Ring Car Following Ring Car Following ModelsModels

byby

Sharon Gibson and Mark McCartneySharon Gibson and Mark McCartney

School of Computing & Mathematics, University of Ulster at JordanstownSchool of Computing & Mathematics, University of Ulster at Jordanstown

Mathematical models which describe how individual drivers follow one another in a stream of traffic.

Many different approaches, including:

Fuzzy logic

Cellular Automata (CA)

Differential equations

Difference equations

Car Following ModelsCar Following Models

Car Following ModelsCar Following Models

Classical stimulus response model (GHR model):

where;

xi(t) is the position of the ith vehicle at time t;

T is the reaction or thinking time of the following driver;

and the sensitivity coefficient is a measure of how strongly the following driver responds to the approach/recession of the vehicle in front.

2

12

1

m

i

i i il

i i

dx t T

d x t dx t T dx t Tdt

dt dt dtx t T x t T

Car Following ModelsCar Following Models

A simpler linear form of the GHR model (SGHR) can be expressed in terms of vehicle velocities as:

where;

ui(t) is the velocity of the ith vehicle at time t.

1i

i i

du tu t T u t T

dt

Ring ModelsRing Models A model in which the last vehicle in the stream is

itself being followed by the ‘lead’ (first) vehicle:

Motivation: ‘Real’ simulations re-use data Idealised as a representation of outer rings Mathematically interesting

01 0

1 .

n

ii i

du tu t T u t T

dt

du tu t T u t T

dt

A Simple Ring ModelA Simple Ring Model If the driver of each vehicle has zero reaction time

model simplifies to:

Implication: The steady state velocity of all vehicles can be found

immediately once we have been given initial velocities.

01 0

1 .

n

ii i

du tu t u t

dt

du tu t u t

dt

A Simple Ring ModelA Simple Ring Model

Need to give the lead car a ‘preferred’ velocity profile, w0(t):

where;the sensitivity coefficient is a measure of how stronglythe lead driver responds to his/her ‘preferred velocity’.

01 0 0 0

1

n

ii i

du tu t u t w t u t

dt

du tu t u t

dt

A Simple Ring ModelA Simple Ring Model For n = 2, the transient velocity of the ith vehicle is of the

form:

where;

and

The post transient velocity of the ith vehicle is dependent on the form of the preferred velocity.

1 21 2

t tc e c e

2 2

1

4

2 2

2 2

2

4

2 2

A Simple Ring ModelA Simple Ring Model Three forms of preferred velocity considered

Constant velocity,

Linearly increasing velocity,

Sinusoidal velocity,

NB. The post transient results hold for a general n vehicles in the system.

0w t U

iu t U

0w t At

i

n iu t At A

0 1 sinw t U t

sin cosi i i iu t x y t z t

ix U

1 2 1 222

0 02 2 2 1 20 0

1 1

iii i j i j

j j

i i j i jj j

i iy z yC C

12 2 12 2

0 02 2 2 2 10 0

1 1

i ii i j i j

j j

i i j i jj j

i iz z yC C

0 22 2 2

1

2 1

U by

a a b

0 22 2 22 1

U az

a a b

1

11 1 2 12

2 2 1 2 10

11

nn n j

j

n jj

na C

1

1 1 22

2 2 1 20

11

nn n j

j

n jj

nb C

where

and

where

Ring Model with Time DelayRing Model with Time Delay

♦ This new ring model when the drivers reaction times are included can be expressed as:

♦ We solve this system of Time Delay Differential Equations (TDDE) numerically using a RK4 routine

01 0 0 0

1

n

ii i

du tu t T u t T w t T u t T

dt

du tu t T u t T

dt

Approximating Time DelayApproximating Time Delay

An approximate solution to the Time Delay Differential Equation (TDDE) form of the Ring Model can be found using a Taylor’s series expansion in time delay, T:

0 11 0

0 00

1 0 10 1

du t du tu t T u t

dt dtdu t dw t

T w t Tdt dt

du t du t du tu t T u t T

dt dt dt

Approximating Time DelayApproximating Time Delay

For n = 2, the transient velocity of the ith vehicle is of the form:

where;

and

If system is to reach steady state then:

1 21 2

t tc e c e

2 2

1

42 2

1 2 1

T

T T T

2 2

2

42 2

1 2 1

T

T T T

2 22 4

2T

Comparison of Zero Time Delay, Taylor’s Series Approximation Comparison of Zero Time Delay, Taylor’s Series Approximation

& RK4 Numerical Methods& RK4 Numerical Methods

0 20 40 60 80 100-5

0

5

10

15

20

25

30

t = 0.1s, = 0.3s-1, = 0.8s-1

u0(t) (T = 0s)

u1(t) (T = 0s)

u0(t) (RK4, T = 0.7s)

u1(t) (RK4, T = 0.7s)

u0(t) (TSE, T = 0.7s)

u1(t) (TSE, T = 0.7s)

velo

city

ui(t

) (m

s-1)

time (seconds)

Stability of the Ring ModelStability of the Ring Model

System is locally stable if each car in the system eventually reaches a steady state velocity. Non-oscillatory motion Damped oscillatory motion

Stability criteria is dependent upon the number of vehicles in the system. General criteria for n = 2:

Criteria for n > 2 currently under investigation – One of the boundaries obtained for n = 3:

Hypothesis: The stable region for each value of n > 2 is bounded by exactly 2 boundaries.

22

2

T

T T

2 2 2

2 2 2

12 6

2 4 4

T T

T T T

Stability of the Ring Model (n=2)Stability of the Ring Model (n=2)Stable Region when T = 0.5s, dt = 0.001s

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lambda

Alp

ha

Stability of the Ring Model (n=3)Stability of the Ring Model (n=3)Stable Region when T = 0.5s and dt = 0.001s

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lambda

Alp

ha

Stability of the Ring Model (n=5)Stability of the Ring Model (n=5)Stable Region when T = 0.5s and dt = 0.01s

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lambda

Alp

ha

Future WorkFuture Work

Investigate discrete time models, as:

Easier to implement (Computationally faster)

Arguably more realistic

More likely to give rise to chaotic behaviour

Stability of the Ring Model (n=2)Stability of the Ring Model (n=2)(Euler Method)(Euler Method)Stable Region when T = 0.5s and dt = 0.001s

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lambda

Alp

ha

Questions?Questions?