Generalized Force Model of Traffic Dynamics

6 Pages·0.32 MB·English

by  Dirk Helbing

Generalized Force Model of Traffic Dynamics
Dirk Helbing and Benno Tilch
II. Institute of Theoretical Physics, University of Stuttgart, 70550 Stuttgart, Germany
http://www.theo2.physik.uni-stuttgart.de/helbing.html
9
9
9 ingtothe respectivemodelunder consideration,starting
1 Floating car data of car-following behavior in cities were with the the same initial velocity and distance like the
compared to existing microsimulation models, after their pa-
n floating car. The average relative quadratic deviation D
rametershadbeencalibratedtotheexperimentaldata. With
a between the simulated and actually measured distance
J theseparametervalues,additionalsimulationshavebeencar- (cf. Eq. (10)) was used as a measure for the goodness of
ried out, e.g. of a moving car which approaches a stopped
8 fit of the respective model [8].
car. Itturnedout that,in orderto manage suchkindsof sit-
1 In order to get improved results, we have developed
uationswithoutproducingaccidents,improvedtrafficmodels
a generalized force model, in which each term and each
] are needed. Good results have been obtained with the pro-
h parameter has a clear meaning. Moreover,by parameter
posed generalized force model.
c calibration it turns out that all model parameters have
e 02.70.Ns,34.10.+x,05.70.Ln,89.40.+k the right order of magnitude. Therefore, it can be eas-
m
ily said, how the parameter values will look like, if the
- speedlimit, theaccelerationcapability,the averagevehi-
t
a cle length, the visibility, or the reaction time is modified
st I. INTRODUCTION (e.g. due to technical measures). In addition, this model
. reaches a better goodness of fit at a reduced number of
t
a During the last five years, gas-kinetic [1,2], fluid- model parameters than the previous models. Finally,
m
dynamic [1–3], and other models have been developed, the generalized force model manages to cope success-
- aiming at an understanding of stop-and-go traffic. The fully with particular situations like vehicles approaching
d
topic is related to the fields of non-linear dynamics [3], standing cars, in which other models produce accidents.
n
o phase transitions [4], and stochastic processes[5]. In ad-
c dition, microscopic traffic models were proposed for the
[ description of interacting driver-vehicle units. They can II. DISCUSSION OF PREVIOUS MODELS

be classifiedinto cellularautomata models [5], whichare
2
v discreteinspaceandtime,andcontinuousmodels[1,6,7]. The first microscopic traffic models were developed in
3 The latter are requiredfor detail studies of car-following the1960ies. Manyofthemarespecialcasesofthefollow-
4 behavior and traffic instabilities, which are necessary for the-leadermodelproposedby Gazis,Herman,andRoth-
2 an investigation of the consequences of technical opti- ery [6]. This assumes that the dynamics of a vehicle α
6
mization measures (e.g. of autopilots for an automatic withvelocityv (t)atplacex (t)isgivenbytheequation
0 α α
control of vehicle acceleration and braking). of motion
8
Therefore, a research group of the Bosch GmbH has
9
dx (t)
/ recently recorded follow-the-leader data by means of a α =v (t) (1)
at floating car α which measured the vehicle speed vα, the dt α
m netto distance s to the car in front, the accelerationa
α α and the acceleration equation
- ofit,andtherelativevelocity∆vα. Byacorrelationanal-
d ysisitwasdemonstratedthat, amongallpossiblecombi- dv (t+T)
α
n nations of subsets of these four quantities, sα, ∆vα, and dt =κ(t+T)[vα−1(t)−vα(t)]. (2)
o
v are the most significant variables for the description
c α Accordingtothis,adriveradaptstothevelocityv (t)
: ofvehicle dynamics[8]. InSec. III, we willfind plausible α−1
v of the car in front, but this is delayed by the adaptation
reasons for this.
i time T ≈ 1.3s. The deceleration is proportional to the
X The follow-the-leader data, if plotted in the sα-∆vα-
relative velocity
plane, show the characteristic oscillation of vehicle mo-
r
a tionaroundstateswith relativevelocityzero(cf. Fig.1),
∆v =v −v , (3)
which was already reported by Hoefs [9]. Except for the α α α−1
previouslymentionedsignificanceanalysis,thedatawere wherethe proportionalityfactorκ reflectsthe sensitivity
alsousedforcalibratingexistingmicrosimulationmodels. to the stimulus ∆v . The sensitivity was assumed to
α
With the resulting optimal sets of parameter values, the dependonthevehiclevelocityandonthebruttodistance
models were simulated for the observed situation. That
is, the first vehicle was moved according to its measured S =x −x (4)
α α−1 α
velocity, and the following vehicle was simulated accord-
in the following way:
1
[vα(t+T)]m thevehiclevelocity(cf.Fig.2a). Theunrealisticallyhigh
κ(t+T)=κ . (5)
0 [S (t)]l accelerations also become obvious in Figure 3, since em-
α
piricalaccelerationsarelimitedto4m/s2 (cf.Fig.2c). A
This choiceallowedto fitallequilibriumvelocity-density
similarproblemoccurs with the decelerationbehavior,if
relations of the form
a standing car(e.g.atthe endof atraffic jam orin front
l−1 1/(1−m) of a red traffic light) is approachedfroma largedistance
ρ by an initially freely moving car. It turns out that the
V (ρ)=v 1− (6)
e 0
" (cid:18)ρmax(cid:19) # movingvehicle reactstoolate to the vehicle atrest. The
values of deceleration are unrealistic large, but still not
byappropriatespecificationoftheexponentslandm(v sufficient to avoid an accident (cf. Fig. 4).
0
= maximum velocity, ρ = spatial vehicle density, ρ These problems are solved by the T3 model
max
= maximum vehicle density). The best fit was reached
dv 1+b v +b s +b v s +b v +b v v
for fractional exponents m ≈ 0.8 and l ≈ 2.8, so that α 1 α 2 α 3 α α 4 α−1 5 α α−1
=
this model has no obvious interpretation. Apart from dt c0+c1vα+c2sα+c3vαsα+c4vα−1+c5vαvα−1
that,themodeldoesnotallowtodistinguishdriverswith
(11)
different preferred velocity, and it cannot describe the
acceleration of a single vehicle correctly. proposed by Bosch [8]. b and c are model parameters.
k k
Onlyafewyearsago,Bando,Sugiyamaetal. proposed The regression model (11), which is based on a ratio-
a very charming microscopic traffic model. Despite its nal function, describes all aspects of vehicle dynamics in
simplicity and its few parameters, their optimal velocity cities realistically (cf. Figs. 2–4),but at the costof addi-
model (OVM) describes many properties of real traffic tional parameters. Whereas the optimal velocity model
flows [7] and is easily interpretable. It is based on the needs only 5 model parameters, the T3 model contains
acceleration equation 11 parameters. If the model equations are scaled to di-
mensionless equations (by scaling space or velocity and
dv (t)
α =κ[V(s )−v (t)], (7) time by characteristic model quantities), the number of
α α
dt
parameters is reduced by 2.
so that the vehicles adapt to a distance-dependent opti- Inthefollowingsection,wewillproposeanalternative
mal velocity model which reaches about the same goodness of fit as
the T3 model, but with a considerably smaller number
V(s )=V +V tanh(C s −C ) (8) of parameters.
α 1 2 1 α 2
with a certain relaxation time τ =1/κ. Here,
III. THE GENERALIZED FORCE MODEL
s =x −x −l =S −l (9)
α α−1 α α−1 α α−1
Motivatedbythe successofso-calledsocialforcemod-
denotes the netto distance,where l means the lengthof
α
els in the description of behavioral changes [11,1], es-
vehicleα. Likethe follow-the-leadermodels,the optimal
pecially of pedestrian dynamics [12,1], we developed a
velocity model is able to describe the formation of stop-
related model for the dynamics of interacting vehicles.
and-gowavesandemergenttrafficjams,butitovercomes
In setting up anequationof motion by specifying the ef-
the afore mentioned problems.
We carried out a calibration of the optimal velocity fectiveaccelerationanddecelerationforces,theapproach
is analogous to the molecular-dynamic method which is
model with respect to the empirical follow-the-leader
data, which we obtained from Bosch. The optimization used for the simulation of many-particle systems [13],
e.g. of driven granular media [14].
procedure based on the evolutionary Boltzmann strat-
egy [10], and the optimization criterion was the average Besides of various methodological similarities in the
theoretical treatment of traffic and granular flows, there
relative quadratic deviation
arealsophenomenologicalanalogiesliketheformationof
1 N s (t)−sm(t) 2 densitywaves[15]. Inbothcases,theinteractionsaredis-
D = α α (10) sipative, i.e. they do not conserve kinetic energy. How-
N sm(t)
t=1(cid:18) α (cid:19) ever, there are also differences. For granular media, the
X
interactionforcesareshort-rangedandbelongtocollision
of the simulated distance s (t) from the measured ve-
α processes,where particles touch and temporarily deform
hicle distance sm(t). The resulting optimal parameter
α eachother. Vehicle interactionsarelong-rangedandcor-
values for city traffic in Stuttgart are κ = 0.85s−1,
respondtodecelerationmaneuvers. Theyareusuallynot
V = 6.75m/s, V = 7.91m/s, C = 0.13m−1, and
1 2 1 related to collisions (i.e. accidents), since the drivers try
C =1.57.
2 to keep a safe distance s from each other which can be
A comparisonwith the data shows that the extremely
considerablylargerthanthevehiclelength(cf.Eq.(16)).
shortrelaxationtimeτ =1/κ=1.17sresultsintoohigh
Therefore,theeffectivespacerequirementsofvehiclesare
values of acceleration, which leads to an overshooting of
2
much larger than their actual size. Moreover, vehicular and τ = 1/κ, we would obtain the optimal velocity
α
interactions do not conserve momentum (in contrast to model, again. We extend this relation by a complemen-
granular ones). tary term which should guarantee early enough and suf-
Another difference between granular and traffic dy- ficient braking in cases of large relative velocities ∆v .
α
namics is, that the laws of granularinteractions are very This term should increase with growing velocity differ-
well known, pretty much like the basic laws of physics, ence ∆v , but it should be only effective, if the velocity
α
whereasthelawsofdriver-vehicledynamics(iftheyexist of the following vehicle is larger than that of the leading
at all) are still to be established. The particular chal- vehicle, i.e. if the Heaviside function Θ(∆v ) is equal to
α
lenge of modeling vehicle dynamics is its dependence on 1. Moreover,the additional deceleration term should in-
factors like perceptions, psychological motivations and crease with decreasing distance s , but vanish for large
α
reactions,orsocialbehaviors. Thus, incontrastto phys- distance s → ∞. The braking time τ′ belonging to
α α
ical processes, driver behavior cannot be expected to be this term should be smaller than τ , since deceleration
α
describeable by a few natural constants. capabilities of vehicles are greater than acceleration ca-
According to the socialforce concept, the amount and pabilities. We chose the following formula which meets
direction of a behavioral change (here: the temporal the above conditions:
changeofvelocity,i.e.theacceleration)isgivenbyasum
of generalized forces. These reflect the different motiva- f = V(sα)−vα0 − ∆vαΘ(∆vα) e−[sα−s(vα)]/R′α.
tions which an individual feels at the same time, e.g. in α,α−1 τα τα′
response to their respective environment. Since these
(15)
forces do not fulfill Newton’s laws like actio = reactio,
theyarecalledgeneralized forces. Alternatively,theyare This formula takes into account that vehicles prefer to
namedbehavioralorsocialforces,sincetheymostlycorre- keep a certain velocity-dependent safe distance
spondtosocialinteractions. Thesuccessofthisapproach
in describing traffic dynamics is based on the fact that s(v )=d +T v , (16)
α α α α
driver reactions to typical traffic situations are more or
lessautomaticanddeterminedbytheoptimalbehavioral where dα is the minimal vehicle distance and Tα is the
strategy (which is the results of an initial learning pro- safetime headway(i.e.aboutthereactiontime). R′ can
α
cess). Adetailledmotivation,description,anddiscussion be interpreted as range of the braking interaction.
of the social force concept is given in Refs. [11,1,12]. We canfurther reduce the number of parameters(and
The driver behavior is mainly given by the motiva- the numerical effort), if we replace the previous V(sα)-
tion to reacha certain desired velocity v0 (which will be function (8) by
α
reflectedby anaccelerationforcef0) andby the motiva-
α
tion to keep a safe distance to other cars β (which will Vα(sα,vα)=vα0 1−e−[sα−s(vα)]/Rα . (17)
be described by repulsive interaction forces f ):
α,β In case of identical mod(cid:8)el parameters of all(cid:9)vehicles, the
dv corresponding equilibrium velocity-distance relation re-
α =f0(v )+ f (x ,v ;x ,v )+ξ (t). (12)
dt α α α,β α α β β α sults from the implicit condition vα = Vα(sα,vα) and is
βX(6=α) depicted in Figure 5.
The traffic model defined by equations (13), (15), and
The fluctuating force ξ (t) may be used to include indi-
α (17) will, in the following,be called the generalized force
vidual variations of driver behavior, but in our present
model of traffic dynamics (GFM). Since all its seven pa-
investigationsitwassettozero. Ifweassumethattheac-
rameters have a clear and measurable meaning, they
celerationforce is proportionalto the difference between
should have the right order of magnitude. A calibra-
desired and actual velocity and suppose that the most
tion with respect to the follow-the-leader data shows
important interaction concerns the car (α−1) in front,
that this is indeed the case. We found the following op-
we end up with
timal parameter values: v0 = 16.98m/s, τ = 2.45s,
α α
dv v0 −v dα = 1.38m, Tα = 0.74s, τα′ = 0.77s, Rα = 5.59m, and
dtα = ατ α +fα,α−1(xα,vα;xα−1,vα−1). (13) Rα′ = 98.78m. Now, the acceleration time τα is more
α than twice as large as in the optimal velocity model, the
The acceleration time τα is a third of the time which brakingtimeτα′ issmallerthanτα,asdemanded,andthe
a freely accelerating vehicle needs to reach 95% of the reactiontimeTα isalsorealistic. NotethattherangeRα
desired velocity. of the acceleration interaction is much shorter than the
Now, we have to specify the interaction force fα,α−1. rangeRα′ ofthedecelerationinteraction. Thisisnotonly
For sensible, it is also the reason for the astonishingly good
agreementwiththeempiricaldata. TableIcomparesthe
f = V(sα)−vα0 (14) minimalvaluesoftheaveragerelativequadraticdeviation
α,α−1 τ D that could be reached for the different discussed traf-
α
fic models by evolutionary parameter optimization [10].
3
(TheadvantageoftheappliedBoltzmannstrategyisthat dvα(t) vα−1(t)−vα(t)
≈ , (23)
this particular gradientmethod escapes local minima by dt τ′′
α
means of a fluctuation term with eventually decreasing
variance.) which coincides with a car-following model in which the
following vehicle adapts to the velocity of the vehicle in
It turns out that the optimal velocity model is consid-
erablyimprovedbytheT3model. Thisisnotsurprising, front. According to the formula for τα′′, the decelera-
tion is the stronger, the closer the vehicles come to each
since the goodness of fit should increase with the num-
ber of model parameters. Nevertheless, the generalized other. Therefore, the limiting cases of the generalized
force model behave very reasonably.
force model reaches the best agreement with the data,
although it includes only two third of the number of pa-
rameters of the T3 model. The simulation results for
IV. SUMMARY AND DISCUSSION
the generalizedforce model are depicted in Figure 2. Fi-
nally,therepresentationoftherelativevehiclemovement
inthe∆v -s -planeshowstheexpectedoscillatorychar- We have calibrated several microscopic traffic models
α α
acter of follow-the-leader behavior, which can cause the to city traffic, compared them with empirical follow-the-
development of stop-and-go traffic (cf. Fig. 6). leader data, and investigated their properties. It turned
In comparison with physical models, the generalized out that one model showed too large accelerations and
force model still seems to contain a lot of parameters. decelerations, but nevertheless caused accidents in cer-
However, let us discuss this in more detail for the previ- tain situations. Another model was a regression model,
ously mentioned molecular-dynamic models of granular sothatthemeaningofthe modelanditsparameterswas
media. If we want to describe interactions of smooth, not clear. Therefore, we developed the generalized force
inelastic, and spherical grains, only, we need to know model, which reached the best agreement with the em-
the particle size and the normal restitution coefficient pirical data, although it had only two parameters more
related to translational energy dissipation. In cases of thantheoptimalvelocitymodelandfourparametersless
rough spheres, we require two additional parameters, a than the T3 model. Another advantage of the general-
tangentialrestitutioncoefficientandafrictioncoefficient ized force model is that all its parameters are easily in-
[16]. Moreover,ifthe grainsare non-spherical,the situa- terpretable and have the expected order of magnitude.
tion becomes even more complicated. In conclusion, the Therefore, it can be immediately said, how the param-
small number of parameters occuring in physical models eters will differ between fast cars, slow cars, and trucks
are often a result of simplifications and idealizations. (the latter being characterizedby small vα0, but large τα
Finally, let us check the plausibility of the generalized and τ′). It can also be predicted what happens, if the
α
force model. Inorderto do this, werewrite the modelin speedlimit(i.e.v0)ischanged,ifthe weatherconditions
α
the form are bad (greater τ′, but smaller v0 and R′ ), if roads
α α α
dv V∗(s ,v ,∆v )−v (t) would be used by vehicles with smaller length lα, if the
α = α α α α α (18) reactiontimeT couldbereduced(bymeansoftechnical
dt τ∗ α
α measures like an autopilot), etc. For these reasons, the
with generalized force model is an ideal tool for carrying out
detailstudiesoftrafficflow,aswellasfordevelopingand
1 1 Θ(∆v )
α
= + (19) testing traffic optimization measures.
τ∗ τ τ′′
α α α The simulation of large vehicle numbers is completely
and analogous to molecular-dynamic simulation studies of
τ′′V +τ Θ(∆v )v many-particle systems, e.g. of granular flows. The pa-
V∗ = α α α α α−1 , (20) rametersofthedifferentdriver-vehicleunitsarespecified
α τ′′+τ Θ(∆v )
α α α individually(α-dependent),then. Inthiscase,onewould
where specify typical parameter values of fast cars, slow cars,
and trucks, and their respective percentages. Alterna-
τ′′ =τ′ exp{[s −s(v )]/R′ }. (21)
α α α α α tively, one could introduce a distribution of each param-
For small velocity differences ∆v or large distances s , eter (e.g. a Gaussian one) around a typical value. Then,
α α
we find the simulation is started with the initial and boundary
conditions of interest. Of course, the model can be also
dv (t) V (s ,v )−v (t)
α ≈ α α α α , (22) extended to a multi-lane model with lane-changing and
dt τ
α overtaking maneuvers [1,17]. This is a topic of current
so that vehicles try to approach the optimal velocity V research.
α
with a relaxation time τ . This corresponds to the opti-
α
mal velocity model, but with a velocity-dependent func-
tionV (s ,v ). Ifa vehicleis faster thanthe leadingve-
α α α
hicle (i.e. ∆v >0) and its distance is sufficiently small,
α
we have
4
Acknowledgments berg and D. E. Wolf (eds.) Traffic and Granular Flow
’97 (Springer, Singapore, 1998).
Theauthorswanttothankforfinancialsupportbythe [16] J. T. Jenkins, J. Appl. Mechanics 59, 120 (1992); S. F.
Foerster, M. Y. Louge, H. Chang, and K. Allia, Phys.
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and by the DFG (Heisenberg scholarship He 2789/1-1).
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Moreover,they are gratefulto the BoschGmbH for sup-
PhysicaA231,534(1996); P.Wagner,K.Nagel,andD.
plyingsomeoftheirfloatingcardataandtoTiloSchwarz
E. Wolf, Physica A 234, 687 (1997).
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5
sαF(tI)G(.b)2,.anTdheactciemleer-adteiponenddveαn(tt)v/edltoc(ict)yavcαco(rtd)in(ag),todtishteanocpe- m/s) 1168
timalvelocitymodel(OVM),thegeneralforcemodel(GFM), y ( 14
and the T3 model in comparison with follow-the-leader data elocit 1102
of city traffic. According to (a), most of the models compare V 8
welltothemeasuredvelocities. OnlytheOVMshowsasignif- ary 6 OVM
icantovershooting,indicatingtoolargeaccelerations. Fitting ation 24 GFTM3
of the vehicle distances is a much harder task, as shown in St 0
(b),buttheT3modelandtheGFMperformwell. In(c),one 0 10 20 30 40 50 60
canseethattheempiricalaccelerations anddecelerationsare Netto Distance (m)
usually limited to the range between −3m/s2 and +4m/s2, FIG. 5. Velocity-distance relations of the different traffic
which is met bythe GFM. models in the stationary case, if all vehicles have identical
Note that the vehicles had to stop three times due to red parameters.
traffic lights (during the periods between 169.9 and 184 sec-
onds, 233.5 and 253.5 seconds, and after t=288s). At time
t=143.8s,thevehiclein front wasturningright,sothat the 4
second car was following another vehicle, afterwards.
3
m/s) 2
y (
14 OVM, Leader ocit 1
2m/s) 1102 OVMT, 3F,o Llleoawdeerr e Vel 0
eration ( 68 T3, Follower Relativ -1
cel 4 -2
Ac 2
0 -3
0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60
Netto Distance (m)
Time (s)
FIG. 3. Acceleration of an unobstructed vehicle and of
a following vehicle according to the optimal velocity model
FIG. 6. The simulation of the follow-the-leader behavior
(OVM)andtheT3model. Initially,bothvehiclesareatrest.
accordingtothegeneralizedforcemodelshowstheoscillatory
nature of the relative motion of vehicles. The above result is
in good agreement with the empirical findings depicted in
Fig. 1.
18
m/s) 111246 OVTM3
y ( 10 GFM TABLEI. Minimalvaluesoftheaveragerelativedeviation
ocit 68 D between empirical data and simulation results that were
Vel 24 (a) reached for the different traffic models by evolutionary pa-
0 rameter optimization.
0 10 20 30 40 50
Model OVM T3 GFM
2m/s) -011 D 0.0586 0.0354 0.0316
n ( -2
eleratio ---543 OVTM3
cc -6 (b) GFM
A -7
0 10 20 30 40 50
Time (s)
FIG. 4. Time-dependent velocity vα(t) (a) and accelera-
tion dvα(t)/dt (b) for a vehicle which approaches a standing
vehicle according to the different discussed simulation mod-
els. Theoptimalvelocitymodelproducesanaccidentattime
t=34.7s.
6