum safety distance of five cases can be obtained in accordance with formula (14), as shown in Figure 6.Figure 6
Minimum safety distance under uniform deceleration state of the front vehicle (VA < VB).


### 4.3.2. The Speed of the Front Vehicle Equals That of the Rear Vehicle

When the speed of both the front vehicle and the rear vehicle is 50–52 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (15), as shown in Figure 7.Figure 7
Minimum safety distance under uniform deceleration state of the front vehicle (VA = VB).


### 4.3.3. The Speed of Front Vehicle Is Larger Than That of the Rear Vehicle

In the third following situation of vehicle A in a uniform deceleration state, when the speed of vehicle A is 60–62 km/h, and the speed of vehicle B is 40–42 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (16), as shown in Figure 8.Figure 8
Minimum safety distance under uniform deceleration state of the front vehicle (VA > VB).As can be seen from Figures6 to 8, the minimum safety distance increases as the speed of vehicle A (or vehicle B) increases. By comparing the changing of the minimum safety distance under uniform deceleration of vehicle A, some conclusions can be get: (1) the minimum safety distance is the maximum when the speed of vehicle A is less than that of vehicle B and is the minimum when the speed of vehicle A speed equals that of vehicle B. (2) The drivers’ characteristics are the most important factor affecting the braking reaction time. The road characteristics and environmental conditions have similar and weak influence. When the speed of vehicle A equals that of vehicle B, the influence of environmental condition is slightly greater than that of road characteristics. (3) In summary, when vehicle A is in a uniform deceleration state, the influence degree of the factors can be sorted: drivers’ characteristics, environmental conditions, and road characteristics. The influence degree of the environmental condition is slightly greater than that of road characteristics under certain special conditions.


## 4.3.1. The Speed of the Front Vehicle Is Less Than That of the Rear Vehicle

When the speed of the front vehicle is 40–42 km/h, and the speed of the rear vehicle is 60–62 km/h. The minimum safety distance of five cases can be obtained in accordance with formula (14), as shown in Figure 6.Figure 6
Minimum safety distance under uniform deceleration state of the front vehicle (VA < VB).


## 4.3.2. The Speed of the Front Vehicle Equals That of the Rear Vehicle

When the speed of both the front vehicle and the rear vehicle is 50–52 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (15), as shown in Figure 7.Figure 7
Minimum safety distance under uniform deceleration state of the front vehicle (VA = VB).


## 4.3.3. The Speed of Front Vehicle Is Larger Than That of the Rear Vehicle

In the third following situation of vehicle A in a uniform deceleration state, when the speed of vehicle A is 60–62 km/h, and the speed of vehicle B is 40–42 km/h, the minimum safety distance of five cases can be obtained in accordance with formula (16), as shown in Figure 8.Figure 8
Minimum safety distance under uniform deceleration state of the front vehicle (VA > VB).As can be seen from Figures6 to 8, the minimum safety distance increases as the speed of vehicle A (or vehicle B) increases. By comparing the changing of the minimum safety distance under uniform deceleration of vehicle A, some conclusions can be get: (1) the minimum safety distance is the maximum when the speed of vehicle A is less than that of vehicle B and is the minimum when the speed of vehicle A speed equals that of vehicle B. (2) The drivers’ characteristics are the most important factor affecting the braking reaction time. The road characteristics and environmental conditions have similar and weak influence. When the speed of vehicle A equals that of vehicle B, the influence of environmental condition is slightly greater than that of road characteristics. (3) In summary, when vehicle A is in a uniform deceleration state, the influence degree of the factors can be sorted: drivers’ characteristics, environmental conditions, and road characteristics. The influence degree of the environmental condition is slightly greater than that of road characteristics under certain special conditions.


## 5. Conclusions

The communication between vehicles in the Internet of Vehicles environment has a great impact on car-following behavior. In order to establish a model that can better describe the car-following behavior in the Internet of Vehicles environment, this paper selects drivers’ response time and road friction coefficient as the research objects according to the analysis of safety influencing factors under the environment of Internet of Vehicles and uses these two parameters to improve the traditional minimum safety distance model. Then, the simulation and analysis of the improved car-following distance models are conducted. The required safe following distance of five different cases was compared and analyzed. The following conclusions can be obtained under the three different driving states of the leading vehicle (vehicle A).(1)
With the increase of vehicle speed, the minimum safe following distance increases correspondingly. When the vehicle speed is relatively high, the safe following distance required should be increased even more when the vehicle speed increases by 1 km/h.(2)
In the same driving state of the leading vehicle A, the safe following distance required by case 5 is the smallest, and that required by situation 1 is the largest by comparing five different driving states.(3)
The three factors affecting the safe car-following can be sorted in accordance with the influence degree on the minimum safety distance from large to small are drivers’ characteristics, environmental conditions, and road characteristics. The influence degree of the environmental condition is slightly greater than that of the road characteristics. However, the influence degree of the two factors can be regarded as approximately the same in most driving conditions.The improved models established in this paper can better describe the driving behavior in the Internet of Vehicles environment and provide guidance for vehicle safety following. The ranking of the influence degree of the influencing factors can provide better theoretical support for the vehicle safe driving assistance system.



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*Source: 8005601-2022-04-23.xml*