study and simulation analysis of vehicle rear-end

Research ArticleStudy and Simulation Analysis of Vehicle Rear-End CollisionModel considering Driver Types

Qiang Luo1 Xinqiang Chen 2 Jie Yuan 1 Xiaodong Zang 1 Junheng Yang1

and Jing Chen3

1School of Civil Engineering Guangzhou University 230 Wai Huan Xi Road Guangzhou Higher Education Mega CenterGuangzhou 510006 China2Institute of Logistics Science and Engineering Shanghai Maritime University 1550 Haigang Ave Shanghai 201306 China3Merchant Marine College Shanghai Maritime University Shanghai 201306 China

Correspondence should be addressed to Jie Yuan yuanjgzhueducn

Received 20 June 2019 Accepted 28 December 2019 Published 23 January 2020

Academic Editor Shamsunnahar Yasmin

Copyright copy 2020 Qiang Luo et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e reasonable distance between adjacent cars is very crucial for roadway traffic safety For different types of drivers or differentdriving environments the required safety distance is different However most of the existing rear-end collision models do notfully consider the subjective factor such as the driver Firstly the factors affecting driving driversrsquo characteristics such as driverage gender and driving experience are analyzed en on the basis of this drivers are classified according to reaction timeSecondly three main factors affecting driving safety are analyzed by using fuzzy theory and the new calculation method of thereaction time is obtained Finally the improved car-following safety model is established based on different reaction time eexperimental results have shown that our proposed model obtained more accurate vehicle safety distance with varied traffickinematic conditions (ie different traffic states varied driver types etc) e findings can help traffic regulation departmentsissue early warnings to avoid potential traffic accidents on roads

1 Introduction

Private car has become affordable for public with the quickdevelopment of economic and promotion of manufacturetechnique An adverse effect of is increasing traffic accidentson roadways which greatly imperil traffic safety and effi-ciency e previous studies have shown that drivers areassumed to the take the major responsibility for the trafficaccidents (ie a few accidents are triggered by vehicle de-fects) Indeed over 80 traffic accidents on roads can beascribed to driver misconduct (eg answering cell phonesmoking and taking naps) [1 2] It is observed that varieddriver characteristics (eg age gender experience andspeed of response) can impose different impact on indi-vidual vehicle maneuver procedure e driver responsetime for different driver types for taking actions againstdangerous driving situations is varied which has attracted alot of research attentions [3ndash5] More specifically the

aggressive drivers require smaller displacement betweenneighboring vehicles when traveling on road while the milddrivers are likely to keep sufficient vehicle headway for thepurpose of avoiding potential traffic accidents Note thatquantifying such vehicle rear end displacement consideringdriver types is not easy which is indeed a hot topic in thetransportation safety research community

Several studies have been conducted to establish mini-mum safety following distance model with vehicle kinematicdata (ie driver types vehicle accelerationdecelerationsdistributions etc) Zhang and Hao deeply analyzed the re-sistance influence on theminimum safety distance which wasinvolved with air resistance road resistance vehicle wheelrolling resistance etc [6] Xu et al proposed a minimumsafety car-following model under different driver states byconsidering the vehicle accelerationdeceleration in vehiclebraking process [7] Spyropoulou proposed a novel vehiclesafety distance model which considered the constraints of

HindawiJournal of Advanced TransportationVolume 2020 Article ID 7878656 11 pageshttpsdoiorg10115520207878656

vehicle speed variation minimal safety distance etc [8] Huet al firstly studied driver behavior differences and empiricaljudgment ratio distributions in an abnormal traffic scenarioand then proposed a car-following model for estimating theminimum safety distance for the emergency evacuation ve-hicle [9] Similar research studies can be found in [10ndash14]

Many studies have been conducted to analyze the rela-tionship between traffic safety and drivers characteristic in-cluding driver personality physical fitness driver distractionetc [15ndash18] Tang and Xia implemented a series of experi-ments to measure the reaction time of four different drivertypes and further studied its impact on the minimal safetydistance of preventing rear-end collision [17] Tang and Xiaanalyzed a variety of factors that affect the driver reaction timeby using fuzzy mathematics theory and then proposed a novelmodel to estimate driver reaction time [17] By consideringinfluence of driver individual differences vehicle brakingperformance and driving states Zhang et al established a car-following model with minimum safety distance for threetypical traffic states [19] Similarly Xue et al proposed a rear-end collision behavior model considering the individual dif-ferences of drivers [20] Some scholars analyzed the rela-tionship between other influencing factors and car safety suchas the speed relation among adjacent vehicles [21 22]

Fuzzy relevant models have been proposed to simulatereal-world traffic situations considering varied driver typesand thus provide optimal traffic control strategies for thepurpose of ensuring traffic safety and efficiency (ie withminimal waiting time short queue length etc) Chai et alproposed a simulation-based approach to measure drivercognitive failures which was implemented with fuzzy logicand cellular automata model [23] Azimirad et al proposed anovel fuzzy traffic controller to formulate and optimize trafficcontrol at isolated signalized intersection [24] Li proposed adynamic fuzzy neural networks traffic flow prediction modelto accurately obtain traffic flow prediction under chaos trafficstate [25] Bocklisch et al proposed an adaptive fuzzy patternclassification model for simulating nonlinear multidimen-sional transition processes which aims to identify the lanechange intentions for varied driver types [26]

e previous studies focusing on vehicle rear-end colli-sion analysis mainly employed the kinematic data to estimateminimal neighboring vehicle distance with the quantitativeindicators (ie speed displacement etc) which did notconsider the qualitative indicators (eg driving behaviordifferences influence) and mainly led to biased results eadvantages of fuzzy theory can help us quantify the physi-ological and psychological characteristics of vehicle driversand thus provide us a more holistic view of the car-followingmodele paper is organized as follows Driver features weredeeply analyzed in Section 2 and the proposed safety modelwas illustrated in Section 3 e experimental results weredescribed in Section 4 and Section 5 concluded the paper

2 Driving Characteristics and Reaction Time

21 Driving Characteristics Analysis e vehicle controllingprocedure consists of environment perception potential riskjudgment and traveling decision making and vehicle maneuver

which can be found in Figure 1 In the perception stage thedriverrsquos control of car speed and perception of surroundingenvironment have a close bearing on the driverrsquos reaction speedto the unexpected situation In the judgment and decision-making stage the driver makes control decisions based ondriving experience to ensure driving safety In the operationstage the driver controls the car accelerating deceleratingturning and braking according to the decision Note that over80 traffic accidents are triggered due to the driver errors[16 27 28] To avoid the potential traffic accidents drivers aresupposed to take early and correct maneuver activities (ieidentify risky behaviors decide the suitable travel behavior takemaneuver activities etc) for each of vehicle maneuver steps

e driverrsquos individual characteristics can be analyzedfrom physiological and psychological factors for example thedriverrsquos age gender and driving age fatigue and emotionswhile driving the driverrsquos own driving style and safety atti-tude traffic regulations awareness level and so on eseresearch studies show that [29ndash31] (1) the reaction speed tothe emergency situation will be significantly reduced with age(2) in terms of emergency response capacity and the ability tocope with the complex external environment the femaledrivers are worse than the male drivers (3) the experienceddriver has the faster reaction in the emergency situation and(4) the adventure driver is more likely to be in danger of rapidacceleration deceleration and changing lanes than the cau-tious driver Based on the above analysis it is necessary toquantify the driving characteristics into quantifiable param-eters such as reaction time which can be introduced into themodeling process of the improved car-following model

22 Relationship Analysis between Driver Characteristics andReaction Time e driver age driving experience durationand fatigue level are considered as crucial elements forquantifying analysis on the relationship between drivercharacteristics and reaction time [32 33] Previous studieshave evaluated the reaction time variation under differenttraffic flow density constraints by building three kinds oftraffic flow density scenarios which suggested that the driverreaction time tends to be shorter in higher traffic flowdensity Factually the driver in a conservative driving style(intends to obtain much larger distance) has the longestreaction time regardless of the vehicle braking manner edriver age has a significant interference to driver reactiontime which is obviously observed in the emergency trafficsituations e statistical distributions of empirical trafficdata suggested that the average and variance of collisionavoidance reaction time of 18 to 30 years old drivers are thesmallest and the drivers over 51 years old have the longestcollision avoidance reaction time Previous studies suggestedthat aging drivers need more reaction time because theyrequire longer time to think carefully about various potentialtravel strategies [18 34 35]

tr 1 +(1 minus C)1minus Q

(1)

where tr is the driverrsquos reaction time C is the responsedegree of different types of drivers under different condi-tions and Q is vehicle braking type

2 Journal of Advanced Transportation

According to the formula the reaction time of severalrepresentative driver types is obtained which is shown inTable 1 (for details see [9])

3 Analysis and Improvement of SafetyDistance Model

e reaction time of various types of drivers is differentwhen they face the same situation so the required safetydistance is also differente improvement of safety distancemodel mainly researches on driverrsquos reaction time andconfirms a more reasonable safety distance When estab-lishing the model the driverrsquos reaction time varies fromdifferent influencing factors Some researchers discoveredthat these influencing factors are not absolutely indepen-dent but have a certain intersection and they have char-acteristics of fuzziness [36 37] erefore this study uses thefuzzy mathematics principle to calculate reaction time ofdifferent types of drivers

31 ReactionTimeAnalysis ere are many factors affectingthe driverrsquos driving behavior and the impact of these factorscan be measured by the reaction time of driversrough in-depth analysis and comparative research three main factorsaffecting response time were selected in this manuscript agedriving age and fatigue Other factors will be considered infuture research

In the fuzzy inference system there are three variablesincluding input variables fuzzy rules and fuzzy output asshown in Figure 2 After determining the influencing factorsthe fuzzy distribution method is used to determine mem-bership function and the Gaussian or semi-Gaussian dis-tributions are selected which include small large andintermediate types that correspond exactly to the conser-vative adventurous and conventional types of driver clas-sification [38 39]

(1) Membership function of agee driverrsquos age can be roughly divided into threeparts including youth middle and old age which aredenoted as A1 A2 and A3 respectively e legaldriver age in China ranges from 18 to 70 which aresupposed by China traffic regulations Followingsuch rule the driver age interval is set as [20 70]emembership function of driverrsquos age is trapezoidalmembership function

(2) Membership function of driving ageAs for the driverrsquos driving age it can be roughlydivided into three levels including low medium andhigh which are denoted as B1 B2 and B3 and the

interval [0 50] is selected More specifically we setthe driving age interval from 0 to 50 considering thata driver can hold a driver license without longer than50 years e membership function of driving age isthe Gaussian membership function

(3) Membership function of fatigue degreee driving time is used to express the driverrsquos fa-tigue It can be roughly divided into mild moderateand high fatigue magnitude respectively which aremarked as C1 C2 and C3 Note that driver fatigueinterval is set as [0 10] following the rules suggestedby the Chinese traffic regulations e driver fatiguesthe membership function is the Gaussian member-ship function

(4) Reaction time variablee inference system finally derives the driverrsquos re-action time according to fuzzy theory Based on alarge number of survey results it is generally believedthat the driverrsquos response time interval is [02 30]which can be roughly divided into short mediumand long erefore the Gaussian-type function isused for the membership function

After inputting the driverrsquos age driving age and fatiguelevel the fuzzification process can be performed to obtainthe approximate value of the driverrsquos reaction time and thenthe center of gravity defuzzification method is used forantifuzzification and the precise value of the reaction timecan be obtained Its expression is

y0 1113938 uc(y)y dy

1113938 uc(y)dy (2)

where 1113938 is the integral of all subsets in the continuousdomain y and y0 means that the area of left and right sides isthe same

32 Improved Safety Distance Model Establishment ereaction time is used to distinguish different types of driversand is used as a parameter in the process of model buildingIn order to analyze different driving conditions compre-hensively three states of static uniform and deceleration ofthe preceding car were analyzed in order to establish thesafety distance model

(1) Static state of the front carWhen the front vehicle is stationary the positionalrelationship between the two vehicles is shown inFigure 3

Perception Operation DrivingEnvironmentalvehicle information

Feedback

Judgment anddecision

Figure 1 Control flow for vehicle maneuver procedure

Journal of Advanced Transportation 3

At position 1 the rear car begins to realize thedanger After the reaction time t the rear car willarrive at position 2e rear car maintains a constantspeed v1 during the whole process and travelingdistance in reaction time t is

S1 v1t (3)

At position 2 the rear car starts to brake at the maximumdeceleration and its speed drops to 0 at position 3 At thistime the distance between the front and rear cars is d0and the distance traveled by rear car during this time is

S2 v212a1

(4)

To avoid collision of two cars the minimum safetydistance needed to maintain is

D1 S1 + S2 + d0 v1t + v212a1 + d0

(5)

where a1 is the maximum braking deceleration of therear car and d0 is the minimum safety distance re-quired for the two cars

(2) Uniform state of the front carWhen two cars are driving at a constant speed acollision occurs only when the speed of the followingvehicle is faster than the speed of the precedingvehicle e positional relationship is shown inFigure 4

Assume that the rear car is in position 1 and the frontcar is in position 3 e rear car will reach position 2after reaction time t and then it starts to brakeDuring the reaction time the rear car maintains aconstant speed and the distance traveled is

S1 v1t (6)

At position 2 assume that the rear car begins to de-celerate until its speed is the same with the front carmeanwhile the front car arrives at position 5 and therear car arrives at position 4 In this process the dis-tance traveled by the rear car and the front car isrespectively

S3 v2t2 v2 v1 minus v2( 1113857

a1

v1v2 minus v22a1

(7)

To avoid collisions of two cars the minimum safetydistance needed to maintain is

D2 S1 + S2 minus S3 + d0 v1t +v21 minus v222a1

minusv1v2 minus v22

a1+ d0

(8)

(3) Deceleration state of the front carWhen the speed of the rear car is faster than that ofthe front car the two cars have the possibility ofcollision In order to prevent this collision the speedof the two cars is equal or the speed is zero when thedistance between two cars is d0 e position rela-tionship is shown in Figure 5

With the previous analysis the rear car travels fromposition 1 to position 2 within the reaction time t edistance traveled is

S1 v1t (9)

At position 2 the rear car begins to brake When thespeed of two cars is equal (set as v) the rear and the front car

Table 1 Reaction time of different types of drivers (s)

Driver types Conservative Cautious Conventional Radical AdventurousC 01 C 03 C 05 C 07 C 09

Hydraulic brakeQ 015 191 174 155 136 144Barometric brakeQ 04 194 181 166 149 125

DefuzzificationFuzzification Fuzzy rules Fuzzy resultInputage drivingage fatiguedegree etc

Outputreaction

time

Figure 2 Fuzzy inference system

Rear car Rear car Front carRear carv

S1 S2 d0

Position 1 Position 2 Position 3 Position 4

Figure 3 Positional relationship (in the static state of the frontcar)


will arrive the position 4 and 5 respectively e traveldistances are as follows

S2 v21 minus v2

2a1

S3 v22 minus v2

2a2

(10)

To avoid collisions the minimum safety distance to bemaintained is

D3 S1 + S2 minus S3 + d0 v1t +a2 minus a1( 1113857v2 + a1v

22 minus a2v

21

2a1a2+ d0

(11)

where a2 is the braking deceleration of the front car and theremaining parameters have the same meaning as mentionedabove

4 Model Simulation and Analysis

41 Improved Model Simulation When the front car is inthree different states the establishedmodel is simulated Due tothe combination of three factors (age driving age and fatiguedegree) and limited space representative combinations areselected for simulation For example the age is 30 45 and 60the driving age is 5 10 and 15 and the values of fatigue degreetake 2 5 and 8 ese numbers are arranged and combinedrandomly Table 2 shows the driver reaction time obtained bysubstituting three groups of factors into the fuzzy system

From Table 2 some conclusions can be obtained asfollows (1) When the driverrsquos age and driving age areconstant the driverrsquos reaction time is directly proportionalto the fatigue degree (2) When the driverrsquos age and fatiguedegree are constant the driverrsquos reaction time is directlyproportional to the driving age (3) When the driverrsquosdriving age and fatigue value are constant the driverrsquos re-action time is directly proportional to the age And between

the ages of 30 and 45 the driverrsquos reaction time grows slowlyand after the age of 45 the growth rate accelerated

42 Comparative Analysis of Different Models

421 Traditional Safety Distance Model So far there aremany classic car safety distancemodels such as theMazdamodelthe Honda model and the models proposed by the domesticresearch institutes and various universities [14] as follows

(a) e Mazda model

Rear car Rear car Front car Rear carv

S1 S2 d0

Front car

D2 S3

Position 1 Position 2 Position 3 Position 4 Position 5

Figure 4 Positional relationship (in the uniform state of the front car)

Rear car Rear car Front car Rear car

Position 1 Position 2 Position 3 Position 4 Position 5v

S1 S2 d0

Front car

D2 S3

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

Table 2 Reaction time in different combinations

Age driving age and fatigue degree Reaction time (s)(30 5 2) 121(30 5 5) 158(30 5 8) 169(30 10 8) 183(30 10 5) 152(30 10 2) 128(45 5 2) 139(45 5 5) 192(45 5 8) 210(45 10 8) 199(45 10 5) 173(45 10 2) 132(45 15 8) 191(45 15 5) 162(45 15 2) 138(60 5 2) 194(60 5 5) 258(60 5 8) 275(60 10 8) 271(60 10 5) 253(60 10 2) 181(60 15 8) 191(60 15 5) 161(60 15 2) 134


d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)

(c) Research institutesrsquo model

S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)

(d) Universitiesrsquo models

dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)

In the formulas d D S0 dw and Dx represent the safetydistance v1 and v2 represent the speed of the leading andfollowing car a1 and a2 represent the acceleration of theleading and following car vr is the relative speed d0 and d

represent the minimum safety spacing l is the length of carτ is the response time of driver and the remaining pa-rameters are the same as before

422 Model Comparison and Analysis e reaction timededuced by the fuzzy theory is substituted into differentsafety distance models Under different reaction times thesafety distances calculated by different models are dif-ferent Four groups of data are randomly selected ascomparisons

(1) Static state of the front carD1 represents the safety distance of modified modelwhen the leader car is in the anchoring state In suchtraffic state the parameters are set as follows theinitial speed of the rear car is 60 kmh and the de-celeration is 4ms2 e safety distance between thetwo cars is 50m More specific parameter settings areshown in Table 3

(a) We employ MATLAB to implement experimentwhen the driver reaction time is set to 121 s andthe corresponding results are shown in Figure 6e comparison result of safety distances fordifferent models is shown in Figure 7

From Figure 6 we can see that the two curves in-tersect at 50m which proves that setting the safetydistance to 50m is not enough and a collision willoccur Under this condition the safety distance ofrear-end collision model is 5489m by calculationFrom Figure 7 it can be found that in all the models

the improved model D1 is the closest to the mini-mum safety distance

(b) We employ MATLAB to implement experimentwhen the driver reaction time is set to 158 sand the corresponding results are shown inFigure 8 e comparison chart of safety dis-tances for different models is shown in Figure 9

From Figure 8 the safety distance is 6106m FromTable 3 it can be seen that the models D1 and D areclose to theminimum safety distance However due tothe need of keeping a certain distance between two carswhen they are still the model D1 is more appropriate

(2) Uniform state of the front carD2 represents the safety distance of the modifiedmodel when the front car is in the uniform state Insuch traffic state the parameters are set as followsthe speed and deceleration of the rear car are 80 kmhand 4ms2 and the speed of the front car is 40 kmhe safety distance between the two cars is 50mMore specific traffic parameters for obtaining safetydistance in the traffic state are shown in Table 4

(a) We employ MATLAB to implement experimentwhen the driver reaction time is set to 139 s andthe corresponding results are shown in Figure 10e comparison chart of safety distances fordifferent models is shown in Figure 11

It can be seen from Figure 10 that since there are nointersections between the two curves it can be seen thatthe safety distance of 50m is sufficient and the cal-culated safety distance to prevent rear end is 3137mFigure 11 shows that the safe distance of model D2 isclosest to 3137m so the model D2 is most suitable

(b) We employ MATLAB to implement experimentwhen the driver reaction time is set to 173 s andthe corresponding results are shown in Figure 12e comparison of the safety distances of dif-ferent models is shown in Figure 13

It can be seen from Figure 12 that since there are nointersections between the two curves it can be seenthat the 50m safety distance is sufficient Accordingto MATLAB calculations the safety distance is3478m if there is no collision From Figure 13 it isfound that the safe distance of model D2 is closest to3137m so the model D2 is most suitable

(3) Deceleration state of the front carD3 represents the safety distance of the modifiedmodel when the front car is in the decelerationstate In this driving condition the parameters areset as follows the speed and deceleration of therear car are 100 kmh and 5ms2 and the speed anddeceleration of the front car are 60 kmh and 3ms2 e safety distance between the two cars is 50me safety distance of different models (in thedeceleration state) is shown in Table 5


(1) When the driverrsquos reaction time is 121 s useMATLAB simulation to obtain results as shown inFigure 14 e comparison of the safety distance ofdifferent models is shown in Figure 15

From Figure 14 there are intersections betweenthe two curves which proves that it is not enoughto set the safety distance to 50 m and there will bea collision After the simulation calculation the

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

Age driving age and fatigue degree (30 5 2) (30 5 5) (30 5 8) (30 10 8)Reaction time (s) 121 158 169 183D1 5989 6606 6789 7022d 6322 6939 7122 7356D 5982 6195 6742 7053S0 9626 9626 9626 9626dw 6137 6987 7236 7596Dx 7012 7012 7012 7012

0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

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

0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models

Figure 7 Safety distance comparison (in the static state on frontcar)

0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m

Figure 8 Distance change (in the static state of front car)

0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models

Figure 9 Safety distance comparison (in the static state of frontcar)


safety distance for preventing rear-end collision is645 m It can be seen from Figure 15 that thesafety distance of model D3 is the closest to645 m us the model D3 is most suitable

(2) We employ MATLAB to implement experimentwhen the driver reaction time is set to 173 s andthe corresponding results are shown inFigure 16

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


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

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

0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models

Figure 11 Safety distance comparison (in the uniform state offront car)

0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models



From Figure 16 it can be seen that there are intersectionsbetween the two curves which proves that setting the safetydistance to 50m is not enough and a collision will occurAfter simulation calculation the safety distance of pre-venting rear-end collision is 7867m As can be seen from

Figure 17 the safety distance of model D3 is the closest to7867m so the model D3 is most suitable

We have evaluated varied minimal car-following dis-tances at different speed variations For instance the min-imal vehicle headways are 3204m and 6386m when the

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


0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

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

0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models

Figure 15 Safety distance comparison (in the deceleration state offront car)

Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models



following car moves at 60 kmh and 90 kmh with theleading vehicle in the uniform state e minimal vehicledistance can be shorter when the leading vehicle is in thedeceleration state and the following car has same speedconstraints (ie 60 kmh and 90 kmh) Vehicle speed in-fluence on the simulation model can be summarized as two-folds (1) the required minimum following distance for thefollowing car is different when the leading car is in differenttraffic states (eg different speeds) and (2) the travelingspeed of the following car is in positive relationship to that ofthe minimal car-following distance (ie higher speeds re-quires larger minimal distance)

5 Conclusions

We have analyzed different driver typesrsquo influence onroadway safety and thus quantified the influence byestablishing a novel car rear-end collision model Weemployed the fuzzy theory to build the reasoning modelwith the input of typical traffic safety factors (ie driverage driving age and fatigue degree) and output thedriver reaction time With the aim of obtaining mini-mum safety following distance we have deeply analyzedthe safety distance between two neighboring vehicleswith leading vehicle at varied states We have estimatedminimal safety distances for the two vehicles with theleading car in varied kinematic states (ie static de-celeration and constant speed) which are indeedcommonly encountered on the roadway traffic scenariosSuppose two of the traffic factors are in constant statesdriver reaction time is proportional to the variation ofthe remaining factor (either in negative or positive)Compared with the traditional rear-end model theproposed safety distance model established can beadapted to different types of drivers which can betterbenefit traffic management and control

Data Availability

All data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partially supported by the GuangzhouMunicipalUniversity Research Project (1201630172) Natural ScienceFoundation of Guangdong Province (2017A030313293) Na-tional Natural Science Foundation of China (51579143 and51709167) and Shanghai Committee of Science andTechnologyChina (18040501700 18295801100 and 17595810300)

References

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real-life field studyrdquo Transportation Research Part CEmerging Technologies vol 19 no 1 pp 145ndash155 2011

[2] J Lai P Zhang H Zhou F Cheng and H Qin ldquoStudy ofrules of traffic accidents in expressway tunnelsrdquo TunnelConstruction vol 37 pp 37ndash42 2017

[3] X Na and D J Cole ldquoGame-theoretic modeling of thesteering interaction between a human driver and a vehiclecollision avoidance controllerrdquo IEEE Transactions on Human-Machine Systems vol 45 no 1 pp 25ndash38 2014

[4] J Engstrom M Ljung Aust and M Vistrom ldquoEffects ofcognitive load and anticipation on driver responses to acritical traffic eventrdquo in Proceedings of the 2018 HumanFactors and Ergonomics Society Annual Meeting vol 62 no 1pp 1584ndash1588 Philadelphia PA USA October 2018

[5] N Arbabzadeh M Jafari M Jalayer S Jiang andM Kharbeche ldquoA hybrid approach for identifying factorsaffecting driver reaction time using naturalistic driving datardquoTransportation Research Part C Emerging Technologiesvol 100 pp 107ndash124 2019

[6] H Y Zhang and X L Hao ldquoSimulation research on mini-mum safety distance of automobile rear-end anti-collisionrdquoComputer Simulation vol 16 pp 146ndash150 2014

[7] L H Xu Q Luo J Wu and Y Huang ldquoStudy of car-fol-lowing model based on minimum safety distancerdquo Journal ofHighway amp Transportation Research amp Development vol 6no 1 pp 95ndash101 2012

[8] I Spyropoulou ldquoSimulation using gipps car-followingmodelmdashan in-depth analysisrdquo Transportmetrica vol 3 no 3pp 231ndash245 2007

[9] H Hu H Wei X Liu and X Yang ldquoCar-following modelfeatured with factors reflecting emergency evacuation situa-tionrdquo Journal of Southeast University (English Edition)vol 24 pp 216ndash221 2008

[10] XWang M Chen M Zhu and P Tremont ldquoDevelopment ofa kinematic-based forward collision warning algorithm usingan advanced driving simulatorrdquo IEEE Transactions on In-telligent Transportation Systems vol 17 no 9 pp 2583ndash25912016

[11] W Honig and T S Kumar ldquoMulti-agent path finding withkinematic constraintsrdquo in Proceedings of the 2016 Twenty-Sixth International Conference on Automated Planning andScheduling London UK June 2016

[12] L Chai B Cai W ShangGuan J Wang and H Wang ldquoBasicsimulation environment for highly customized connected andautonomous vehicle kinematic scenariosrdquo Sensors vol 17no 9 p 1938 2017

[13] X Chen S Wang C Shi H Wu J Zhao and J Fu ldquoRobustship tracking via multi-view learning and sparse represen-tationrdquo Journal of Navigation vol 72 no 1 pp 176ndash192 2019

[14] X Chen and Y Yang ldquoShip type recognition via a coarse-to-fine cascaded convolution neural networkrdquo Journal of Nav-igation 2020 In press

[15] H Liu H Wei Z Yao Q Ai and H Ren ldquoEffects of collisionavoidance system on driving patterns in curve road conflictsrdquoProcediamdashSocial and Behavioral Sciences vol 96 pp 2945ndash2952 2013

[16] M R R Shaon X Qin Z Chen and J Zhang ldquoExploration ofcontributing factors related to driver errors on highwaysegmentsrdquo Transportation Research Record Journal of theTransportation Research Board vol 2672 no 38 pp 22ndash342018

[17] Y S Tang and D H Xia ldquoStudy the influence of differentdriverrsquos reaction time to the automobile anti-collision safety


distancerdquo Science Technology amp Engineering vol 16 no 1pp 250ndash254 2016

[18] Q Xue Y Zhang X Yan and B Wang ldquoAnalysis of reactiontime during car-following process based on driving simula-tion testrdquo in Proceedings of the 2015 International Conferenceon Transportation Information and Safety (ICTIS) pp 207ndash212 Wuhan China June 2015

[19] J Zhang YWang and G Lu ldquoImpact of heterogeneity of car-following behavior on rear-end crash riskrdquo Accident Analysisamp Prevention vol 125 pp 275ndash289 2019

[20] Q Xue X Yan X Li and Y Wang ldquoUncertainty analysis ofrear-end collision risk based on car-following driving simu-lation experimentsrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5861249 13 pages 2018

[21] L Eboli G Mazzulla and G Pungillo ldquoCombining speed andacceleration to define car usersrsquo safe or unsafe driving be-haviourrdquo Transportation Research Part C Emerging Tech-nologies vol 68 pp 113ndash125 2016

[22] Y Luo Y Xiang K Cao and K Li ldquoA dynamic automatedlane change maneuver based on vehicle-to-vehicle commu-nicationrdquo Transportation Research Part C Emerging Tech-nologies vol 62 pp 87ndash102 2016

[23] C Chai Y D Wong and X Wang ldquoSafety evaluation ofdriver cognitive failures and driving errors on right-turnfiltering movement at signalized road intersections based onfuzzy cellular automata (FCA) modelrdquo Accident Analysis ampPrevention vol 104 pp 156ndash164 2017

[24] E Azimirad N Pariz and M B N Sistani ldquoA novel fuzzymodel and control of single intersection at urban trafficnetworkrdquo IEEE Systems Journal vol 4 no 1 pp 107ndash1112010

[25] H Li ldquoResearch on prediction of traffic flow based on dy-namic fuzzy neural networksrdquo Neural Computing and Ap-plications vol 27 no 7 pp 1969ndash1980 2016

[26] F Bocklisch S F Bocklisch M Beggiato and J F KremsldquoAdaptive fuzzy pattern classification for the online detectionof driver lane change intentionrdquo Neurocomputing vol 262pp 148ndash158 2017

[27] M R R Shaon X Qin A P Afghari S Washington andM D Mazharul Haque ldquoIncorporating behavioral variablesinto crash count prediction by severity a multivariate mul-tiple risk source approachrdquo Accident Analysis amp Preventionvol 129 pp 277ndash288 2019

[28] K Wang T Bhowmik S Yasmin S Zhao N Eluru andE Jackson ldquoMultivariate copula temporal modeling of in-tersection crash consequence metrics a joint estimation ofinjury severity crash type vehicle damage and driver errorrdquoAccident Analysis amp Prevention vol 125 pp 188ndash197 2019

[29] L I Gang and H Han ldquoStudy on classification and identi-fication methods of driver steering characteristicsrdquo Journal ofHebei University of Science amp Technology vol 36 pp 559ndash5652015

[30] A Mcdowell D Begg J Connor and J Broughton ldquoUnli-censed driving among urban and rural maori drivers NewZealand drivers studyrdquo Traffic Injury Prevention vol 10 no 6pp 538ndash545 2009

[31] H Hu X Liu and X Yang ldquoA car-following model ofemergency evacuation based on minimum safety distancerdquoJournal of Beijing University of Technology vol 33pp 1070ndash1074 2007

[32] S Feng Z Li Y Ci and G Zhang ldquoRisk factors affecting fatalbus accident severity their impact on different types of busdriversrdquo Accident Analysis amp Prevention vol 86 pp 29ndash392016

[33] G Zhang K KW Yau X Zhang and Y Li ldquoTraffic accidentsinvolving fatigue driving and their extent of casualtiesrdquo Ac-cident Analysis amp Prevention vol 87 pp 34ndash42 2016

[34] C Wu X Yan and X Ma ldquoA new car-following model basedon multi-agent systemrdquo in Proceedings of the 2007 Interna-tional Symposium on Distributed Computing and Applicationsto Business Engineering and Science (DCABES 2007) vol 2pp 101ndash107 Yichang China August 2007

[35] K Yi and Y D Kwon ldquoVehicle-to-vehicle distance and speedcontrol using an electronic-vacuum boosterrdquo JSAE Reviewvol 22 no 4 pp 123ndash129 2001

[36] X Zhao S Wang J Ma Q Yu Q Gao and M YuldquoIdentification of driverrsquos braking intention based on a hybridmodel of GHMM and GGAP-RBFNNrdquo Neural Computingand Applications vol 31 no S1 pp 161ndash174 2019

[37] N Monot X Moreau A Benine-Neto A Rizzo andF Aioun ldquoComparison of rule-based and machine learningmethods for lane change detectionrdquo in Proceedings of the 201821st International Conference on Intelligent TransportationSystems (ITSC) pp 198ndash203 Honolulu HI USA November2018

[38] G Castignani R Frank and T Engel ldquoAn evaluation study ofdriver profiling fuzzy algorithms using smartphonesrdquo inProceedings of the 2013 21st IEEE International Conference onNetwork Protocols (ICNP) pp 1ndash6 Goettingen GermanyOctober 2013

[39] S Fernandez and T Ito ldquoDriver classification for intelligenttransportation systems using fuzzy logicrdquo in Proceedings of the2016 IEEE 19th International Conference on IntelligentTransportation Systems (ITSC) pp 1212ndash1216 Rio de JaneiroBrazil November 2016


vehicle speed variation minimal safety distance etc [8] Huet al firstly studied driver behavior differences and empiricaljudgment ratio distributions in an abnormal traffic scenarioand then proposed a car-following model for estimating theminimum safety distance for the emergency evacuation ve-hicle [9] Similar research studies can be found in [10ndash14]

Many studies have been conducted to analyze the rela-tionship between traffic safety and drivers characteristic in-cluding driver personality physical fitness driver distractionetc [15ndash18] Tang and Xia implemented a series of experi-ments to measure the reaction time of four different drivertypes and further studied its impact on the minimal safetydistance of preventing rear-end collision [17] Tang and Xiaanalyzed a variety of factors that affect the driver reaction timeby using fuzzy mathematics theory and then proposed a novelmodel to estimate driver reaction time [17] By consideringinfluence of driver individual differences vehicle brakingperformance and driving states Zhang et al established a car-following model with minimum safety distance for threetypical traffic states [19] Similarly Xue et al proposed a rear-end collision behavior model considering the individual dif-ferences of drivers [20] Some scholars analyzed the rela-tionship between other influencing factors and car safety suchas the speed relation among adjacent vehicles [21 22]

Fuzzy relevant models have been proposed to simulatereal-world traffic situations considering varied driver typesand thus provide optimal traffic control strategies for thepurpose of ensuring traffic safety and efficiency (ie withminimal waiting time short queue length etc) Chai et alproposed a simulation-based approach to measure drivercognitive failures which was implemented with fuzzy logicand cellular automata model [23] Azimirad et al proposed anovel fuzzy traffic controller to formulate and optimize trafficcontrol at isolated signalized intersection [24] Li proposed adynamic fuzzy neural networks traffic flow prediction modelto accurately obtain traffic flow prediction under chaos trafficstate [25] Bocklisch et al proposed an adaptive fuzzy patternclassification model for simulating nonlinear multidimen-sional transition processes which aims to identify the lanechange intentions for varied driver types [26]

e previous studies focusing on vehicle rear-end colli-sion analysis mainly employed the kinematic data to estimateminimal neighboring vehicle distance with the quantitativeindicators (ie speed displacement etc) which did notconsider the qualitative indicators (eg driving behaviordifferences influence) and mainly led to biased results eadvantages of fuzzy theory can help us quantify the physi-ological and psychological characteristics of vehicle driversand thus provide us a more holistic view of the car-followingmodele paper is organized as follows Driver features weredeeply analyzed in Section 2 and the proposed safety modelwas illustrated in Section 3 e experimental results weredescribed in Section 4 and Section 5 concluded the paper

2 Driving Characteristics and Reaction Time

21 Driving Characteristics Analysis e vehicle controllingprocedure consists of environment perception potential riskjudgment and traveling decision making and vehicle maneuver

which can be found in Figure 1 In the perception stage thedriverrsquos control of car speed and perception of surroundingenvironment have a close bearing on the driverrsquos reaction speedto the unexpected situation In the judgment and decision-making stage the driver makes control decisions based ondriving experience to ensure driving safety In the operationstage the driver controls the car accelerating deceleratingturning and braking according to the decision Note that over80 traffic accidents are triggered due to the driver errors[16 27 28] To avoid the potential traffic accidents drivers aresupposed to take early and correct maneuver activities (ieidentify risky behaviors decide the suitable travel behavior takemaneuver activities etc) for each of vehicle maneuver steps

e driverrsquos individual characteristics can be analyzedfrom physiological and psychological factors for example thedriverrsquos age gender and driving age fatigue and emotionswhile driving the driverrsquos own driving style and safety atti-tude traffic regulations awareness level and so on eseresearch studies show that [29ndash31] (1) the reaction speed tothe emergency situation will be significantly reduced with age(2) in terms of emergency response capacity and the ability tocope with the complex external environment the femaledrivers are worse than the male drivers (3) the experienceddriver has the faster reaction in the emergency situation and(4) the adventure driver is more likely to be in danger of rapidacceleration deceleration and changing lanes than the cau-tious driver Based on the above analysis it is necessary toquantify the driving characteristics into quantifiable param-eters such as reaction time which can be introduced into themodeling process of the improved car-following model

22 Relationship Analysis between Driver Characteristics andReaction Time e driver age driving experience durationand fatigue level are considered as crucial elements forquantifying analysis on the relationship between drivercharacteristics and reaction time [32 33] Previous studieshave evaluated the reaction time variation under differenttraffic flow density constraints by building three kinds oftraffic flow density scenarios which suggested that the driverreaction time tends to be shorter in higher traffic flowdensity Factually the driver in a conservative driving style(intends to obtain much larger distance) has the longestreaction time regardless of the vehicle braking manner edriver age has a significant interference to driver reactiontime which is obviously observed in the emergency trafficsituations e statistical distributions of empirical trafficdata suggested that the average and variance of collisionavoidance reaction time of 18 to 30 years old drivers are thesmallest and the drivers over 51 years old have the longestcollision avoidance reaction time Previous studies suggestedthat aging drivers need more reaction time because theyrequire longer time to think carefully about various potentialtravel strategies [18 34 35]

tr 1 +(1 minus C)1minus Q

(1)

where tr is the driverrsquos reaction time C is the responsedegree of different types of drivers under different condi-tions and Q is vehicle braking type













y0 1113938 uc(y)y dy

1113938 uc(y)dy (2)





Feedback





S1 v1t (3)


S2 v212a1

(4)


D1 S1 + S2 + d0 v1t + v212a1 + d0

(5)




S1 v1t (6)


S3 v2t2 v2 v1 minus v2( 1113857

a1

v1v2 minus v22a1

(7)



minusv1v2 minus v22

a1+ d0

(8)



S1 v1t (9)






Outputreaction

time



S1 S2 d0





S2 v21 minus v2

2a1

S3 v22 minus v2

2a2

(10)



22 minus a2v

21

2a1a2+ d0

(11)








(a) e Mazda model


S1 S2 d0

Front car

D2 S3





S1 S2 d0

Front car

D2 S3





d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)


S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)


dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)





















0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References























































y0 1113938 uc(y)y dy

1113938 uc(y)dy (2)





Feedback





S1 v1t (3)


S2 v212a1

(4)


D1 S1 + S2 + d0 v1t + v212a1 + d0

(5)




S1 v1t (6)


S3 v2t2 v2 v1 minus v2( 1113857

a1

v1v2 minus v22a1

(7)



minusv1v2 minus v22

a1+ d0

(8)



S1 v1t (9)






Outputreaction

time



S1 S2 d0





S2 v21 minus v2

2a1

S3 v22 minus v2

2a2

(10)



22 minus a2v

21

2a1a2+ d0

(11)








(a) e Mazda model


S1 S2 d0

Front car

D2 S3





S1 S2 d0

Front car

D2 S3





d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)


S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)


dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)





















0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References













































S1 v1t (3)


S2 v212a1

(4)


D1 S1 + S2 + d0 v1t + v212a1 + d0

(5)




S1 v1t (6)


S3 v2t2 v2 v1 minus v2( 1113857

a1

v1v2 minus v22a1

(7)



minusv1v2 minus v22

a1+ d0

(8)



S1 v1t (9)






Outputreaction

time



S1 S2 d0





S2 v21 minus v2

2a1

S3 v22 minus v2

2a2

(10)



22 minus a2v

21

2a1a2+ d0

(11)








(a) e Mazda model


S1 S2 d0

Front car

D2 S3





S1 S2 d0

Front car

D2 S3





d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)


S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)


dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)





















0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References













































S2 v21 minus v2

2a1

S3 v22 minus v2

2a2

(10)



22 minus a2v

21

2a1a2+ d0

(11)








(a) e Mazda model


S1 S2 d0

Front car

D2 S3





S1 S2 d0

Front car

D2 S3





d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)


S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)


dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)





















0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References












































d a2v

21 minus a1v

22

2a1a2+ v1t1 + vrt2 + d0 (12)

(b) e Honda model

D (12 + l)v1 +v21 minus v222a

+ d (13)


S0 03195v1 + 0034vr +vr 2v1 minus vr( 1113857

25406φ+ 5 (14)


dw τvc +v2c2ac

+ doff

Dx v2c

2 00826vc + 06177( 1113857+ 36

(15)





















0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References
















































0

10

20

30

40

50

60

Dist

ance

(m)

Time t(s)

5489m

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

10

20

30

40

50

60

70

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10

6106m


0

20

40

60

80

100

120

D1 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models







0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References
















































0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


0

20

40

60

80

100

120

140

160

180

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

140

160

D2 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models








0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References

















































0

20

40

60

80

100

120

Dist

ance

(m)

Time t(s)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models


Time t(s)

0

20

40

60

80

100

120

140

Dist

ance

(m)

Rear carFront car

0 1 2 3 4 5 6 7 8 9 10


0

20

40

60

80

100

120

D3 d D S0 Dw Dx

Safe

ty d

istan

ce (m

)

Different models




5 Conclusions


Data Availability




Acknowledgments


References













































5 Conclusions


Data Availability




Acknowledgments


References












































study and simulation analysis of vehicle rear-end

Documents