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The Less-than-perfect Driver: A Model of Collision-inclusive Car-

following Behavior

Wuping Xin, PhD CandidateResearch and Development Division

KLD Associates, Inc47 Mall Drive Suite 8

Commack, NY 11725

Tel: 631-543-6500-227(ext)Fax: 631-543-4330

Email: wupingxin@kldassociates.com

John Hourdos, PhDMinnesota Traffic Observatory

University of Minnesota

500 Pillsbury Dr. SE

Minneapolis, MN 55455

Tel: 612-626-5492Fax: 612-626-7750

E-mail: hourdos@umn.edu

Panos Michalopoulos, ProfessorDepartment of Civil Engineering

University of Minnesota500 Pillsbury Dr. SE

Minneapolis, MN 55455Tel: 612-625-1509

Fax: 612-626-7750

E-mail: micha001@umn.edu

Gary Davis, Professor

Department of Civil Engineering

University of Minnesota

500 Pillsbury Dr. SEMinneapolis, MN 55455

Tel: 612-625-2598

Fax: 612-626-7750

E-mail: dtrips@umn.edu

Submitted for Presentation and Publication

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ABSTRACTIn this paper a new behavioral car-following model is presented. Unlike traditional car-followingmodels which preclude vehicle collisions, the proposed model aims to emulate less-then-perfect everyday driving, while striving to capture both safe and unsafe driver behavior. Most

importantly, a realistic perception-response process is incorporated with the model based on

developments from visual perception studies. In particular, driver inattention is characterized bya driver-specific variable called scanning interval. This variable when coupled with the drivers

visual perception-response process, actually results in variable reaction times that are not only

dependent on each drivers individual characteristics, but also on instantaneous traffic conditionssuch as speed and density. This enables closer emulation of real life human driving and its

interactions with surrounding vehicles. Stated otherwise, both inter- and intra-drivervariations

in reaction time are captured in a plausible and coherent manner, while in earlier studies, reactiontime is either presumed fixed, or of limited variability. Furthermore, parameters of this model

have a direct physical and behavioral meaning; this implies that vehicle collisions, if any, can be

analyzed with respect to behavioral patterns rather than simply being treated as numerical

artifacts. In all, 54 detailed and accurate vehicle trajectories extracted from 10 real life crashes

were employed to test the models capability of replicating freeway rear-end collisions. Inaddition high resolution crash-free trajectory data were employed to validate the model against

normal driving behavior. Test results indicate that the proposed model is able to replicate bothnormal as well as unsafe driving behavior that could lead to vehicle collisions. Finally, the

feasibility of integrating the proposed model with existing micro-simulators is briefly discussed.

The outcome of this work can eventually facilitate studying crash mechanisms at a highdefinition microscopic level as well as enabling safety-related system design improvements and

evaluation through micro-simulation software.

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INTRODUCTION

In recent years, considerable research has been conducted for the development, implementation,and evaluation of innovative ITS technologies aiming to improve traffic operations and safety.As part of this process, micro-simulation has become an increasingly indispensable tool for

system design and evaluation. However, while being successful in many applications, micro-

simulation has been insufficient for safety-related studies, i.e., its applications have long beenlimited to collision-free scenarios. This is because existing micro-simulators employ car-

following and lane-changing models that are limited to the emulation ofnormal driver behavior.

Collisions are deliberately precluded as they are hard to model. To be sure, realistic collision-

inclusive car-following models pertinent to both safe and unsafe driving behavior are still lacking.

In recent years, limited research efforts have focused on enhancing existing micro-simulation

models by developing conflict statistics and safety surrogates (Archer and Kosonen 2000;Hunuenin et al. 2005; Kosonen 1999; Kosonen and Ree 2000; Ledoux and Archer 1999;

Mehmood et al. 2001; Torday et al. 2003). However, due to the basic assumption that followers

will always maintain a safe distance, it has been difficult to test and validate the correlation

between extracted conflict statistics/surrogate measures with actual crash potential. This

inevitably limits the reliability of safety-performance conclusions obtained from crash-freemicro-simulation models.

In this paper, a new behavioral collision-inclusive car-following model is presented. This

model should eventually help to enable micro-simulation as an effective and reliable tool to

analyze crash-prone designs and aid in safety evaluations. Unlike traditional car-followingmodels that deliberately prohibit vehicle collisions, the proposed model aims to replicate both

safe and unsafe driver behavior. This is accomplished by employing a realistic perception-

response process based on developments in visual perception studies. Furthermore driverinattention is characterized by a driver-specific variable called scanning interval. This variable,when coupled with the perception-response process actually results in variable reaction times not

only dependent on individual driver characteristics, but also instantaneous traffic conditions such

as speeds and density. This allows closer emulation of real life human driving and itsinteractions with the surrounding vehicles. Model parameters have a direct physical and

behavioral meaning which allows vehicle collisions to be introduced as a result of behavioral

patterns rather than re-determined simple numerical artifacts. 54 vehicle trajectories extracted

from 10 real life crashes and near-crashes were employed to test the models capability ofreplicating freeway rear-end collisions. High resolution crash-free trajectory data were also used

to validate the model against collision-free driving behavior. Test results indicate that theproposed model is able to replicate both normal as well as unsafe driving behavior that could

lead to vehicle collisions.

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addition, an enhanced perception-response process is integrated into the model based on research

results from visual perception studies. This essentially results in variable reaction times

dependent not only on driver characteristics but also instantaneous traffic conditions. Stateotherwise, both inter- and intra-drivervariations in reaction time are captured in a plausible and

logical manner, while in earlier studies, reaction time is either presumed constant, or of limitedvariability. Finally, the proposed model was tested using both non-crash and crash data. This

should assist in better understanding of unsafe car-following behavior and provide insights for

crash-mechanisms at the microscopic (individual vehicle) level.

BACKGROUND

Car-following models have been the subject of numerous research efforts since the 1950s. Themotivation arose from the need to model individual vehicle movements as realistically as

possible. In conjunction with lane-changing algorithms, car following modeling forms the

cornerstone of microscopic simulation tools and is of particular importance for studying rear-endcollisions. Briefly, most of existing car-following models can be represented as:

( ) ( , , , , ; )na t f x v v a a t + = (1)or ( ) ( , , , , ; )na t f x v v a a t + = + (2)

with1 1

( ) ( ) ( )n n n

x t x t l x t

=

1( ) ( )

n nv v t v t

=

1( ) ( ) ( )n na t a t a t =

where1n

a

is acceleration of lead vehicle;n

a is acceleration of following vehicle; a represents

relative acceleration between lead and following vehicles; 1nl is the length of lead vehicle; nl isthe length of following vehicle; 1n represents the index of lead vehicle; n represents the index

of following vehicle; is the reaction delay of following vehicle; 1nv is the instantaneous speed

of lead vehicle;n

v is the instantaneous speed of following vehicle; v is relative speed between

lead and following vehicles;1n

x

is the position of lead vehicles bumper;n

x is the position of

following vehicles bumper; x is the bumper to rear distance between lead and following

vehicles; is noise term.

These models can be either deterministic as described by Equation (1), or stochastic as in

Equation (2) where a certain noise term is added, making them discontinuous in time. In

addition, taking the driver and vehicle as one integrated unit, most current car-following models

incorporate reaction time delay into the dynamics. This reaction time delay is defined as the timelag between a stimuli effected by the lead vehicle (e g braking) and the response by the subject

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psycho-physical modeling that cannot be solely represented by Equation (1) or (2). Specifically,

this type of models assumes vehicle acceleration is more or less constant until a certain action-

point is reached (therefore, these models are sometimes referred to as action point models).Todesiev (Todesiev 1963) was among the first using action points to characterize car-following

process; while the most successful action-point models include Wiedsmann (Wiedemann and

U.Reiter 1992) and Fritzsche model (Fritzsche 1994); variants of these two models are

implemented in two widely-used commercial micro-simulators, i.e., VISSIM and

PARAMICS , respectively.

As pointed out earlier, existing car-following models have been successful at reproducing

many observed features of traffic flow and normative driving behavior. However, when vehiclecollisions and safety become important, these models cannot capture intricacies of individual

driver behavior. Specifically, performance limitations, judgment errors, and variable reaction

times that account for both intra- and/or inter-driver variations are not well integrated or simply

treated as additive random noise. Further, lapse of attention is yet to be properly addressed in a

logical and coherent way, and at best treated as random delayed reaction to an event in front. For

instance, in the VISSIM simulator, diver inattention is artificially generated by a predefined

probability and added as additional reaction delay (PTV 2007). In this paper, a new model is

proposed aiming to replicate both safe and unsafe driver behavior while taking into account theweakness and risks of human driving. To that end, a critical modeling effort is on incorporating a

realistic perception-response process based on developments in visual perception studies. As will

become clearer later, this actually results in both intra- and inter-driver variations in reaction

times, hence allowing closer emulation of real life human driving and its interactions with the

surrounding traffic dynamics.

Once again it should be stressed that classical car-following models have been developed

on a crash-free basis, under the assumption (explicitly or implicitly)that the following vehicle

always tries to maintain a safety distance from the leader. Albeit in the past years considerable

research efforts have been devoted to investigate the so-called instability of classical car-

following models (e.g., the GM-family models), and actually identified certain unstable

parameters that could result in collision situations (Castillo et al. 1994; Gazis et al. 1959; Gazis

et al. 1961; Herman et al. 1959; Wilson 2001; Zhang and Jarrett 1997), still crashes are not their

intended modeling objectives while vehicle collisions generated with unstable parameters are

more of modeling outliers rather than logical products of behavioral patterns.

MODEL

Figure 1 (a) illustrates the conceptual framework of the proposed model Essentially the

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Key Parameters and Implications

1) Maximum Comfortable Deceleration Rate ( 2/ft s ) nb This is the maximum deceleration rate an individual driver is willing to use in non-emergency

situations. It is bounded by the vehicles physical limits for decelerating. Usually this ranges

from 9~15 ft/s^2 (0.20 ~0.35 g). Particularly, the Traffic Engineering Handbook by ITE (ITE

1999a) suggests 10 2/ secft as comfortable deceleration rate. Meanwhile, theAmerican

Association of State Highway and Transportation Officials (AASHTO) recommends 11.22/ secft as the value for comfortable deceleration rate for most drivers (AASHTO 1990).

Regardless, in this research, the maximum comfortable deceleration rate is not determined apriori, but as a result from model calibration subject to the feasible box constraints (i.e., simple

inequality constraints).

2) Maximum Comfortable Acceleration Rate ( 2/ft s ) na This is the maximum acceleration rate a driver is willing to use to achieve desired free-flow

speed or to catch up with the lead vehicle. This parameter is bounded by the vehicles physical

limits for accelerating.

3) Perceptual Threshold of Visual Expansion Rate (radian/sec) C&

This parameter characterizes the mechanism a driver uses to identify relative motion.

Specifically, the visual stimulus to detect relative motion has been found in visual perception

studies to be related to a continuously changing optic array characterized by the visual angle

subtended by the image of the lead vehicle on the retina, and the expansion rate of this angle &

(Evans and Rothery 1974; Gibson 1982; Lee 1976; Lee and Reddish 1981; Michaels 1965;Michaels and Cozan 1963; Sun and Frost 1998). Mathematically, and & can be specified as

(See Figure 1 (b) ):

/ 2tan( )

2 ( )

W

D t

= (3)

sin( ) ( )D t

V

=&

(4)

where

: Visual angle subtended by the image of the lead vehicle on retina;

& : Angular velocity, i.e., the expansion rate of ;

W : Vehicle width;

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( )D t

V

=

&(6)

2

VW

D

=& (7)

In order to perceive relative motion, the angular velocity & must be above a certain

threshold value C&

. This means, in order for a following driver to perceive relative velocity in

relation to the preceding vehicle,2

| |W VC

D

& must be satisfied. Different values of this threshold

have been reported in the literature ranging from 1 ~ 40 410 / secradian (Brackstone et al.

2000; Evans and Rothery 1973; Evans and Rothery 1977; Hoffmann 1968; Michaels and Cozan

1963; Wiedemann and U.Reiter 1970s).

It is very important to point out that the perception process characterized by this perceptual

threshold actually entails a logical and coherent mechanism for variable reaction time. This

implication is best explained through the following example. Assuming a driver (vehicle) has a

characteristic perceptual threshold of 8

4

10

/ secradian . Both the subject vehicle and thepreceding vehicle are traveling with a constant speed of 90 ft/sec and distance headway 180 ft.

The width of the preceding vehicle is 5 ft. At time 1 0sect = , the preceding vehicle starts braking

at -10 ft/s 2 . It is not until 2t = 2.1 sec (when 2( )

( )

V tW

D t

= 8 410

) that the following vehicle

begins to perceive the relative motion. This means the subject vehicles actual reaction time to

the braking event of the preceding vehicle is at least 2.1 sec. Note2

t varies as a function of

initial instantaneous speeds as well as distance headway. In other words, the actual reaction timeof the subject vehicle is dependent both on the instantaneous speeds as well as on local density.

Considering in real life traffic the instantaneous speeds and distance headways are actually time-

varying, the visual perception process described above essentially provides a logically coherent

and plausible mechanism that can better explain variable reaction times observed in reality. Most

importantly, this mechanism, with its mathematical simplicity can be easily implemented in car-

following models without the need to store large amount of historical data. Typically,

implementing variable reaction times, even with limited variability, requires storing historical

data of vehicles past states thus has significant performance impacts.

4) Weber RatioD

C

This parameter characterizes another mechanism a driver uses to identify relative motion when

the relative speed is very small. Essentially, Weber Ratio refers to theJust Noticeable Change of

Di t (JNCD) h th bj t hi l i hi th hi l i f t (L i 1998)

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relative motion more promptly than a driver with greater C&

andD

C , i.e., the former driver has an

effectively smaller reaction time than the latter.

5) Oscillation Acceleration ( 2/ft s )osc

a and Oscillation Deceleration Rates ( 2/ft s )osc

b

When a driver can not perceive any relative motion between the subject vehicle and the vehicle

in front, he will use oscillation acceleration and deceleration rates to maintain the distance

headway in relation to the front vehicle. Essentially these two parameters are necessary to

capture the oscillatory spirals that have long be recognized as typical car-following behavior.

6) Desired Following Gap Time (sec) gt This is the following gap time (time headway) a driver wishes to maintain. This parameter

characterizes the aggressiveness and safety potential of the driver. For instance, a less-aggressive

driver would prefer greater following time headway to allow for sufficient reaction time in case

of emergent braking, while an aggressive driver would prefer shorter time headway and choose

to follow the lead vehicle closely.

7) Error of Desired Following Gap Timeg

This parameter refers to the percentage error in relation to the actual time headway when a driver

evaluates the latter.

8) Scanning Interval (sec)scan

t

Scanning interval is the time interval a driver samples his/her front view situation while driving.

This reflects the attention level and alertness of the driver. Scanning interval is a driver-specific

as well as traffic-situation specific parameter.

9) Mechanical Response Delay (sec) dt Mechanical response delay is the time delay involved in the drivers implementation of a control

decision, i.e., the time lag between the drivers start of the control action and the time such

control action takes effect.

Note that a unique feature of the proposed model is the notion of effective reaction time.

Although there is no explicit reaction time parameter associated with each driver-vehicle-unit,the perceptual thresholds as well as scanning intervals plus mechanical response delay

effectively render a variable reaction time that is dependent on instantaneous speed and local

density. In other words, reaction time is an implicit time-dependent variable embedded in the

model logic. This provides greater flexibility and modeling realism regarding the drivers

perception response process than conventional approach assuming fixed reaction time.

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Component. Then a control decision is determined by the Decision Making Component in terms

of acceleration or deceleration to minimize any deviation between its current state and desired

state. Finally, control decisions are implemented via the Vehicle Mechanics Component after amechanical response delay. These four components compose the proposed DUV model. Note

that situational factors do not include multilane effects, while the impacts of personal factors and

environmental factors on drivers perception-decision process are not explicitly incorporated in

the model. Once more experimental data and relevant research results from human factors or

psychology studies become available, the model can be expanded to take into account these

missing factors.

Information Acquisition Component (IAC)TheInformation Acquisition Component(IAC) samples surrounding traffic environment every

scanning interval. The inputs to this component include subject vehicles instantaneous speed,

acceleration, preceding vehicles speed and acceleration, and relative speed and distance

headway between the subject vehicle and the preceding vehicle. IAC translates these inputs into

three variables:

Visual Expansion Rate & Change of Distance Instaneous Time Gap

Visual Expansion Rate is computed as in Equation(7). Change of Distance is simply the

difference between current distance headway and the distance headway at previous scanning

interval.Instaneous Time Gap is derived by dividing current distance headway by subject

vehicles instantaneous speed.

Driving Strategy Component (DSC)

The specific driving strategy is to maintain a desired following gap time gt subject to safetyconstraints. Due to human drivers inability to keep exact following gap time, it is further

assumed that a driver will allow an error ofg

while maintaining his/her desired following gap

time. This means, if the instantaneous gap time is within [(1 ) , (1 ) ]g g g g

t t + , the driver would

simply identify it withg

t .

Decision Making Component (DMC)Decision Making Component(DMC) determines an appropriate control maneuver (acceleration

or deceleration or cruising) based on the information fromInformation Acquisition Component

andIndividual Driving Strategy Component. Specifically, at each scanning interval, DMC does

the following Boolean checking:

1) True or false: Visual Angle Expansion Rate exceeds the threshold C&

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driving with a comfortable following gap time thus there is no need to accelerate to catch up with

preceding vehicle or decelerate to avoid a looming collision. However, small displacements of

gas pedal, wind or pavement friction could still effect an actual small acceleration or decelerationfor the subject vehicle. Usually this rate is in the range of 0.5~0.8 2/ft s (Brackstone et al. 2000;

Brackstone et al. 2002; McDonald et al. 1997; Sultan et al. 2004; Wang 2005). In this study, it is

termed as Oscillation Acceleration/Deceleration Rate because apart from having the preceding

vehicle changes its speed drastically, the subject vehicle would keep employing this small rate

until its perceptual threshold is exceeded and then the subject vehicle will reverse the sign of the

acceleration rate until the perceptual threshold is exceeded again.

When neither of the above checking results true, Gipps-like car following rules areused:

( ) min ( ), ( )a bn g n g n g

v t t v t t v t t + = + + (8)

max max

( ) ( )( ) ( ) 2.5 [1 ] 0.025a n nn g n n g

n n

v t v t v t t v t a t

V V+ = + + (9)

22 2 1

1 0

1

( )( ) {2[ ( ) ( ) )] ( ) }

b nn g g n g n n n n g n

n

v tv t t t b t b b x t x t s t v t

b

+ = + + + (10)

where

gt is the drivers desired following gap time;

na is the maximum comfortable acceleration rate for vehicle n;

nb is the maximum comfortable braking rate for vehicle n;

nx , 1nx

are the position of vehicle n and n-1, respectively;

1n

b

is nth drivers estimation for ( 1)n th vehicle maximum comfortable braking rate;max

nV is the desired free flow speed of vehicle n;

0s is the distance headway at standstill.

Essentially Equations (8)(9)(10) reflect the speed the subject driver wishes to achieve

within his/her desired following gap time. In these equations,g

t is added as time delay rendering

the process a time-delay process. Note this is differentfrom the original Gipps car-following

model which uses drivers fixed reaction time as the time delay term. To be sure,g

t in Equation

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Vehicle Mechanics Component (VMC)

A suitable acceleration or deceleration rate determined by theDecision Making Component(DMC) will be implemented by Vehicle Mechanics Componentafter a mechanical response

delay dt .

Variable Reaction Time

As mentioned earlier, a unique feature of the proposed model is the actual variable reaction time

inherently implementedfor each individual driver. When combined with scanning intervalscan

t

and mechanical response delay dt , a drivers actual reaction time to an event in front becomes abounded time-varying variable:

d scant actual reaction time Nt (11)

where N= 1, 2, 3, depending on at which scanning interval the perceptual thresholds are

exceeded.

TESTING

DATATwo high resolution (0.1 sec) trajectory datasets are used in testing the proposed model. The first

is collision-free vehicle trajectories that were collected in a test track using Real-time Kinetic

GPS (RTKTPS) by Hokkaido University, Japan (Ranjitkar 2004). The unique feature ofRTKGPS dataset is that the lead vehicles speed profiles were carefully controlled during each

run to emulate various real-life car-following situations. The other is crash trajectories collected

in this study from a high crash-rate freeway section of I-94WB in Minneapolis, Minnesota.

Figure 2 (a) gives an aerial view of this section. The location is a three-lane section ofapproximately 1900 feet in length. Average daily volume is approximately 80,000 vehicles per

direction. Figure 2 (b) provides a summary of the 2002 crash statistics both for this section as

well as other high crash-rate locations in the Twin Cities area (Hourdos 2005). As shown inFigure 2 (b), the regional average crash rate in the Twin Cities area is 0.96 million-vehicle-miles

(MVM) in 2002, whereas the selected section has the highest 4.81 MVM as compared to theother locations. This is almost four times higher than the regional average level and effectivelyequivalent to one crash every two days.

Specifically, inasmuch as crashes are rare and random events, in order to maximize thenumber of crashes collected, an integrated video collection system was developed in the

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the wireless communication infrastructure. At the supervision station, crash recordings are

inspected to identify any crashes or near-crashes that have occurred during the day. Identified

crash recordings are processed to extract raw trajectories by using a video-based vehicle trackingprogram calledNG-VIDEO (Kovvali et al. 2007). It should be stressed that several non-trivial

subtasks have to be manually accomplished prior to applying theNG-VIDEO program. Thesesubtasks include Camera Calibration, World Coordinate Matching, and Video Orthorectification.

Camera Calibration means obtaining the intrinsic parameters (focal length, aspect ratio, radial

distortion etc.) of the surveillance cameras. Within the context of this study, this involves usingthe subject camera to snap a planar pattern shown at various orientations, and estimating the

cameras intrinsic parameters from the snapshots. Moreover, World Coordinate Matching is the

process of mapping pixel coordinates from the crash recordings to world coordinates in real life.This is accomplished with the aid of the ArcGIS software and high-resolution (0.5 foot pixel

image resolution) ortho-aerial images. Finally, Video Orthorectification requires registering the

images of crash videos to the coordinate system in the aerial image. After orthorectification,terrain features in the video images are removed and vehicles appear to be moving on a plane

surface. Once these subtasks are accomplished, the outputs are fed toNG-VIDEO. The latter

generates raw trajectories data and stores them in a MySQL database.

Out of 700 hours of video recordings collected, the final crash dataset includestrajectories of 54 vehicles involved in 6 crashes and 4 near-crashes that occurred at the site fromMay to August 2006. These trajectories are at 0.1 second resolution, and each trajectory is about

90 seconds. Figure 2(c) illustrates the extracted trajectories of 8 vehicles that had been involved

in one such crash. The crash occurred on the rightmost lane between Portland Ave and the

flyover to I-94 Westbound at 16:15:48 pm, July 12, 2006. Note that the numbering of vehicleIDs is not necessarily in ordinal order; while in this figure, Vehicle 02 is the second, Vehicle 07

the third vehicle in the platoon and Vehicle 30 is the last one. Similar plots for other

crashes/near-crashes are available but not presented here due to space limitations. Detailedinformation about the crash data collection and processing methodology can be found in (Xin et

al. 2007b)

MODEL CALIBRATIONParameters of the proposed model (represented by a parameter vector ) were calibrated by

minimizing the following objective function:

2

1

2 2

( ) ( |

( ), ( ) |

( ) ( | )

Nobs sim

tsim obs

modelN N

obs sim

x t x t

x t x t=

) =

(12)

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The objective is equivalently the sum of Theils U statistics with respect to vehicle

position data. Generalized Reduced Gradient (GRG) and Branch and Bound approaches were

implemented for searching solutions that minimize the objective function. Also the optimizationis started with different starting parameters to increase the possibility of finding global minima

(for each individual vehicle trajectory, 5000 different initial starts were employed which wasfound to give a good balance between computational time and convergence of the solution). The

feasible region of parameter space is constrained with the lower-bound and upper-bound of each

parameter determined from physical limits. For example, a feasible range for Maximum

Comfortable Acceleration would be around 11 2/ft s (ITE 1999). Finally, the calibration is

conducted for each following vehicle, i.e., the observed lead vehicle trajectory was employed as

fixed input to the model, and the parameters of the model are optimized so that the model-generated trajectory for the follower is as close as possible to the observed data.

It should be noted that often an objective function like Equation (13) is used forcalibration. However, pooling of data as in Equation (13) for calibration may add new

dimensions of difficulty for global optimization, because the observations are correlated (as they

are sequential position outputs of car-following model). In addition, there is also a potential

problem ofheteroscedasticity, i.e., the error associated with the predicted position will dependon its value. Therefore, Theils U statistics as expressed in Equation (12) is used as the objective

function for calibration, where the error is somehow normalized to offset the above mentioned

effects.

MODEL VALIDATION

Measures of Effectiveness

The purpose of validation is to test the models capability of replicating real-life scenarios with

calibrated parameters. The following metrics are used to evaluate the deviation between model

generated trajectories and real-life trajectories:1) Root Mean Square Error (RMSE)

( )2

1

1 N

i ii

RMSE x yN =

=

(13)

wherexi is the model predicted value indexed by i;

yi is the actual observation indexed by i;N is the number of total observations.

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The test track RTKGPS car-following data are used to validate the model against normal car-

following behavior. The validation was accomplished by calibrating each DUV using only part

of its real-life trajectory but validating it using the entire trajectory in full length. For example aDUV in the platoon has 15 minute-long trajectory at 0.1 sec resolution. One-minute data out of

this entire 15 minutes trajectory is used to calibrate the DUV individually, and then the entire 15minutes trajectory is used to validate the model in platoon situation.

Validation against Crash DataValidation against crash data was conducted by testing if the model can generate the crash event

by collective simulation. This means, each DUV is calibrated individually with its preceding

vehicles observed trajectory as the input; then all DUVs involved in a crash (usually 5~6vehicles with the last two directly crashing each other) are simulated collectively with only theplatoon leader using observed trajectory as the predefined input. Other vehicles in the platoon are

simply updated sequentially using their respective calibrated parameters. If the crash event can

be generated as occurred in reality, then the validity of the model is verified.

RESULTS

Figure 3 (a) illustrates the simulation results of non-crash trajectories. In this figure, sevenvehicles indexed from G03 to G10 were simulated using the proposed model. Solid black lines

represent ground-truth vehicle trajectories while dotted colored lines stand for trajectories

predicted by the model. As shown in this figure, the model can replicate theses trajectories fairlywell. Additionally, Table 1(a) summarizes the calibrated DUV parameters. The goodness-of-fit

measures between real trajectories and model-predicted trajectories are summarized in Table

1(b). The measures presented in the table suggest the model can replicate the trajectories and thespeed variations closely.

As mentioned earlier, in this study a total of 54 vehicle trajectories from 10 crashes/near-

crashes were used to test the model. However due to space limitations, only one crash case ispresented here. The model is equally successful in replicating other crash cases given well-

calibrated parameters. Interested readers should refer to (Xin et al. 2007a) for more detailed

results. Specifically, the presented crash, i.e., Vehicle 30 colliding into Vehicle 16, occurred at16:15pm, July 12, 2006 on the leftmost lane of the I94WB freeway section. At the time the

weather conditions were dry and traffic was slow-and-go (see Figure 2(c) ). Figure 3 (b) depicts

the simulated crash trajectories vis--vis real crash trajectories. In this figure, the black solidlines are real-life trajectories while the colored dotted lines represent model predicted trajectories.

As shown in this figure, the proposed model successfully replicated the crash occurrence.

Additionally Table 2 presents the statistics of calibrated parameters values for all 54 crash

trajectories. More detailed discussion about the distribution of the calibrated parameter valuesf hi d h h b f d i (Xi l 2007 )

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existing micro-simulation models only describe normative car-following behavior devoid of

weaknesses and risks associated with real-life everyday driving. To date, realistic car-following

models pertinent to the less-then-perfect nature of driver behavior are lacking. In this paper anew car-following model is presented which is capable of more realistically emulating real-life

car-following behavior with its risks and imperfections. The model capitalizes on real-life crashdata collected from a US freeway as well as test track car-following data from Hokkaido

University in Japan to assist model development, calibration and validation. Test results based

on actual crash data indicate that the model is capable of replicating both normal drivingbehavior and unsafe behavior that could lead to collisions.

Prior to closing, it should be noted that due to space limitations a detailed discussion ofsome important issues related to model calibration/validation, e.g., parameter identifiability andsensitivity, as well as in-depth analysis of the parameters distributions are not presented. Also,

the models capability to generate well-known macroscopic traffic phenomena such as

shockwaves, the fundamental diagram, is not examined in this paper. Such topics are beyondthis papers initial scope and are the subjects of several others currently under way. Furthermore,

research efforts to integrate the proposed model with existing commercial simulators such as

AIMSUN or VISSIM are in progress. These simulators provide flexible APIs that can be used to

facilitate integrating the proposed model. However, since the proposed model is collision-inclusive, once integrated with a full-featured simulator environment, more issues need to be

resolved before the simulator can be employed in actual safety evaluations or safety-related

applications. One critical issue would be potentially long simulation running time or significantlyincreased number of replications in order to get meaningful safety indicators. A potential

solution to this is the employment of parallel computing techniques. Finally it is hoped that this

research will contribute our understanding of car-following behavior while improving micro-simulation modeling to facilitate assessing freeway safety concepts at a high definition

microscopic level.

ACKNOWLEDGEMENTSThis study utilized the advanced detection and surveillance stations maintained by the Minnesota TrafficObservatory while support for this study was provided by the ITS Institute of the University of Minnesota.

The authors wish to thank Professor Takashi Nakatsuji and Dr. Prakish Ranjitkar from Hokkaido

University, Japan for providing the RTKGPS data.

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Figure 1 (a). Conceptual Framework of the Proposed Car-Following Modeling

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Figure 1 (b). Visual angle subtended by the image of an object on the retina and its expansion rate.

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Figure 2 (a). Data collection site: a high crash-rate freeway section of Interstate-94 WB

Figure 2(b) Ten Highest Crash Sections in 2002(Source: Mn/DOT Crash Facts 2002)

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Figure 3 (a). Validation results using collision-free data (lead vehicle is G03 not shown in the figure)

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Statistics

Parameters

Mean StandardDeviation StandardError of

the Mean

25%Percentile

75%Percentile

Inter-Quartile

Range

Minimum Maximum Range Median Variance Coefficientof

Variance

Kurtosis

Max Comft Accel na 9.47927 2.40595 0.80198 7.0303 11.3775 4.3472 6.01744 11.7485 5.73106 11.1362 5.78859 0.25384 -2.00512

Max Comft Decel nb -15.19791 3.75494 1.25165 -17.3094 -14.1512 3.1582 -21.4175 -7.95936 13.45815 -15.6733 14.09957 -0.24707 1.34193

Osc Accelosca

0.98065 0.55385 0.18462 0.59419 0.95337 0.35918 0.5095 1.96256 1.45306 0.75521 0.30675 0.56478 0.23958

Osc Deceloscb -0.83362 0.1814 0.06047 -0.8792 -0.7552 0.124 -1.1163 -0.5977 0.5186 -0.8127 0.03291 -0.2176 -0.38262

Desired Fol Gapgt 1.28736 0.48314 0.16105 1.21169 1.62736 0.41567 0.23443 1.95112 1.71669 1.26694 0.23342 0.3753 2.59624

Gap Errorg

0.52333 0.28516 0.09505 0.29351 0.83923 0.54572 0.11743 0.88589 0.76846 0.52513 0.08132 0.5449 -1.50472

Scanning Intervalscant

0.87778 0.54263 0.18088 0.6 1.2 0.6 0.1 1.9 1.8 0.8 0.29444 0.61818 0.37272

Mechnical Delay dt 0.16667 0.0866 0.02887 0.1 0.2 0.1 0.1 0.3 0.2 0.1 0.0075 0.51962 -1.07937

Expansion Rate ThresC&

0.00421 5.6044E-4 1.8681E-4 0.00361 0.00466 0.00104 0.00343 0.00485 0.00142 0.00432 3.1409E-7 0.13305 -1.70277

Table 1(a) Statistics of calibrated parameters for non-crash data(RTKGPS data): sample size 9

Table 1(b). Validation Statistics (G03 is platoon leader)

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Statistics

Parameters

MeanStandardDeviation

StandardError of

the Mean

25%

Percentile

75%

Percentile

Inter-Quartile

Range

MinimumMaximum Range Median Variance

Coefficientof

Variance

Kurtosis

Max Comft Acceln

a 6.92531 1.95146 0.28167 5.2196 7.62579 2.40619 5.00569 11.7824 6.77671 6.14407 3.80821 0.28179 0.01718

Max Comft Decel nb -13.7712 4.11654 0.59417 -17.4338 -11.2517 6.1821 -17.99 -5.11 12.8852 -15.07755 16.94589 -0.29892 -0.82675

Osc Accelosc

a 0.87709 0.25366 0.03661 0.6983 1.06468 0.36638 0.50042 1.4479 0.94748 0.81979 0.06434 0.2892 -0.41101

Osc Deceloscb

-1.0489 0.36772 0.05308 -1.30114 -0.92893 0.37221 -1.49995 0.922 2.42195 -1.05869 0.13522 -0.35058 17.13454

Desired Fol Gapgt

1.1504 0.76785 0.11083 0.56342 1.25661 0.69319 0.5 3.49641 2.99641 0.96833 0.5896 0.66746 1.88519

Gap Errorg

0.42476 0.24519 0.03539 0.23924 0.60246 0.36322 0.01397 0.86882 0.85485 0.40122 0.06012 0.57724 -0.97086

Scanning Intervalscant 0.52292 0.46184 0.06666 0.2 0.7 0.5 0.1 1.9 1.8 0.35 0.21329 0.88319 0.63199

Mechnical Delaydt

0.2625 0.20999 0.03031 0.1 0.3 0.2 0.1 0.8 0.7 0.2 0.0441 0.79996 0.26576

Expansion Rate ThresC& 0.0028 0.00121 1.7524E-4 0.00161 0.00381 0.0022 0.00101 0.00491 0.0039 0.00293 1.474E-6 0.43365 -1.26009

Table 2. Statistics of calibrated parameters for crash data( I94 crash trajectories)

Sample size: 54 drivers

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