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# Chapter 14 - Car Following Models Notes | EduRev

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## : Chapter 14 - Car Following Models Notes | EduRev

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Transportation Systems Engineering 14. Car Following Models
Chapter 14
Car Following Models
14.1 Overview
Longitudinal spacing of vehicles are of particular importance from the points of view of safety,
capacity and level of service. The longitudinal space occupied by a vehicle depend on the
physical dimensions of the vehicles as well as the gaps between vehicles. For measuring this
longitudinal space, two microscopic measures are used- distance headway and distance gap.
Distance headway is de?ned as the distance from a selected point (usually front bumper) on
the lead vehicle to the corresponding point on the following vehicles. Hence, it includes the
length of the lead vehicle and the gap length between the lead and the following vehicles.
14.2 Car following models
Carfollowingtheoriesdescribe howonevehicle followsanothervehicle inanuninterrupted ?ow.
Various models were formulated to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like Pipes, Forbes, General Motorsand Optimal velocity
model are worth discussing.
14.2.1 Notation
Before going in to the details, various notations used in car-following models are discussed here
with the help of ?gure 14:1. The leader vehicle is denoted as n and the following vehicle as
(n+1). Two characteristics ataninstant t areofimportance; locationandspeed. Locationand
speed of the lead vehicle at time instant t are represented by x
t
n
and v
t
n
respectively. Similarly,
the location and speed of the follower are denoted by x
t
n+1
and v
t
n+1
respectively. The following
vehicle is assumed to accelerate at time t+?T and not at t, where ?T is the interval of time
required for a driver to react to a changing situation. The gap between the leader and the
follower vehicle is therefore x
t
n
-x
t
n+1
.
Dr. Tom V. Mathew, IIT Bombay 14.1 February 19, 2014
Page 2

Transportation Systems Engineering 14. Car Following Models
Chapter 14
Car Following Models
14.1 Overview
Longitudinal spacing of vehicles are of particular importance from the points of view of safety,
capacity and level of service. The longitudinal space occupied by a vehicle depend on the
physical dimensions of the vehicles as well as the gaps between vehicles. For measuring this
longitudinal space, two microscopic measures are used- distance headway and distance gap.
Distance headway is de?ned as the distance from a selected point (usually front bumper) on
the lead vehicle to the corresponding point on the following vehicles. Hence, it includes the
length of the lead vehicle and the gap length between the lead and the following vehicles.
14.2 Car following models
Carfollowingtheoriesdescribe howonevehicle followsanothervehicle inanuninterrupted ?ow.
Various models were formulated to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like Pipes, Forbes, General Motorsand Optimal velocity
model are worth discussing.
14.2.1 Notation
Before going in to the details, various notations used in car-following models are discussed here
with the help of ?gure 14:1. The leader vehicle is denoted as n and the following vehicle as
(n+1). Two characteristics ataninstant t areofimportance; locationandspeed. Locationand
speed of the lead vehicle at time instant t are represented by x
t
n
and v
t
n
respectively. Similarly,
the location and speed of the follower are denoted by x
t
n+1
and v
t
n+1
respectively. The following
vehicle is assumed to accelerate at time t+?T and not at t, where ?T is the interval of time
required for a driver to react to a changing situation. The gap between the leader and the
follower vehicle is therefore x
t
n
-x
t
n+1
.
Dr. Tom V. Mathew, IIT Bombay 14.1 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
n
Direction of traffic
n+1
v
n
x
n
y
n+1
v
n+1
x
n
-x
n+1
Figure 14:1: Notation for car following model
14.2.2 Pipe’s model
The basic assumption of this model is “A good rule for following another vehicle at a safe
distance is to allow yourself at least the length of a car between your vehicle and the vehicle
ahead for every ten miles per hour of speed at which you are traveling” According to Pipe’s
car-following model, the minimum safe distance headway increases linearly with speed. A
disadvantage of this model is that at low speeds, the minimum headways proposed by the
theory are considerably less than the corresponding ?eld measurements.
14.2.3 Forbes’ model
In this model, the reaction time needed forthe following vehicle to perceive the need todeceler-
ateand applythe brakes isconsidered. That is, the time gapbetween the rearofthe leader and
the front of the follower should always be equal to or greaterthan the reaction time. Therefore,
the minimum time headway is equal to the reaction time (minimum time gap) and the time
required for the lead vehicle to traverse a distance equivalent to its length. A disadvantage of
this model is that, similar to Pipe’s model, there is a wide di?erence in the minimum distance
headway at low and high speeds.
14.2.4 General Motors’ model
The General Motors’ model is the most popular of the car-following theories because of the
following reasons:
1. Agreement with ?eld data; the simulation models developed based on General motors’
car following models shows good correlation to the ?eld data.
2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-
density relationship can be derived from General motors car following model.
Dr. Tom V. Mathew, IIT Bombay 14.2 February 19, 2014
Page 3

Transportation Systems Engineering 14. Car Following Models
Chapter 14
Car Following Models
14.1 Overview
Longitudinal spacing of vehicles are of particular importance from the points of view of safety,
capacity and level of service. The longitudinal space occupied by a vehicle depend on the
physical dimensions of the vehicles as well as the gaps between vehicles. For measuring this
longitudinal space, two microscopic measures are used- distance headway and distance gap.
Distance headway is de?ned as the distance from a selected point (usually front bumper) on
the lead vehicle to the corresponding point on the following vehicles. Hence, it includes the
length of the lead vehicle and the gap length between the lead and the following vehicles.
14.2 Car following models
Carfollowingtheoriesdescribe howonevehicle followsanothervehicle inanuninterrupted ?ow.
Various models were formulated to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like Pipes, Forbes, General Motorsand Optimal velocity
model are worth discussing.
14.2.1 Notation
Before going in to the details, various notations used in car-following models are discussed here
with the help of ?gure 14:1. The leader vehicle is denoted as n and the following vehicle as
(n+1). Two characteristics ataninstant t areofimportance; locationandspeed. Locationand
speed of the lead vehicle at time instant t are represented by x
t
n
and v
t
n
respectively. Similarly,
the location and speed of the follower are denoted by x
t
n+1
and v
t
n+1
respectively. The following
vehicle is assumed to accelerate at time t+?T and not at t, where ?T is the interval of time
required for a driver to react to a changing situation. The gap between the leader and the
follower vehicle is therefore x
t
n
-x
t
n+1
.
Dr. Tom V. Mathew, IIT Bombay 14.1 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
n
Direction of traffic
n+1
v
n
x
n
y
n+1
v
n+1
x
n
-x
n+1
Figure 14:1: Notation for car following model
14.2.2 Pipe’s model
The basic assumption of this model is “A good rule for following another vehicle at a safe
distance is to allow yourself at least the length of a car between your vehicle and the vehicle
ahead for every ten miles per hour of speed at which you are traveling” According to Pipe’s
car-following model, the minimum safe distance headway increases linearly with speed. A
disadvantage of this model is that at low speeds, the minimum headways proposed by the
theory are considerably less than the corresponding ?eld measurements.
14.2.3 Forbes’ model
In this model, the reaction time needed forthe following vehicle to perceive the need todeceler-
ateand applythe brakes isconsidered. That is, the time gapbetween the rearofthe leader and
the front of the follower should always be equal to or greaterthan the reaction time. Therefore,
the minimum time headway is equal to the reaction time (minimum time gap) and the time
required for the lead vehicle to traverse a distance equivalent to its length. A disadvantage of
this model is that, similar to Pipe’s model, there is a wide di?erence in the minimum distance
headway at low and high speeds.
14.2.4 General Motors’ model
The General Motors’ model is the most popular of the car-following theories because of the
following reasons:
1. Agreement with ?eld data; the simulation models developed based on General motors’
car following models shows good correlation to the ?eld data.
2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-
density relationship can be derived from General motors car following model.
Dr. Tom V. Mathew, IIT Bombay 14.2 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
In car following models, the motion of individual vehicle is governed by an equation, which
is analogous to the Newton’s Laws of motion. In Newtonian mechanics, acceleration can be
both the external force as well as those arising from the interaction with all other particles in
the system. This model is the widely used and will be discussed in detail later.
14.2.5 Optimal velocity model
The concept of this model is that each driver tries to achieve an optimal velocity based on the
distance to the preceding vehicle and the speed di?erence between the vehicles. This was an
alternative possibility explored recently in car-following models. The formulation is based on
the assumption that the desired speed v
n
desired
depends on the distance headway of the nth
vehicle. i.e.v
t
n
desired
= v
opt
(?x
t
n
) where v
opt
is the optimal velocity function which is a function
of the instantaneous distance headway ?x
t
n
. Therefore a
t
n
is given by
a
t
n
= [1/t][V
opt
(?x
t
n
)-v
t
n
] (14.1)
where
1
t
is called as sensitivity coe?cient. In short, the driving strategy of n
th
vehicle is that,
it tries to maintain a safe speed which in turn depends on the relative position, rather than
relative speed.
14.3 General motor’s car following model
14.3.1 Basic Philosophy
The basic philosophy of car following model is from Newtonian mechanics, where the acceler-
ation may be regarded as the response of a matter to the stimulus it receives in the form of
the force it receives from the interaction with other particles in the system. Hence, the basic
philosophy of car-following theories can be summarized by the following equation
[Response]
n
a [Stimulus]
n
(14.2)
for the nth vehicle (n=1, 2, ...). Each driver can respond to the surrounding tra?c conditions
only by accelerating or decelerating the vehicle. As mentioned earlier, di?erent theories on car-
following have arisen because of the di?erence in views regarding the nature of the stimulus.
The stimulus may be composed of the speed of the vehicle, relative speeds, distance headway
etc, and hence, it is not a single variable, but a function and can be represented as,
a
t
n
= f
sti
(v
n
,?x
n
,?v
n
) (14.3)
Dr. Tom V. Mathew, IIT Bombay 14.3 February 19, 2014
Page 4

Transportation Systems Engineering 14. Car Following Models
Chapter 14
Car Following Models
14.1 Overview
Longitudinal spacing of vehicles are of particular importance from the points of view of safety,
capacity and level of service. The longitudinal space occupied by a vehicle depend on the
physical dimensions of the vehicles as well as the gaps between vehicles. For measuring this
longitudinal space, two microscopic measures are used- distance headway and distance gap.
Distance headway is de?ned as the distance from a selected point (usually front bumper) on
the lead vehicle to the corresponding point on the following vehicles. Hence, it includes the
length of the lead vehicle and the gap length between the lead and the following vehicles.
14.2 Car following models
Carfollowingtheoriesdescribe howonevehicle followsanothervehicle inanuninterrupted ?ow.
Various models were formulated to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like Pipes, Forbes, General Motorsand Optimal velocity
model are worth discussing.
14.2.1 Notation
Before going in to the details, various notations used in car-following models are discussed here
with the help of ?gure 14:1. The leader vehicle is denoted as n and the following vehicle as
(n+1). Two characteristics ataninstant t areofimportance; locationandspeed. Locationand
speed of the lead vehicle at time instant t are represented by x
t
n
and v
t
n
respectively. Similarly,
the location and speed of the follower are denoted by x
t
n+1
and v
t
n+1
respectively. The following
vehicle is assumed to accelerate at time t+?T and not at t, where ?T is the interval of time
required for a driver to react to a changing situation. The gap between the leader and the
follower vehicle is therefore x
t
n
-x
t
n+1
.
Dr. Tom V. Mathew, IIT Bombay 14.1 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
n
Direction of traffic
n+1
v
n
x
n
y
n+1
v
n+1
x
n
-x
n+1
Figure 14:1: Notation for car following model
14.2.2 Pipe’s model
The basic assumption of this model is “A good rule for following another vehicle at a safe
distance is to allow yourself at least the length of a car between your vehicle and the vehicle
ahead for every ten miles per hour of speed at which you are traveling” According to Pipe’s
car-following model, the minimum safe distance headway increases linearly with speed. A
disadvantage of this model is that at low speeds, the minimum headways proposed by the
theory are considerably less than the corresponding ?eld measurements.
14.2.3 Forbes’ model
In this model, the reaction time needed forthe following vehicle to perceive the need todeceler-
ateand applythe brakes isconsidered. That is, the time gapbetween the rearofthe leader and
the front of the follower should always be equal to or greaterthan the reaction time. Therefore,
the minimum time headway is equal to the reaction time (minimum time gap) and the time
required for the lead vehicle to traverse a distance equivalent to its length. A disadvantage of
this model is that, similar to Pipe’s model, there is a wide di?erence in the minimum distance
headway at low and high speeds.
14.2.4 General Motors’ model
The General Motors’ model is the most popular of the car-following theories because of the
following reasons:
1. Agreement with ?eld data; the simulation models developed based on General motors’
car following models shows good correlation to the ?eld data.
2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-
density relationship can be derived from General motors car following model.
Dr. Tom V. Mathew, IIT Bombay 14.2 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
In car following models, the motion of individual vehicle is governed by an equation, which
is analogous to the Newton’s Laws of motion. In Newtonian mechanics, acceleration can be
both the external force as well as those arising from the interaction with all other particles in
the system. This model is the widely used and will be discussed in detail later.
14.2.5 Optimal velocity model
The concept of this model is that each driver tries to achieve an optimal velocity based on the
distance to the preceding vehicle and the speed di?erence between the vehicles. This was an
alternative possibility explored recently in car-following models. The formulation is based on
the assumption that the desired speed v
n
desired
depends on the distance headway of the nth
vehicle. i.e.v
t
n
desired
= v
opt
(?x
t
n
) where v
opt
is the optimal velocity function which is a function
of the instantaneous distance headway ?x
t
n
. Therefore a
t
n
is given by
a
t
n
= [1/t][V
opt
(?x
t
n
)-v
t
n
] (14.1)
where
1
t
is called as sensitivity coe?cient. In short, the driving strategy of n
th
vehicle is that,
it tries to maintain a safe speed which in turn depends on the relative position, rather than
relative speed.
14.3 General motor’s car following model
14.3.1 Basic Philosophy
The basic philosophy of car following model is from Newtonian mechanics, where the acceler-
ation may be regarded as the response of a matter to the stimulus it receives in the form of
the force it receives from the interaction with other particles in the system. Hence, the basic
philosophy of car-following theories can be summarized by the following equation
[Response]
n
a [Stimulus]
n
(14.2)
for the nth vehicle (n=1, 2, ...). Each driver can respond to the surrounding tra?c conditions
only by accelerating or decelerating the vehicle. As mentioned earlier, di?erent theories on car-
following have arisen because of the di?erence in views regarding the nature of the stimulus.
The stimulus may be composed of the speed of the vehicle, relative speeds, distance headway
etc, and hence, it is not a single variable, but a function and can be represented as,
a
t
n
= f
sti
(v
n
,?x
n
,?v
n
) (14.3)
Dr. Tom V. Mathew, IIT Bombay 14.3 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
where f
sti
is the stimulus function that depends on the speed of the current vehicle, relative
position and speed with the front vehicle.
is based on two assumptions; (a) higher the speed of the vehicle, higher will be the spacing
between the vehicles and (b) to avoid collision, driver must maintain a safe distance with the
Let ?x
t
n+1
is the gap available for (n+1)
th
vehicle, and let ?x
safe
is the safe distance, v
t
n+1
and v
t
n
are the velocities, the gap required is given by,
?x
t
n+1
= ?x
safe
+tv
t
n+1
(14.4)
where t is a sensitivity coe?cient. The above equation can be written as
x
n
-x
t
n+1
= ?x
safe
+tv
t
n+1
(14.5)
Di?erentiating the above equation with respect to time, we get
v
t
n
-v
t
n+1
= t.a
t
n+1
a
t
n+1
=
1
t
[v
t
n
-v
t
n+1
]
General Motors has proposed various forms of sensitivity coe?cient term resulting in ?ve gen-
erations of models. The most general model has the form,
a
t
n+1
=

a
l,m
(v
t
n+1
)
m
(x
t
n
-x
t
n+1
)
l


v
t
n
-v
t
n+1

(14.6)
and can take values from -2 to +2, and a is a sensitivity coe?cient. These parameters are to
be calibrated using ?eld data. This equation is the core of tra?c simulation models.
Incomputer, implementation ofthesimulationmodels, threethingsneedtoberemembered:
1. A driver will react to the change in speed of the front vehicle after a time gap called the
reaction time during which the follower perceives the change in speed and react to it.
2. The vehicle position, speed and acceleration will be updated at certain time intervals
depending on the accuracy required. Lower the time interval, higher the accuracy.
3. Vehicle position and speed is governed by Newton’s laws of motion, and the acceleration
is governed by the car following model.
Dr. Tom V. Mathew, IIT Bombay 14.4 February 19, 2014
Page 5

Transportation Systems Engineering 14. Car Following Models
Chapter 14
Car Following Models
14.1 Overview
Longitudinal spacing of vehicles are of particular importance from the points of view of safety,
capacity and level of service. The longitudinal space occupied by a vehicle depend on the
physical dimensions of the vehicles as well as the gaps between vehicles. For measuring this
longitudinal space, two microscopic measures are used- distance headway and distance gap.
Distance headway is de?ned as the distance from a selected point (usually front bumper) on
the lead vehicle to the corresponding point on the following vehicles. Hence, it includes the
length of the lead vehicle and the gap length between the lead and the following vehicles.
14.2 Car following models
Carfollowingtheoriesdescribe howonevehicle followsanothervehicle inanuninterrupted ?ow.
Various models were formulated to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like Pipes, Forbes, General Motorsand Optimal velocity
model are worth discussing.
14.2.1 Notation
Before going in to the details, various notations used in car-following models are discussed here
with the help of ?gure 14:1. The leader vehicle is denoted as n and the following vehicle as
(n+1). Two characteristics ataninstant t areofimportance; locationandspeed. Locationand
speed of the lead vehicle at time instant t are represented by x
t
n
and v
t
n
respectively. Similarly,
the location and speed of the follower are denoted by x
t
n+1
and v
t
n+1
respectively. The following
vehicle is assumed to accelerate at time t+?T and not at t, where ?T is the interval of time
required for a driver to react to a changing situation. The gap between the leader and the
follower vehicle is therefore x
t
n
-x
t
n+1
.
Dr. Tom V. Mathew, IIT Bombay 14.1 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
n
Direction of traffic
n+1
v
n
x
n
y
n+1
v
n+1
x
n
-x
n+1
Figure 14:1: Notation for car following model
14.2.2 Pipe’s model
The basic assumption of this model is “A good rule for following another vehicle at a safe
distance is to allow yourself at least the length of a car between your vehicle and the vehicle
ahead for every ten miles per hour of speed at which you are traveling” According to Pipe’s
car-following model, the minimum safe distance headway increases linearly with speed. A
disadvantage of this model is that at low speeds, the minimum headways proposed by the
theory are considerably less than the corresponding ?eld measurements.
14.2.3 Forbes’ model
In this model, the reaction time needed forthe following vehicle to perceive the need todeceler-
ateand applythe brakes isconsidered. That is, the time gapbetween the rearofthe leader and
the front of the follower should always be equal to or greaterthan the reaction time. Therefore,
the minimum time headway is equal to the reaction time (minimum time gap) and the time
required for the lead vehicle to traverse a distance equivalent to its length. A disadvantage of
this model is that, similar to Pipe’s model, there is a wide di?erence in the minimum distance
headway at low and high speeds.
14.2.4 General Motors’ model
The General Motors’ model is the most popular of the car-following theories because of the
following reasons:
1. Agreement with ?eld data; the simulation models developed based on General motors’
car following models shows good correlation to the ?eld data.
2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-
density relationship can be derived from General motors car following model.
Dr. Tom V. Mathew, IIT Bombay 14.2 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
In car following models, the motion of individual vehicle is governed by an equation, which
is analogous to the Newton’s Laws of motion. In Newtonian mechanics, acceleration can be
both the external force as well as those arising from the interaction with all other particles in
the system. This model is the widely used and will be discussed in detail later.
14.2.5 Optimal velocity model
The concept of this model is that each driver tries to achieve an optimal velocity based on the
distance to the preceding vehicle and the speed di?erence between the vehicles. This was an
alternative possibility explored recently in car-following models. The formulation is based on
the assumption that the desired speed v
n
desired
depends on the distance headway of the nth
vehicle. i.e.v
t
n
desired
= v
opt
(?x
t
n
) where v
opt
is the optimal velocity function which is a function
of the instantaneous distance headway ?x
t
n
. Therefore a
t
n
is given by
a
t
n
= [1/t][V
opt
(?x
t
n
)-v
t
n
] (14.1)
where
1
t
is called as sensitivity coe?cient. In short, the driving strategy of n
th
vehicle is that,
it tries to maintain a safe speed which in turn depends on the relative position, rather than
relative speed.
14.3 General motor’s car following model
14.3.1 Basic Philosophy
The basic philosophy of car following model is from Newtonian mechanics, where the acceler-
ation may be regarded as the response of a matter to the stimulus it receives in the form of
the force it receives from the interaction with other particles in the system. Hence, the basic
philosophy of car-following theories can be summarized by the following equation
[Response]
n
a [Stimulus]
n
(14.2)
for the nth vehicle (n=1, 2, ...). Each driver can respond to the surrounding tra?c conditions
only by accelerating or decelerating the vehicle. As mentioned earlier, di?erent theories on car-
following have arisen because of the di?erence in views regarding the nature of the stimulus.
The stimulus may be composed of the speed of the vehicle, relative speeds, distance headway
etc, and hence, it is not a single variable, but a function and can be represented as,
a
t
n
= f
sti
(v
n
,?x
n
,?v
n
) (14.3)
Dr. Tom V. Mathew, IIT Bombay 14.3 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
where f
sti
is the stimulus function that depends on the speed of the current vehicle, relative
position and speed with the front vehicle.
is based on two assumptions; (a) higher the speed of the vehicle, higher will be the spacing
between the vehicles and (b) to avoid collision, driver must maintain a safe distance with the
Let ?x
t
n+1
is the gap available for (n+1)
th
vehicle, and let ?x
safe
is the safe distance, v
t
n+1
and v
t
n
are the velocities, the gap required is given by,
?x
t
n+1
= ?x
safe
+tv
t
n+1
(14.4)
where t is a sensitivity coe?cient. The above equation can be written as
x
n
-x
t
n+1
= ?x
safe
+tv
t
n+1
(14.5)
Di?erentiating the above equation with respect to time, we get
v
t
n
-v
t
n+1
= t.a
t
n+1
a
t
n+1
=
1
t
[v
t
n
-v
t
n+1
]
General Motors has proposed various forms of sensitivity coe?cient term resulting in ?ve gen-
erations of models. The most general model has the form,
a
t
n+1
=

a
l,m
(v
t
n+1
)
m
(x
t
n
-x
t
n+1
)
l


v
t
n
-v
t
n+1

(14.6)
and can take values from -2 to +2, and a is a sensitivity coe?cient. These parameters are to
be calibrated using ?eld data. This equation is the core of tra?c simulation models.
Incomputer, implementation ofthesimulationmodels, threethingsneedtoberemembered:
1. A driver will react to the change in speed of the front vehicle after a time gap called the
reaction time during which the follower perceives the change in speed and react to it.
2. The vehicle position, speed and acceleration will be updated at certain time intervals
depending on the accuracy required. Lower the time interval, higher the accuracy.
3. Vehicle position and speed is governed by Newton’s laws of motion, and the acceleration
is governed by the car following model.
Dr. Tom V. Mathew, IIT Bombay 14.4 February 19, 2014
Transportation Systems Engineering 14. Car Following Models
Therefore, the governing equations of a tra?c ?ow can be developed as below. Let ?T is
the reaction time, and ?t is the updation time, the governing equations can be written as,
v
t
n
= v
t-?t
n
+a
t-?t
n
×?t (14.7)
x
t
n
= x
t-?t
n
+v
t-?t
n
×?t+
1
2
a
t-?t
n
?t
2
(14.8)
a
t
n+1
=

a
l,m
(v
t
n+1
)
m
(x
t-?T
n
-x
t-?T
n+1
)
l

(v
t-?T
n
-v
t-?T
n+1
) (14.9)
The equation 14.7 is a simulation version of the Newton’s simple law of motion v = u+at and
equation 14.8 is the simulation version of the Newton’s another equation s = ut+
1
2
at
2
. The
acceleration of the follower vehicle depends upon the relative velocity of the leader and the
follower vehicle, sensitivity coe?cient and the gap between the vehicles.
Numerical Example
Let a leader vehicle is moving with zero acceleration for two seconds from time zero. Then he
accelerates by 1 m/s
2
for 2 seconds, then decelerates by 1m/s
2
for 2 seconds. The initial speed
is 16m/s and initial location is 28m from datum. A vehicle is following this vehicle with initial
speed 16 m/s, and position zero. Simulate the behavior of the following vehicle using General
Motors’ Car following model (acceleration, speed and position) for 7.5 seconds. Assume the
parameters l=1, m=0 , sensitivity coe?cient (a
l,m
) = 13, reaction time as 1 second and scan
interval as 0.5 seconds.
Solution The ?rst column shows the time in seconds. Column 2, 3, and 4 shows the accel-
eration, velocity and distance of the leader vehicle. Column 5,6, and 7 shows the acceleration,
velocity and distance ofthe follower vehicle. Column 8gives the di?erence invelocities between
the leader and follower vehicle denoted as dv. Column 9 gives the di?erence in displacement
between the leader and follower vehicle denoted as dx. Note that the values are assumed to be
the state at the beginning of that time interval. At time t=0, leader vehicle has a velocity of
16 m/s and located at a distance of 28 m from a datum. The follower vehicle is also having the
same velocity of 16 m/s and located at the datum. Since the velocity is same for both, dv =
0. At time t = 0, the leader vehicle is having acceleration zero, and hence has the same speed.
The location of the leader vehicle can be found out from equation as, x = 28+16×0.5 = 36
m. Similarly, the follower vehicle is not accelerating and is maintaining the same speed. The
location of the follower vehicle is, x = 0+16×0.5 = 8 m. Therefore, dx = 36-8 =28m. These
steps are repeated till t = 1.5 seconds. At time t = 2 seconds, leader vehicle accelerates at the
rate of 1 m/s
2
and continues to accelerate for 2 seconds. After that it decelerates for a period
Dr. Tom V. Mathew, IIT Bombay 14.5 February 19, 2014
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