Jam avoidance with autonomous systems

1.1 MB·

by  Antoine Tordeux

Jam avoidance with autonomous systems
AntoineTordeuxandSylvainLassarre
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Abstract Many car-following models are developed for jam avoidance in high-
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ways.Twomechanismsareusedtoimprovethestability:feedbackcontrolwithau-

] tonomousmodelsandincreasingoftheinteractionwithincooperativeones.Inthis
h paper,wecomparethelinearautonomousandcollectiveoptimalvelocity(OV)mod-
p
els. We observe that the stability is significantly increased by adding predecessors
-
c ininteractionwithcollectivemodels.Yetautonomousandcollectiveapproachesare
o close when the speed difference term is taking into account. Within the linear OV
s
. models tested, the autonomous models including speed difference are sufficient to
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maximisethestability.
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1 Introduction
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1 Recently, many car-following models have been developed for jam avoidance in
v highways.Modelshaveequilibriumhomogeneoussolutionswhereallvehiclespeeds
3 and spacings are constant and equal. ‘Jam avoidance property’ is investigated
1
through analysis of the stability of such solutions. Most of the approaches are ex-
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tended versions of the optimal velocity (OV) model [1]. The basic model is solely
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0 based on the distance spacing with the predecessor (local next-neighbour interac-
. tion). Several studies shown that speed and spacing feedback mechanisms in au-
1
0 tonomous OV models allow to improve the stability of the homogeneous solution
6 and to avoid jam formation [10, 9, 2, 16, 6]. Similar results are obtained with the
1 intelligentdriver(ID)modelforspecificparametervalues[8,7].
:
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X AntoineTordeux
Ju¨lich Supercomputing Centre, Forschungszentrum Ju¨lich, Germany and Computer Simula-
r
tion for Fire Safety and Pedestrian Traffic, Bergische Universita¨t Wuppertal, Germany, e-mail:
a
[email protected]
SylvainLassarre
GRETTIA/COSYS–IFSTTAR,Francee-mail:[email protected]
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2 AntoineTordeuxandSylvainLassarre
Severalvehiclesintheneighbourhoodaretakenintheinteractionforcollective
(or cooperative) systems. Many studies show improvements of the stability if the
number of predecessors in interaction increases [11, 3, 15, 4]. Comparable results
areobtainedwithsymmetricinteraction(interactionwithpredecessorsandfollow-
ers,seeforinstance[13,12]).Oppositelytoautonomousmodelsforwhichthevari-
ables can be directly measured, cooperative systems require that the vehicles are
connectedtocommunicatetheirstates.Thismakesdifficulttheirimplementation.
Inthispaper,autonomouslinearOVmodelsandextendedoneswithspeeddiffer-
encetermarecomparedtotheircollectiveversionsincludingseveralpredecessorsin
interaction.BothextendedandcollectiveOVmodelsdescribesignificantimprove-
mentofthestability.Moreprecisely,weobservethatthenumberofpredecessorsin
interactioninthecollectivemodelsandthespeeddifferencetermintheautonomous
approacheshavesimilarrolesinthedynamics.Thepaperisorganizedasfollow.The
linearjamavoidancemodelsareintroducedinsection2.Theresultsofsimulation
experiment of a jam are presented in section 3, while the Lyapunov exponents of
thedifferentautonomousandcollectivemodelsarecalculatedinsection4.Thesec-
tion5givesconclusionandoutlook.
2 Linearjamavoidancemodels
Theoptimalvelocitymodelis
x¨ (t)= 1(cid:0)V(∆x (t)) x˙ (t)(cid:1), (1)
n T n − n
with x (t) the position of the vehicle n at time t, ∆x(t)=x (t) x (t) the dis-
n n+1 n
−
tancespacingwithx >x thepredecessorposition(seefigure1),andT >0the
n+1 n
relaxation(orreaction)time[1].Ajamavoidanceshouldhavestablehomogeneous
solution. More precisely it should be locally stable with no oscillation (LSNO) to
avoidcollisionandgloballystable(GS),seeforinstance[14,Chapter15].Thecon-
ditionsensureacollision-freeconvergenceofthesystemtothehomogeneoussolu-
tionforanyinitialcondition.WithOVmodeltheOVmodel,thelinearLSNOand
GSconditionsarerespectively:
V < 1 and V < 1 . (2)
(cid:48) 4T (cid:48) 2T
Notethatthefirstconditionimpliesthesecond.Thefullvelocitydifference(FVD)
isanextendedOVmodelincludingspeeddifferenceterm[5]:
x¨ (t)= 1(cid:0)V(∆x (t)) x˙ (t)(cid:1)+ 1∆x˙ (t). (3)
n T1 n − n T2 n
ItincludestworelaxationtimesT >0andT >0.ThemodelisthesameastheOV
1 2
model at the limit T ∞. For the FVD model, the LSNO and GS conditions are
2
→
respectively:
Jamavoidancewithautonomoussystems 3
(cid:16) (cid:17)2
V < 1 1+T1 and V < 1 + 1. (4)
(cid:48) 4T1 T2 (cid:48) 2T1 T2
These conditions are simplyV <1/T if T =T =T (the first inequality implies
(cid:48) 1 2
the second if T <3T ). Clearly, the speed difference has a stabilization effect on
1 2
thedynamics.TheLSNOandGSconditionsalwaysholdatthelimitT 0.
2
→
∆x =x x
n n+1 n
−
n n+1 n+2
x x x
n n+1 n+2 Space
Fig.1 Notationsused.xnisthepositionand∆xnisthespacingofthevehiclen.
Themodels(1)and(3)areautonomousones:theyaresolelybasedondistance
spacing and speed difference with the predecessor. Collective models depend on
severalpredecessorsinfront.Generally,collectiveOVmodelshavetheformx¨ (t)=
n
∑Kk=1Fk(cid:0)∆xn,k(t),x˙n(t),∆x˙n+k(t),whereKisthenumberofpredecessorstakinginto
accountand∆x =x x isthedistancetothevehiclen+k.F representsthe
n,k n+k k k
−
influenceofthevehiclek ontheaccelerationrateoftheconsideredvehicle.Inthe
Multi-anticipative(MA)model[11],thisinfluenceis
F = αk(cid:0)V(cid:0)∆x (t)/k(cid:1) x˙ (t)(cid:1). (5)
k T n,k − n
Thevelocitydifferencemulti-anticipative(VDMA)modelincludesspeeddifference
terms
F =α (cid:104)1(cid:0)V(cid:0)∆x (t)/k(cid:1) x˙ (cid:1)+ 1∆x˙ (t)(cid:105). (6)
k k T1 n,k − n T2 n+k
Herethepositivecoefficients(αk)aresuchthat∑kαk =1.Theyspecifytheinter-
action with the predecessors. In the following we set α =1/K for all k (uniform
k
interaction)inordertomaximisethestability[11,3].NotethattheMAmodelisthe
OVoneandtheVDMAmodelistheFVDoneforK=1,whiletheVDMAmodel
istheMAoneatthelimitT ∞.Thetestedmodelsareresumedistable1.
2
→
Name Acronym Type Parameter
Optimalvelocity OV Autonomous V ,T
(cid:48)
Fullvelocitydifference FVD Autonomous V ,T,T
(cid:48) 1 2
Multi-anticipative MA Collective V ,T,K
(cid:48)
Velocitydifferencemulti-anticipative VDMA Collective V ,T,T,K
(cid:48) 1 2
Table1 Name,acronym,typeandparametersofthetestedmodels.
4 AntoineTordeuxandSylvainLassarre
3 Simulationofajam
Inthissection,themodels(1),(3),(5)and(6)aresimulatedwithperiodicbound-
ary conditions from jam initial conditions by using explicit Euler schemes with
time step 0.001 s. N =20 vehicles are considered with the settings:V =1 s 1 ,
(cid:48) −
T =T =0.25s(fix),andT =2,0.5,0.1s,K=2,4,10veh(tested).Thesettings
1 2
aredefinedsothattheLSNOandGSconditionsoccurforanymodel.Thetrajecto-
riesobtainedwithOVandFVDautonomousmodelsarepresentedinfigure2.The
convergence speed to the homogeneous solutions increases as T 0. The same
2
→
phenomenonoccurswithMAmodelasK ∞,seefigure3.However,thereisno
→
clearimprovementsofthestabilitywiththeVDMAmodelifT issufficientlysmall
2
(seefigure4).
0 0
5 OVM 5 FVDM
40 40 T2=2
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
0 0
5 5
FVDM FVDM
40 T2=0.5 40 T2=0.1
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
Space(m) Space(m)
Fig.2 TrajectorieswiththeOVandFVDmodelsfromjaminitialconfiguration.
Thespeedofconvergenceofthesystemtotheuniformsolutioncanbequantified
byspacingstandarddeviationsequence(Lyapunovfunction):
(cid:113)
σ∆x= N1 ∑Nn=1(∆xn−∆x¯n)2 with ∆x¯n= N1 ∑Nn=1∆xn. (7)
Inthefigure5,thelogarithmsofthespacingstandarddeviationareplottedaccord-
ingtothetimeforthedifferentmodels.Weobservelinearevolution,meaningthat
thedeviationtendtozerowithexponentialspeed.Asexpected,theslopeofthelog-
Jamavoidancewithautonomoussystems 5
0 0
5 OVM 5 MAM
0 0 K=2
4 4
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
0 0
5 5
MAM MAM
0 K=4 0 K=10
4 4
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
Space(m) Space(m)
Fig.3 TrajectorieswiththeOVandMAmodelsfromjaminitialconfiguration.
0 0
5 5
FVDM VDMAM
40 T2=0.1 40 K=2
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
0 0
5 5
VDMAM VDMAM
0 K=4 0 K=10
4 4
)
(s 30 30
e
m
0 0
Ti 2 2
0 0
1 1
0 0
-200 -100 0 100 200 -200 -100 0 100 200
Space(m) Space(m)
Fig.4 TrajectorieswiththeFVDandVDMAmodelsfromjaminitialconfiguration.
6 AntoineTordeuxandSylvainLassarre
arithm(i.e.theconvergencespeed)increasesasT decreaseswiththeautonomous
2
models (see figure 5, top left panel), while the speed depends on the number of
predecessorsininteractionK withthecollectiveMAmodel(seefigure5,topright
panel).Asweobservedpreviously,thespeeddoesnotchangesignificantlyifK in-
creases with VDMA model (see figure 5, bottom left panel). In fact the speeds of
convergenceofFVD,MA,andVDMAmodelsareclose(seefigure5,bottomright
panel);theyarestronglyfasterthantheconvergencespeedofordinaryOVmodel.
Suchresultssuggestthatspeeddifferencetermwiththeautonomousmodelsandthe
numberofpredecessorsininteractionwiththecollectiveoneshavesimilarrolesin
thedynamics.Theconvergencespeedtothehomogeneoussolutionismaximisedas
T 0orasK ∞.
2
→ →
4 4
2 2
)
x
∆ 0 0
σ
(
g 2 2
o - OVM - OVM
l 4 FVDM T2=2 4 MAM K=2
- FVDM T2=0.5 - MAM K=4
6 FVDM T2=0.1 6 MAM K=10
- -
0 10 20 30 40 50 0 10 20 30 40 50
4 4
2 2
)
x
∆ 0 0
σ
(
g 2 2
lo 4- FVVDDMMAM TK2==20.1 4- OFVVDMM T2=0.1
- VDMAM K=4 - MAM K=10
6 VDMAM K=10 6 VDMAM K=10
- -
0 10 20 30 40 50 0 10 20 30 40 50
Time(s) Time(s)
Fig.5 SequencesofthespacingstandarddeviationlogarithmwithOV,FVD,MAandVDMA
models.
4 Lyapunovexponents
Thesolutionofthelinearsystemsarelinearcombination(LC)ofexponentialterms
(cid:0) (cid:1)
x (t)=LC exp(λt),texp(λt) (8)
n l l
with(λ)theLyapunovexponentsofthesystem(i.e.theeigenvaluesofthesystem
l
Jacobian matrix). In our stable case, all the exponents have strictly negative real
parts,exceptedoneequaltozero.Moreover,wecanexpectthantheconvergenceto
thehomogeneoussolutiongetsfasterastheexponentsgotheleftoftheimaginary
axis.Withtheoptimalvelocityweinvestigate,theexponentsare:
Jamavoidancewithautonomoussystems 7
λl = 12∑Kk=0βkιlk±12(cid:104)(cid:0)∑Kk=0βkιlk(cid:1)2−4∑Kk=1αk(1−ιlk)(cid:105)1/2 (9)
with ιl = exp(2iπl/N), l = 1,...N, N being the vehicles number, αk = k1T1VK(cid:48),
β = 1 1 and β = 1 for all k =1,...N. The Lyapunov exponents are
0 −T1 − T2 k −KT2
plottedinfigure6forthedifferentmodels.Weobservethattheyconvergetoadou-
ble mode pattern as T 0 with the autonomous FVD model, and K ∞ with
2
→ →
the MA collective model. They remain double mode with the collective VDMA
modelasK increases.Suchresultsconfirmqualitativelytheonesobservedbysim-
ulation.Thespeeddifferencebehaveinthedynamicsasthenumberofpredecessors
ininteraction.Alsoincreasingtheinteractionseemsnotnecessarytomaximisethe
stability.
2 OVM 1.5 FVDM T2=2 FVDM T2=0.5 10 FVDM T2=0.1
(λ)l=1-10 -0.50.5 0.01.0 05
-2 -1.5 -1.5 -10
-4 -3 -2 -1 0 -4 -3 -2 -1 0 -7 -5 -3 -1 -20 -10 -5 0
OVM MAM K=2 MAM K=4 MAM K=10
2 1.0 0.4 0.2
)1
(λl0 0.0 0.0 0.0
=
-2-1 -1.0 -0.4 -0.2
-4 -3 -2 -1 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0
FVDM T2=0.1 VDMAM K=2 VDMAM K=4 VDMAM K=10
10
)5 5 5 5
(λl0 0 0 0
=
-10 -5 -5 -5
-20 -10 -5 0 -20 -15 -10 -5 0 -15 -10 -5 0 -15 -10 -5 0
(λ) (λ) (λ) (λ)
< l < l < l < l
Fig.6 LyapunovexponentsforOV,FVD(toppanels),MA(middlepanels)andVDMA(bottom
panels)modelswithN=100.
5 Conclusion
TheconvergencestothehomogeneoussolutionoflinearjamavoidanceOVmodels
arecompared.WeobservedthatextendingtheOVmodelwithspeeddifferenceterm
significantlyimprovesthestability.Inasimilarway,theaddingofneighboursinin-
teractiongivesstabilityenhancements.However,increasingtheinteractiondoesnot
improvethestabilitywiththeextendedOVmodel.Thissuggeststhatboththenum-
berofpredecessorsininteractioninthecollectivemodelsandthespeeddifference
termintheautonomousapproachesallowtomaximisetheconvergencespeedtoho-
8 AntoineTordeuxandSylvainLassarre
mogeneoussolutions.Also,theconnectionbetweenthevehicles,hardtoimplement,
maynotbenecessarytooptimisethestabilityandefficientlyavoidjamformation.
Further investigations remain to be carried out to validate this hypothesis. For in-
stance,theinfluenceofthegeometry,initialconditionsorvehicledensityhavetobe
investigated. The shape of the Lyapunov exponents and their impact on the stabil-
ityarenotexplicit.Furthermore,non-linearmodelsmaypresentbetterconvergence
speed than the basic linear models we analysed. These subjects will the topic of
futureworks.
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