Unformatted text preview: Introduction to Transportation Engineering
Car Following Models
Dr. Antonio A. Trani
Professor of Civil and Environmental Engineering
Virginia Polytechnic Institute and State University Blacksburg, Virginia
2009 Virginia Tech 1 Approaches to Modeling Trafﬁc
Microscopic
Attempts to look into individual driving behaviors
Vehiclefollowing models Macroscopic
Looks at the trafﬁc as a ﬂuidﬂow or heattransfer
phenomena
Vehicles are not identiﬁed individually but as a group of
entities moving on the system Technically, both microscopic and macroscopic models
consider the human in their solutions Virginia Tech 2 Car Following Models
• This problem has been studied for the past 45 years in
trafﬁc ﬂow theory
• Started with the work of Gazis and Herman
• Models vehicles individually
• The vehicle and human interactions are modeled
explicitly
• A driver follows another vehicle by judging:
a) Distance
b) Speed difference
c) Reaction time
d) Vehicle performance
Virginia Tech 3 Car Following Model
Datum Point ˙˙ ˙
x fc x fc Following
Car ˙
x lc x fc x lc
Lead Car • Two cars follow each other on the same road
• The driver in the lead car has a speed known speed proﬁle
independent of the following car
• The driver in the following car adjusts to the behavior of
the front car
Virginia Tech 4 Car Following Models
Nomenclature: x fc and x lc are the positions of the following and lead cars,
respectively ˙
˙
x fc and x lc are the speeds of the following and lead cars,
respectively ˙˙
x fc is the acceleration of the following car. This is the
control variable that the driver adjusts to keep up with the
lead car (and avoid a collision) Virginia Tech 5 Car Following Models
Assume the velocity proﬁle of the lead car is known as a
function of time.
Proposed Driving Rule # 1
The acceleration proﬁle of the “following” car is just a
function of the relative speeds of the two cars; ˙
˙
˙˙
x fc ( t + τ ) = k ( x ( t ) lc – x ( t ) fc ) (1) where: k is a gain constant of the response process ˙˙
x fc ( t + τ ) is the acceleration of the following vehicle
Virginia Tech 6 ˙
x lc ( t ) is the speed of the leading vehicle
˙
x fc ( t ) is the speed of the following vehicle Virginia Tech 7 Car Following Models
Assume the velocity proﬁle of the lead car is known as a
function of time.
Proposed Driving Rule # 2
The acceleration proﬁle of the “following” car is a
function of the relative speeds and the relative distance
between the two vehicles; ˙
˙
˙˙ ( t + τ ) = k ( x lc ( t ) – x fc ( t ) )
x fc
( x lc ( t ) – x fc ( t ) ) (2) where: k is a gain constant of the response process Virginia Tech 8 ˙˙
x fc ( t + τ ) is the acceleration of the following vehicle
adjusted for time lag τ
˙
x lc ( t ) is the speed of the leading vehicle
˙
x fc ( t ) is the speed of the following vehicle
x lc ( t ) is the position of the leading vehicle
x fc ( t ) is the position of the following vehicle Virginia Tech 9 How Do We Use This Model?
• The models presented in equations (1) and (2) can be
tested against critical maneuvers performed by the lead
vehicle
• The critical maneuvers tests conditions where a driver in
the lead vehicle “brakes hard”, accelerates quickly, or
follows an erratic velocity proﬁle
• The second order differential equation can be solved
numerically if desired of with the use of the Matlab
toolbox called Simulink
• Simulink is a toolbox designed to solve differential
equations of motion Virginia Tech 10 SetUp of the Problem
• Assume the velocity proﬁle of the leading vehicle is
known (we “drive” this car to test the response of the
following vehicle) • Initially, assume no time lags in the acceleration response
of the following vehicle ( τ = 0 ) • Test an emergency braking maneuver executed by the
ﬁrst vehicle at 3 m/s2 • Test a new scenario with a deterministic time lag
response time of 0.75 seconds • Verify that both cars do not collide Virginia Tech 11 Simulink
• Simulink is a powerful toolbox to solve systems of
differential equations • Simulink has applications in Systems Theory, Control,
Economics, Transportation, etc. • The Simulink approach is to represent systems of ODE
using block diagram nomenclature • Simulink provides seamless integration with MATLAB.
In fact, Simulink can call any MATLAB function • Simulink interfaces with other MATLAB toolboxes such
as Neural Network, Fuzy Logic, and Optimization
routines Virginia Tech 12 Simulink Building Blocks
• Simulink has a series of libraries to construct models • Libraries have object blocks that encapsulate code and
behaviors • Connectors between blocks establish causality and ﬂow
of information in the model Virginia Tech 13 Simulink Interface
• The main application of Simulink is to model continuous
systems • Perhaps systems that can be described using ordinary
differential equations Virginia Tech 14 Typical Simulink Libraries
Shown are some typical Simulink libraries Virginia Tech 15 Sample Simulink Library (Windows and UNIX)
The new Simulink interface in Windows uses standard
graphical interfaces (Javabased interface) Virginia Tech 16 Example 1. FirstOrder Kinematic Model
We would like to solve the ﬁrstorder differential equation
shown below in Simulink V
d  = k 1 – k 2 V
dt (3) where: V is the speed of the vehicle, k 1 and k 2 are model
constants.
The values of the model parameters are: k 1 = 4.0 and k 2 = 0.1 with units for V in m/s and for dV in m/s2.
dt Virginia Tech 17 Simulink Model
The following plot shows the Simulink model solution for
the ﬁrst order differential equation V
d  = k 1 – k 2 V
dt
Equation Simulink Model
Virginia Tech 18 Procedure to Create a Simulink Model
The Simulink blocks needed for this model are found in
four distinct Simulink libraries:
• Constant block
• Product in the Simulink Sources library and Subtraction blocks in the Simulink Math Operations library
• Integrator block in Simulink Continuous library Virginia Tech 19 Procedure to Create a Simulink Model (cont.) • Scope block in Simulink Sinks library Once the blocks are available in the new model window,
they can be “wired”
Wiring a model inplies connecting the output connection
of each block with the input connection of another block Input connection
of integrator block Output connection
of integrator block Virginia Tech After wiring 20 Creating the Simulink Model
• Recognize the number of terms inside the differential
equation to be solved. For the example below we have
two terms in the righthand side
• Each term requires a series of operations to evaluate it.
Your Simulink model will require one mathematical
block for each operation. For example, the product
operation between k 2 and V requires a Simulink block V
d  = k 1 – k 2 V
dt Virginia Tech 21 Completing the Simulink Model
• In the model of interest we need to integrate the output of
the term k 1 – k 2 V because this is the rate of change of
vehicle speed with respect to time. We accomplish this
using an integrator block (see ﬁgure) V
d  = k 1 – k 2 V
dt Virginia Tech 22 Completing the Simulink Model
• The output of the integrator block is V (the vehicle
speed) which is the signal needed by the product block
multiplying k 2 and V . This forms a ﬁrstorder negative
feedback loop. V
d  = k 1 – k 2 V
dt Virginia Tech 23 Completing the Simulink Model
• The output of the integrator block ( V ) can be displayed in
a scope block. This scope block is labeled “Plot Velocity”
in the model V
d  = k 1 – k 2 V
dt Virginia Tech 24 Simulation Model Settings
• Before a differential equation is solved numerically we
need to deﬁne the simulation conﬁguration parameters
• These parameters are needed to tell the solver the initial
and ﬁnal times of the simulation and the numerical
integration procedure to be used Virginia Tech 25 Simulation Model Settings (cont.)
• For this simple differential equation we use a start time of
zero (t=0) and 40 seconds as the stop time (t=40). This
means the equation will be solved between these limits.
• The numerical differential equation solver used is a
variable step ODE45 solve (default) Virginia Tech 26 Initial Conditions
• Solving a differential equation requires the speciﬁcation
of initial conditions (Initial Value Problems). Initial
conditions are required for all state variables of the
system (e.g., variables to be integrated)
• There is only one state variable in this system ( V ).
Specify initial conditions by opening the integrator block Virginia Tech 27 Simulating and Getting Results
• To solve the differential equation just “start” the model
from the “Simulation” pull down menu (in Windows
there is “play” button in the Simulink window)
• See the model results opening each scope block Note: All plots
in a scope are
functions of time
Virginia Tech 28 Example 2  Car Following Problem
Solve the secondorder differential equation of motion
representing the acceleration proﬁle of the “following”
vehicle (acceleration is function of the relative speeds of
the two cars) ˙
˙
˙˙
x fc ( t + τ ) = k ( x ( t ) lc – x ( t ) fc ) (4) where: k is a gain constant of the response process ˙˙
x fc ( t + τ ) is the acceleration of the following vehicle
˙
x lc ( t ) is the speed of the leading vehicle
Virginia Tech 29 ˙
x fc ( t ) is the speed of the following vehicle Virginia Tech 30 Simulink Representation of the CarFollowing
Problem Virginia Tech 31 CarFollowing Model Output
The time space diagram below illustrates an emergency
braking maneuver for the leading vehicle
Space (m) Following vehicle
Lead vehicle Time (s) Virginia Tech 32 CarFollowing Model Output
Velocity proﬁles for lead and following vehicles
Speed (m/s)
Following vehicle
Lead vehicle Time (s) Virginia Tech 33 CarFollowing Experiment
Goal: to determine the practical capacity of an interstate
highway using a carfollowing model
Method:
a) Simulate a critical maneuver (such as hard braking)
b) Determine if vehicles collide given known initial
positions and speeds (initial conditions)
c) Repeat the experiment in part (b) for various initial
conditions Virginia Tech 34 CarFollowing Experiment
d) Assuming drivers use common sense, the minimum
spacing is the distance between cars that avoids the
collision by some desired margin
NOTE: In real practice, several critical manuevers could
be tried to obtain a safe spacing between successive cars.
The issue is whether people driving are so predictable. Virginia Tech 35 Car Following Experiment
Assumptions:
a) Cars travel at 30 m/s initially
b) Lead car initial position is at datum point (0 meters)
c) Lead car brakes at 3 m/s2 to avoid an obstacle
d) Following car brakes behind lead car
e) Car size is 5.8 meters in length
f) Minimum desired distance between cars at end of
critical maneuver is one car length (5.8 meters) Virginia Tech 36 Car Following Example
Following
Car Lead Car ˙˙ ˙
x fc x fc ˙
x lc x fc x lc
Datum Point Minimum safe distance between two stopped vehicles is
(11.6 meters  two car lengths) Virginia Tech 37 Car Following Experiment
Table containing outcomes of carfollowing experiment
Following Car Initial
Position (m) from
Datum Point Distance Between Front
Bumpers at Closest
Point of Approach (m) Remark 100 70 Safe 90 60 Safe 80 50 Safe 70 40 Safe 60 30 Safe 50 20 Safe 40 10 No 30 0 No Virginia Tech 38 Interpretation of Results
The experiment suggest that the critical spacing to avoid
an accident is between 40 meters and 50 meters
Further reﬁnement of the experiment suggests 42 meters
is the critical spacing to avoid an accident
Starting with an initial condition of 30 m/s for the lead
and following cars, this implies a headway of 1.4 seconds
per successive vehicle
The approximate capacity of the highway for this critical
maneuver would be 2,514 vehicles per hour per lane
NOTE: This value is optimistic since no time lags have
been factored in the manmachine system modeled
Virginia Tech 39 CarFollowing Model with Delay
A pure transport delay block is added to the original
model simulating the transport lag dynamics of a manmachine system Virginia Tech 40 Output of the CarFollowing Model with Delay
• If the following car is 50 meters behind the lead car,
clearly a collision (spacing < 11.6 m.) occurs if τ =1.5s. Space (m) Following vehicle
Lead vehicle Time (s)
Virginia Tech 41 Car Following Experiment (Again)
Repeat (for homework the experiment) including the
time lag (to be conservative use a 2.5 second reaction
time)
Other Assumptions:
a) Cars travel at 30 m/s initially
b) Lead car initial position is at datum point (0 meters)
c) Lead car brakes at 3 m/s2 to avoid an obstacle
d) Following car brakes behind lead car Virginia Tech 42 Car Following Experiment (Again)
e) Car size is 5.8 meters in length
f) Minimum desired distance between cars at end of
critical maneuver is one car length (5.8 meters) Virginia Tech 43 ...
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 Fall '08
 KATZ
 Virginia Polytechnic Institute

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