Ring car following models

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1 Δεκ 2013 (πριν από 3 χρόνια και 6 μήνες)

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Ring Car Following
Models

by

Sharon Gibson and Mark McCartney


School 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 Models

Car Following Models


Classical stimulus response model (GHR model):






where;



x
i
(t)

is the position of the
i
th

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
1
2
1
m
i
i i i
l
i i
dx t T
d x t dx t T dx t T
dt
dt dt dt
x t T x t T




 
 
 
 
 
 
 
  
 
 
 
Car Following Models


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





where;


u
i
(t)

is the velocity of the
i
th

vehicle at time
t.







1
i
i i
du t
u t T u t T
dt


   
 
 
Ring 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













0
1 0
1
.
n
i
i i
du t
u t T u t T
dt
du t
u t T u t T
dt




   
 
 
   
 
 
A 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.













0
1 0
1
.
n
i
i i
du t
u t u t
dt
du t
u t u t
dt




 
 
 
 
 
 
A Simple Ring Model


Need to give the lead car a ‘preferred’ velocity profile,
w
0
(t)
:








where;


the sensitivity coefficient


is a measure of how strongly


the lead driver responds to his/her ‘preferred velocity’.

















0
1 0 0 0
1
n
i
i i
du t
u t u t w t u t
dt
du t
u t u t
dt
 



   
   
   
 
 
 
A Simple Ring Model


For n = 2, the transient velocity of the
i
th

vehicle is of the
form:



where;



and






The post transient velocity of the
i
th

vehicle is dependent on
the form of the preferred velocity.

1 2
1 2
t t
c e c e
 

2 2
1
4
2 2
  
 

   
2 2
2
4
2 2
  
 

   
A 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.





0
w t U



i
u t U



0
w t At



i
n i
u t At A
 
 
  
 
 






0
1 sin
w t U t
 
 






sin cos
i i i i
u t x y t z t
 
  
i
x U





1
2 1 2
2
2
0 0
2 2
2 1 2
0 0
1 1
i
i
i i j i j
j j
i
i j i j
j j
i i
y z y
C C
  
   
 
 

 
 
  
 
 
  
 
 
   
 
     
   
   
 
     
   

     
   
 
 
 




1
2 2 1
2
2
0 0
2 2
2 2 1
0 0
1 1
i
i
i i j i j
j j
i
i j i j
j j
i i
z z y
C C
  
   
 
 

 
 
  
 
 
  
 
 
   
 
     
   
   
 
     
   

     
   
 
 
 








0
2
2 2 2
1
2 1
U b
y
a a b
  
    
 

    






0
2
2 2 2
2 1
U a
z
a a b
  
    

 
    






1
1
1 1 2 1
2
2 2
1 2 1
0
1
1
n
n n j
j
n j
j
n
a
C
 
  

 

 
   
 
  

 
 
 
   

 
 
 
   
 

   
 
 
 







1
1 1 2
2
2 2
1 2
0
1
1
n
n n j
j
n j
j
n
b
C
 
  

 
 
  
 
 

 
 
 
   

 
 
 
   
 

   
 
 
 

where

and

where

Ring 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

















0
1 0 0 0
1
n
i
i i
du t
u t T u t T w t T u t T
dt
du t
u t T u t T
dt
 



       
   
   
   
 
 
Approximating 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 1
1 0
0 0
0
1 0 1
0 1

du t du t
u t T u t
dt dt
du t dw t
T w t T
dt dt
du t du t du t
u t T u t T
dt dt dt
   
   
   
   
   
   
Approximating Time Delay


For n = 2, the transient velocity of the
i
th

vehicle is of the
form:


where;



and






If system is to reach steady state then:

1 2
1 2
t t
c e c e
 



2 2
1
4
2 2
1 2 1
T
T T T
  
 

  

   

  


2 2
2
4
2 2
1 2 1
T
T T T
  
 

  

   

  
2 2
2 4
2
T
   

  

Comparison of Zero Time Delay, Taylor’s Series
Approximation & 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
u
0
(t) (T = 0s)
u
1
(t) (T = 0s)
u
0
(t) (RK4, T = 0.7s)
u
1
(t) (RK4, T = 0.7s)
u
0
(t) (TSE, T = 0.7s)
u
1
(t) (TSE, T = 0.7s)
velocity u
i
(t) (ms
-1
)
time (seconds)
Stability 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.


2
2
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)

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
Alpha
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
Alpha
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
Alpha
Future 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)

(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
Alpha
Questions?