Overview of the traffic system. ⢠A master model and its controlled expansion. ⢠Two-phase vs. three-phase traffic models. ⢠Occam's razor. ⢠Model extensions ...
Effective Modelling of Traffic Dynamics: Classification and Unification Bo Yang and Christopher Monterola Complex Systems Group Institute of High Performance Computing A*STAR, Singapore
EF Legara, A*STAR 2012
Outline: • •
•
•
Overview of the traffic system A master model and its controlled expansion • Two-phase vs. three-phase traffic models • Occam's razor • Model extensions and tuning Microscopic empirical data collection • Video-taping of the traffic flow • Image processing and machine learning • Data processing and averaging Empirical verification of traffic theories and models
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An overview of the traffic system theoretical modeling A One-Dimensional Driven System
n-1
n
n+1
{ {
•
hn
hn
1
vn = vn+1
vn
nearest neighbour, anisotropic non-linear interactions in a dissipative media EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
An overview of the traffic system •
theoretical modeling
• • •
no symmetry non-identical components stochasticity and time dependence EF Legara, A*STAR 2012
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An overview of the traffic system •
empirical observations
Kerner et.al. 2002
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An overview of the traffic system •
empirical observations
Kerner et.al. 1998, 2002
B.S. Kerner, Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-phase Traffic Theory
D. Helbing et.al.,An Analytical Theory of Traffic Flow, A selection of articles by Dirk Helbing reprinted from The European Physical Journal B
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An overview of the traffic system •
empirical observations
Yang Bo et.al arXiv. 1504.01256 EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
An overview of the traffic system •
microscopic models l
an (t + T ) = cvn (t + T ) hn m vn
D.C. Gazis, R. Herman, R.W. Rothery, Oper. Res. 9 (1961) 545–567 L.C. Edie, R. Herman, R.W. Rothery, Oper. Res. 9 (1961) 66 - 76 .
G.F. Newell, Transp. Res. B 36 (2002) 195–205.
B.S. Kerner, Physica A 392 (2013) 5261–5282
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An overview of the traffic system •
microscopic models
Helbing et.al 2000
an = a0 (V (hn )
g ( vn ) =
8 < :
vn + g ( vn ))
⇥ ( vn ) vn vn 1 ⇥ ( vn ) vn + 2 ⇥ (
Bando et.al 1997 Jiang et.al. 2001 Gong et.al. 2008
vn ) vn
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EF Legara, A*STAR 2012
An overview of the traffic system •
microscopic models
Kerner et.al 2001 EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
An overview of the traffic system •
microscopic models • •
•
How do we properly characterise the differences between two traffic models? Is there a standard way of extending an existing traffic model or construction of a new traffic model? Is there a standard way in selecting the best traffic model based on the experimental data?
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A master model •
a renormalisation-like approach
an = Fn,{si } ({ti })
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
an = Fn,{si } ({ti }) unimportant factors
important factors
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
an = Fn,{si } ({ti }) unimportant factors
important factors
1 X a ¯n = Fn,{si } ({ti }) = fn ({ti }) N0 {si }
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
an = Fn,{si } ({ti }) unimportant factors
important factors
1 X a ¯n = Fn,{si } ({ti }) = fn ({ti }) N0 {si }
1 X a ¯n = fk ({ti }) = f0 ({ti }) N k
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach 1 X a ¯n = fk ({ti }) = f0 ({ti }) N k
{ti } = {hn ,
v n , vn }
deterministic, time an = f0 (hn , vn , vn ) independent identical drivers EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
stochasticity, inhomogeneity, time dependence, vehicle/ driver diversity
ensemble average identical drivers, time independent, homogeneous traffic lanes
simplest possible approximation of f¯ EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
an = f0 (hn ,
v n , vn )
the “ground state” of the traffic dynamics
f0 (hn , 0, vn ) = 0
f0 (hn , 0, Vop ) = 0
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A master model •
a controlled expansion
a ˜n =
X
p,q
p,q
⇣
˜n p,q h
⌘
⇣
⌘⇣ ˜ n v˜n h
⇣
˜n Vop h
1 @ p+q f˜ = p!q! @ p v˜n @ q v˜n
⌘⌘p
v˜nq
˜n) v ˜n =V˜op (h v ˜n =0
Yang Bo et.al PRE 92, 042802 (2015)
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A new perspective in modelling a universal mathematical structure b)
0
v
aΔv=0,h
aΔv=0,h
a)
v
0
c)
0
v
d)
aΔv=0,h
aΔv=0,h
•
v
0
Yang Bo et.al PRE 92, 042802 (2015)
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EF Legara, A*STAR 2012
A new perspective in modelling •
a universal mathematical structure b)
0
v
aΔv=0,h
aΔv=0,h
a)
v
0
two-phase traffic models
0
v
d)
aΔv=0,h
aΔv=0,h
c)
v
0
Yang Bo et.al PRE 92, 042802 (2015)
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A new perspective in modelling b)
0
v
aΔv=0,h
a)
aΔv=0,h
v
0
c)
0
v
d)
aΔv=0,h
•
a universal mathematical structure two-phase vs. three-phase
aΔv=0,h
•
v
0
⇣
a ˜n ⇠ v˜n
⇣
˜n Vop h
⌘⌘p
Yang Bo et.al PRE 92, 042802 (2015) EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A new perspective in modelling • •
a universal mathematical structure two-phase vs. three-phase •
• •
All microscopic traffic models are defined by the optimal velocity (OV) and a set of expansion coefficients (EC). the two-phase and three-phase traffic models can be unified by a “common language”. The simplification of OV and ECs can be empirically verified.
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A new perspective in modelling •
special cases - IDM ✓
Model Parameters -0.8 -0.6 -0.4 -0.2
0.0
an = a 1
-1.2
-1.0
λ10 λ20 λ30 λ40 λ01 λ02 λ11 λ12 λ21 λ22 0
20
40 60 Headway h (m)
80
100
an =
vn v0
p=4,q=2 X
◆
✓
h⇤ (vn , vn ) hn
p,q (vn
Vop (hn ))
p
◆2 !
vnq
p=1,q=0
an =
10
(hn ) (vn
Vop ) +
01
(hn )
Yang Bo et.al PRE 92, 042802 (2015) EF Legara, A*STAR 2012
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vn
EF Legara, A*STAR 2012
A new perspective in modelling •
special cases - Occam’s razor
The best model should be as simple as possible (but not simpler) an =
X
p,q (hn ) (vn
Vop )
p
q vn
p,q
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EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor Model tuning an = (Vop (hn ) g ( vn ) =
1
vn ) + g ( vn ) vn +
2|
vn |
hmax , hmin , n0 emergent quantities from non-linear interactions Yang Bo et.al arXiv:1504.01256 Yang Bo et.al arXiv:1407.3177 EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor Model tuning
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A new perspective in modelling
∆h (m) 10 15 20 25 30 35
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●●● ● Single perturbation ●● ● Random perturbation ● ● ● ●
5
•
special cases - Occam’s razor numerical simulation
0
•
27.5
28.5 29.5 30.5 Average headway (m)
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EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor numerical simulation
Boomerang effect
Pinch effect
merging of numerous narrow jams
Yang Bo et.al arXiv. 1504.01256 EF Legara, A*STAR 2012
A*STAR 2015
EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor numerical simulation
interplay of congested flow upstream of a bottleneck and the wide moving jams Yang Bo et.al arXiv. 1504.01256 EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor numerical simulation frequency of jam evolutions and the width of the “synchornized flow”
robustness of the wide moving jams
flow fluctuation of the dense traffic Yang Bo et.al arXiv. 1504.01256 EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A new perspective in modelling • •
special cases - Occam’s razor numerical simulation
arbitrary profile of the congested flow EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
A new perspective in modelling •
systematic extension of the model
• • •
More realistic optimal velocity function Making the coefficients of expansion density dependent Include higher order terms
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Empirical verification of microscopic traffic models from the detailed acceleration patterns Bo Yang and Christopher Monterola Complex Systems Group Institute of High Performance Computing Ji Wei Yoon UC Berkeley
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Empirical verifications •
• • •
•
understanding human driving behaviours
Video-taping of the traffic flow with high frame-rate camera Machine learning to identify vehicles in the video Virtual sensors to measure the velocity, acceleration as well as headway and approach velocity Data aggregation and statistical averaging
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Empirical verifications •
image processing and machine learning
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EF Legara, A*STAR 2012
Empirical verifications •
edge detections and quantitative measurements
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EF Legara, A*STAR 2012
Empirical verifications •
systematic errors
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EF Legara, A*STAR 2012
A master model •
a renormalisation-like approach
an = Fn,{si } ({ti }) unimportant factors
important factors
1 X a ¯n = Fn,{si } ({ti }) = fn ({ti }) N0 {si }
1 X a ¯n = fk ({ti }) = f0 ({ti }) N k
EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
Empirical verifications tentative results
0
2
●
● 4
6
8
10
● ●
−0.2
● ●
●
●
0.2
●
0.0
●
●
●
0
2
● 4
●
6
8
10
● ● ●
−0.2
0.0
●
Acceleration (m/s/s)
0.2
0.4
Headway =5 m, Approach velocity = 0 m/s
0.4
Headway =4 m, Approach velocity = 0 m/s
● ●
−0.4
●
−0.4
Acceleration (m/s/s)
•
Velocity (m/s)
Velocity (m/s)
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EF Legara, A*STAR 2012
Empirical verifications tentative results
0
2
● 4
●
● 6
8
10
12
●
−0.2
●
● ●
0.2
●
0.0
●
●
0
2
●
●
4
● 6
●
8 ●
10
12
●
●
−0.2
0.0
●
Acceleration (m/s/s)
0.2
0.4
Headway =7 m, Approach velocity = 0 m/s
0.4
Headway =6 m, Approach velocity = 0 m/s
●
●
−0.4
●
−0.4
Acceleration (m/s/s)
•
Velocity (m/s)
●
Velocity (m/s)
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EF Legara, A*STAR 2012
Empirical verifications tentative results
0.4
Headway =10 m, Approach velocity = 0 m/s
0.4
Headway =8 m, Approach velocity = 0 m/s
●
● ●
0
2
4
6
●
●
8
●
10
12
14
●
−0.2
●
0.2
● ●
0
2
4
6
●
●
8
Velocity (m/s)
●
● 10
12
14
● ●
●
−0.4
−0.4
●
●
0.0
0.0
●
−0.2
●
Acceleration (m/s/s)
0.2
●
Acceleration (m/s/s)
•
Velocity (m/s)
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EF Legara, A*STAR 2012
Empirical verifications tentative results
0.4
Headway =11 m, Approach velocity = 0 m/s
0.2
●
0.0
● ●
●
●
●
● ●
2
4
6
8
10
12
14
●
−0.2
0
●
−0.4
Acceleration (m/s/s)
•
●
Velocity (m/s)
EF Legara, A*STAR 2012
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EF Legara, A*STAR 2012
Empirical verifications •
tentative results - summary
•
•
•
The renormalisation-like procedure does yield welldefined master microscopic model from the microscopic empirical data The empirical data shows evidence of the multitude of steady-states (or at least very long lasting states) that can be characterised as the states in the “synchronized phase” There are strong dependence of the expansion of coefficients on the density.
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Thank you very much!
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A new perspective in modelling
0.020
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0.000
0.005
0.010
0.015
Time (min)
0.020
0.5 −0.5 −1.5
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0.000
2
0.015
0
0.010
Time (min)
−2
0.005
−4
0
5
10
20
30
0.000
Acceleration (m/s/s)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Acceleration (m/s/s)
20 10 0
5
Velocity (m/s)
30
special cases - IDM
Velocity (m/s)
•
0.005
0.010
0.015
Time (min)
0.020
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0.000
0.005
0.010
0.015
Time (min)
0.020
Yang Bo et.al PRE 92, 042802 (2015) EF Legara, A*STAR 2012
A*STAR 2015
2.0 1.5
Coexistence curve Neutral stability line λ(1) λ(2)
0.5 -4
-2
0 h0
2
4
0.0
0.5 0.0
b)
λ 1.0
2.0
Coexistence curve Neutral stability line λ(1) λ(2) λ(3)
λ 1.0
a)
1.5
EF Legara, A*STAR 2012
-4
-2
EF Legara, A*STAR 2012
0 h0
2
4
A*STAR 2015
2.0 0 50
150 250 car index n
0 50
150 250 car index n
0 50
150 250 car index n
-2
-2
hn 0 1 2
0.0
hn 1.0
1.0 -1.5
hn 0.0
150 250 car index n
hn 0 1 2
0 50
