計測自動制御学会東北支部第264回研究集会（2011．3．11）
資料番号264−13

ScientincStudyofRoadTrafncFlow（2）・MeasurementandExampleofDataAnalysis・

TasukuTaka由ProfessorEmeritus．TbhokuUniversity

Kevwords：rOadtra仔ic．t，ime−domain，Weberllaw、Fechner slaw．
高木相

連絡先：〒981−0952 仙台市青葉区中山5−2−20（Nakayama5−2・20．Sendai、JapanE−mail：t・aSuku＠sirius．∝n．ne．Jp

AsmentionedinSection3．ourindependent（九ntinued from ScientiBc Shldy of Road
parameterStOl光meaSuredareTime・Headway
Tra臨c Flow O） presented atthe previousT（s） and Car・Speed V bl／s）．The
meeting of262nd SICE Tbhoku atlもhoku
Distance・HeadwayXcanl光givenfromTand
UnlversityonDec．23，2010、WhenSectionsl．2
VasX≠Ⅳthatwasshownin（3・1）．Wbhave
and 3 were pre紺nted．Thus in this祀enintheSection3thatthecardriversen諾S
pre紆ntadonweshal1l光ginfromtheSection4．
Xthroughdriver■seyeandtheXIxcomesthe
Thecontentofthepreviouspresentationisas
physicalstimulitothedriver、andthecar印eed
鮎uows：

ⅡRoadTr曲別owFtnlmentals
fromWbberFetlmerLaw
3．W挑即−F∝h皿erLawin批）adlla瓜（：nOW

Vis contro11ed bythe driverls sensationfor
maintaining saR）tyAlthoughVis a physicalquan1ity car driver cannot senseit as aphysicalquantity．Drivers canreCOgnizethe
physicalquantityof speed only by a印eed

3．1HumanFactorsinroadtra臨cflow
3．2Carfo1lowingmode1

meterthatinstalledinfrontJOfthedriver．Fromsuch a sihlation wastakeninto account．tx）th

3．3RelationshipbetweenXandVas

parameterSXandVhavetherelationdlipof

Weber・FeclmerLaw
3．4Mathematical1毎ri丘cat，ionofV・Ⅹ

Wel姫r・Feclmerlaw each other、aS We SaWin

Weber・FeclmerRelationship
3．5Feclmer sLawin鮎adTra腿cFlowand
MeaningofβV＝1
Appendix3・1

Appendix3・2

Section3byinvokingtheWel光r−Feclmerlawin
Section2．
4．2MeasufementS

Inordertopursuetheinves廃ationsforbasic
characterizations onthe road tra伍cflow weneedtomeasurelx）thTandVsimultaneously．
Weused avideocalneraandmanualwatches．

4．Measurements and Example of Data
A皿dy祀S
4．1Para皿eterSXandV

Simple equil）ment WaS COnStruCted for
operatingthemanualwatches，OneforTandanotherforVbcauseitwasalmostimpossible
tooperatethetwowatchessimultaneouslyTheequlPmentmade possible to ol）eratethe
WatChessimultaneouslywithoneleverwitha

The person、Whoismeasurlng，WatChesthe
videoimageinwhichcarscon血uouslynow．As

manual operation． The l）rinciple of mentionedalx）Ve，thepositionsAandBhave
measurements and construction of thebeensetinadvanCeOntheroadinvideoimage
equpmentisre印eCtivelyshovmasfo1lows：with the distanCe OflOm（払r urban OrFig．4・1showstwo cars．deslgnated byiand
suburban rOad）or 2hl（forI五gh 印eed
i＋1．foregolnganditsfo1lowingonerespectivelyexpressway）．The same pulse waveformis
Tbp丘gure showstheAmeinst且nt Whenthe
ShovmwithinFig．4−2thatisshovminFig．4−1．

foregomgvehicleitouched atthe designatedpoint A，middleis whenitouched B，andbottomiswheni＋1touchedA．Thepoint′Cis
thepositionofforegolngCari．wheni＋1touclled
A．TherundistanceofthecariisXiWhichistherundistancewithinthetimeuntili＋1touches
thepointA．Thedistancet貯tWeenAandBisL

Whichis setlOm for ordinary urban Or
Suburban rOad、and 20m for high 印eed

expresswayforavoidingmeasurlngerrOr（lueto
highspeed．ThepositionsAandB are setin
advanceonthevideoimageappropriately
Thepulsewaveontheleftsideinthis丘gureShowsthe血1ings whenthe cariandi＋1

Fig．4−2Equpmentfor opera血g two

moved atthe designated points A and B．
respectively：bAiswhenitouchedAtiBWheni

watcheswithonelever

touchedB，andtl＋1A Wheni＋1touchedA．ThusThewatchTxcanSenSeOnlywhentheleverthetime−headwayTiLis

On．then血1e−headways（ri．i＝1．2． −）aremeasuredandstoredinthememory．
The watchTiLSenSeStX）thOnandO且二sothe

TiL＝tiB−tiA．
Thespeedcan1光Calculatedby（4・1）

vehicleruntimeofdistanCeLismeasuredand

Vi＝・（4・カStOred．The used watchis

SEIKO Super

AndXiisAth1ete ．which canmemOrize300data．The
Xi＝ViT．
speedcanl光Obtainedby（4・2）．
The equipmentis showninFig．4・2．which Anexampleoftheactual1yrecordedwaveform（4・3）

With the pen recorder．together withthemakes possible to operatethe two watches
Sim山taneouslylA

・1

B±ヱ」

ト1Fig．4・3Exampleofrecordeddata
readoutdatawrittenontherecordingpaper、isShowninFig．4・3which wasrecordedbythe

Fig．4・1Modeldescription of two cars

SuCCeSSivelyrumgOnrOa（12recorderintheequipmentshovminFig．4・2．

Table4・2Exanpleofapartofcompileddata

The numericaldataindicated by arrowsin吠s） T（s） ∨（m／s） X（m）

Fig．4・3aresummari次dinTbble4・1．Likethis
example．thevideodatacanbecompiled．The

50 161．47 2．（光11．11

22．96

9．70

26．35

finaldatanecessitatedfor analyzingtheflow

52

171．41

7．239．90

71．58

al・eTVandX．Thesedatacanl驚Obtainedwith

53

175．10

3．69

11．49

42．41

re印eCttOelap虻dtimethatcan1尤ObtaiJledby
summingupofTl．Theelap紆dtimebwhenthe
CarijusttouchedthepointAcanl芳Obtainedby

ti＝∑；＝1Ti．

（4一め

Thble 4・1Data obtained from recorded
WaVeformshowninFig．4−3
CarTime・tim badway

180．35 5．24

10．36

54．35

55

183．48 3．13

8．52

26．67

56

187．52 4．04

10．10

40．87

57

193．27 5．74

8．84

50．82

58

196．87 3．∝〉

10．83

38．98

59

198．74 876

12．04

22．61

60

200．13 3！氾

9．20

12．79

4−3DataplotBviaelap能d血eFig．4・4 showsthe measured data which（TL（s）） （Ⅹ（Id）

汀G））

7．74

54均）e∝l 10mI▼un msb11（℃・No． beadway （Ⅴ（m／s））

16

164．18 2．71

51

14．9

0．67

From（4・カ】15．3

includethe data of％ble 4−2plottedwith

Fromrespectofelapsed血1et（＝tiinTable4−2）（a）
t・T（b）t−Ⅴ and k）t・X，re印eCdvley The（4・3）ThenumtX？rOfmeasure（1carisatmostupto ab配issasoftheatx）Vethree丘guresareofthe
300bcauseofthememorylimit．Thenum1x）rSame 就ale fX）that we can COmpare their
Ofcarsof300isenoughforanaly栄Sfortime 鈷atweseachother．
domainanalysis、butforthere訂11arStatistical Ifwel00k（alOi）and（c），WeWⅧ＝光abletosee
analysis．we need more numt光r Of carsis
SOmetI血ginterestedasfo1lows：necessaryHowever．sincewehaveaplentyof
investigation of statistical data analyses1）FromvariationsoftheTime−HeadwayT
withrespecttDtimet、WemaynOtl鷲able

performedso払rintheworld．weneednothere

tDdeduceusefu1inR）rmationavailablefor

tDCOnCentratetOaCCumulatesomanydata．

practicalapplications．althoughTisone

We saw that the necessitate data aretime・headway T 印eed V and distance
headwayX（＝TⅥ．Table4・2isapartofEXCELdataofTVandXwithrespecttoelap紀dtime
ti，WhereiisthecarNo．The data shownin

Table4・2weretakenfromthevideoatthe

Ofthe most es鎗ntialparameterS．The

essentialreasonis．asmendonedagainin
thefo1lowiJlg2）、thatT一光COmeSlargein
bothcasesofheavyorcongestingtra伍ccondit;ion and of low density and highSpeedcondition．

Parking area at the Tbhoku E叫）reSSWay
AzumainMiyagiPre飴ctwe．Japan．

2）Atx）Vel）suggestthatanotherparameter

Visneededand．ifwel00katthe丘gtueOi）．wewingetmoreinformadonalx）ut
thetra伍cnow e．g．whenthelow speed

InThble4・2．i：carNo．．も：elap謝dtime．T（d：Time・Headway（second）．V（m／g）：speedbleter／紆COnd）．X（m）：Distance・Headway（meted．

con山1ueSWeCanSeeitasthesituadonof
congesdonOlighdensi扉．Wbsha11note

1もtalnumbrsofcarsmeasured are290．All

datawereanalyzedandgraphsoft−Tt・Ⅴandt・ⅩweremadeandshowedinFigs．4・4．whichthattime・headwayT（Fig．4・4（a））tx）COmeS

largeinca紆Ofcongestion、butital

happens to l光COme large whenareshowninthenextSub舘Ction．

now・densityislow（high印eed）．

g）Figure（d may have compiled al1
information on the road tra伍c now3becau栄the twoes舘ndalparameterSTadmitβV＝1，themaximumnOWCOnditionas

andVareinvoIvedink）astheproductthat providesthe distance・headway X

we have seeninSubsection3．5，Thiscondition（ゴⅠ Ⅵ．

sutIiect（Section5）．

Should k scrutinizedlater as the special
Inthissub舘Clion．wewinseethegraphsin
whichthe datalike Table4−2are eml貨dded

t−T（Tohokul∃Ⅹp，Azuma）
41intoIx）thⅤ拒ber11aw（3・3）and Feclmerlslaw（3−7）showninSection．3．

2

1
1

4．4・1Ⅵ屯br slawnUOO︵S︶↑TheⅥ毎払r−slawinthe pre艶nt CaSe WaS
formula（3・3）inSection3．whichisshownagainhereasfo1lows：ハb

讐＝β△Ⅴ・（3・3）（fromSecdon3）

4

ApartofthedatatoIx）emtX？ddedisshown
Table4・3asaneXamplewhichiscalculated
fromTable4・2by；
こよ主∴ぎ三：≡：二！、．主‡三；．；ヰ葦

20Fig．4・4（a）t・Tplot
t−Ⅴ（TohokuExp．Azumz）
5

4一40…−△Ⅵ＝Ⅵ−Ⅵ．1（4・5）、△Xi＝Xi−Xi．1（4・6）．

andニ ●

55△Xand△X／X

302︵ぜ∈︶A03

Table4・3ExamⅠ）1eofcalculationsofAV．
二二こ三コ

AV5ロ Ⅴ（m点） Ⅹ（m）

△X

△X／X

0り一l

50lt（sト ■500

11．1122．鮎

1．41・3．39・0．14751

9．70

26．35

・0．2・45．23・1．716

52

9．90

71．58

・1．59

29．17

0．407

53

11．49

42．41

1．13・11．94・0．281

54

10．36

54．35

1．84

27．鎚

55

8．52

26．67

・1．58・14．2・0．532

56

10．10

40．87

1．26・9．95イ）．243

57

臥84・1．99

11．84

0．232

58

10．83

59

12．04

60

9．20

1000

Fig．4・4（b）t−Vplot′t・Ⅹ（TohokuExp．Azuma）︵∈︶X

150 卜一−−− 100L・㌻i

500

50

t（s）1000

50．82
38．98

・1．21

22．61

12．79

2．84

1．41

0．509

16．37

0．419

9．82・3．39

0．434・0．147

Fig．4・4（c）t′・Ⅹplot
For example．incase ofi＝50andi＋1＝51inTable4−2．Ⅵ＝11．11．Ⅵ＋1＝9．7、then△Ⅵ＝1．41．．
△Xi＝・3．39．△X／X＝・0．147．ThenumlガrOfdata4．4Emtx）dding data intD Wbtx）r law andFed皿er 1aw

AsmentionedinSection3．the relationshipbetweenTandVobystheWらtx）r・Feclmerlaw
and this was mathematically proved汀we4is290（i＝1〜29〔》．
juldatai＝1〜290were eml貨dded into thethe foI・mula shown in Fig．4・6is the one
automaticallycalculatedbycomputer‥Ⅵ毎1x）r．SlawwhichisshowninFig．4・5．
仙叫腑bb）△XノX＝0．0554△V−0．0176

V・X【Mo叫

X＝13．254e…86V

ヨ00

Fig．4・5Data eml貨ddedintoⅥ毎tx）rlslaw背＝βAV）
一40

10 20 30
AsshowninFig．4・5．a11data are confinedVtm川withnthelinesofβヰ）．03andO．3．Thegothiclmelx）tWeenβ＝0．03and O．3isthe averaged
Fig、4−6DataembeddedintoFeclmer slaw
lme of whichβis O．0554（β＝0．0554）as
収＝XnexprβⅤ））
ShoⅥ1bythec舵臨（：ientofAVintheformula
inthefigtlre．NotethatnumtX）rO．0176inthe050

Oneofthemostcomprehensivewaysshowmg
the accuracyis to show the averaged

formula mustlx！equalto zero whichistheerrorcKX：urredfromourmeasurements．

● Wbbr、slawd（光Sn tcaretheexistingof△Ⅴ＝0′ Wh元h means 血e flowis

characteristicsofthe dataWhenthe curveis

exponential．

lam血abng∴謝）CaⅡed血∽血∂由血肝semi・logarithmic graphexaminationisavai1able．Inthepresentcase．

Thelaminatenowconditionisthemost

wetakethenaturallogOn）forXBVandploton

es艶ntialconditionwhen we considered

the V・lm（‰v）plane（semi・logarithmic駆Ction

theroadqualityofwhichdetails

paper）．If the Feclmerls exponential

mentionedinSection5．

relationshipis preservedforthe road tralBc

4．4・2Fedmer slaw

且owtx）tWeen印eedVandhO（av）．theV−lnO（av）

TheFeclmer slawwasshowninformula（3−7）
OfSection3．Againweshal1showhere，thatisrelationship should t＊COme lmear On the艶mi・logarithmic甜Ction paper．Then．we try
thisforexaminationofthelmearityFig．4・8is

X＝X。eXp（βⅤ）（3・7）（fromSKtion3）．

Fig．4・6showstheresultoftheV−Ⅹplots、in theresult，Whichshowsveryg00drelationship
Whichthe approximatedexponentialcuⅣeis with more than 0．95 0f correlation：払ctor．
Fig．4・7istheoriginalVXav relationship．
sbowm
ThecurveinFig．4・6istheapproximatedone，Anyotherroadtra伍cflowsweremeasured
Ofwhichβ＝0．0686．FromtheoneglanceofFig． and examine（lbythe same way mentioned4−6．coml）aringwiththe case ofWel光ー−slawatx）Ve．The other examples are shown inAppendix 4・1．where the urban road andShowIlinFig．4，5．weseethattheFeclmer、slaw
suburbanrOadareexamined．Theresultsd氾W
COntains more comprehensive and u求fu1
inlbrmationfol・the road tra用．cnow．Thus．wethata山relationshipsl光tWeenVandln（カareShould makethe approximated curve morenearlylmearwithcorrelation鈷ctor ofmore

COnCreteWayinthemeaningofthe accuracy：
thatistosayweshouldexaminetheaccuracy
Ofexponentialfunctionforapproximationsincet血anO．85．

5

4．5．DiscusgionsonTra伍cFlowandβ

Ⅹav−Vav（TohoknExp．Azumz）

4．5．1βas鑑aling払ck汀

13，491eO・椚54Vav￣▼】

Ⅵ毎 have seen in Section 3 that the「【【▼【 100OA帽粥

relationshipstx）tWeenthedistance−headwayX
and the speed V ol治yS the exponentialfunctionswhichcant把Writteningeneralas﹁110尺U︵占 ︶L臣︻ 卜200「一−−−「−−−−￥札三5X＝Xoexp（βⅤ）（5・1）（（3・7）inSection3）．

10 200which丘rstappearedin（3・7）inSection3．In
this simple equation，βisthe紀almg鈷ctor．

30

However．in the actual tra用．c now theC眈伍cientβhasanimportantrole．

Vav（m／s）

4．5．2βinⅥ毎tx）r slawandFeclmer slaw

Ⅵ屯have seenthe dataemt把dded丘guresin
Fig．4・5 for Wel光r slaw and Fig．4・6 for

Fig．4・7 V。V・Xav relationship on

linear section paper

Feclmerlslaw．Vh shouldnoticethatthereare

bigdj飽rencesIxtweenthetwofi訂1reSnOtOnly
the shape of graph butal師the numerical

Vav・h（Xav）（TohokuExp．Azumza）

6 と11n（Ⅹ。，）＝0．0754V。Ⅴ＋2．602

Valuesofβ．
（1）Sba匹Ofgr叫血︵＞出さul

The shape di飽rencecomesfromthedatato
beeml光〔1ded．Wb shouldnoticethatinca譲∋Of

theWbbr slawinFig．4・5．thedataof△V＝Oare omitted．tx）CauSe nO data exist when

△Ⅴ＝0．OnthecontraryFig．4・6forFeclmerls
010

2030

lawincludesalldataareeml光ddedeveninthe

40case of△Ⅴ＝0．What we canSeefromlx）thVav（m／s）

cases ofⅥ毎br．s and Feclmer．sis that theⅥ毎br■slaw dealsthe flow dynamics which

excludesthestaticstatesuchasstoppedrowof
CarSinthe congestion．Onthe contrarythe

Fig．4・8 Vav・1n（X。V）relationship on鎗mi−logsectionpaper

From th0紺 reSults．we are convinced ofFeclmer、slaw does notexcludethe static flow

Saylng that the relationshipl光tWeen thewhichisthe case ofX＝COnStantthatmeanS

distance・headway（カand印eed（Ⅵo一光ySthe

AV＝0．Aswe wiu seelaterthatthe static

StateOfnowisimportantintheroadtra伍c．
Wbbr・Feclmerlawfromthedataanaly栄S（seeAppendix4・1．theendofthisSection）．whichis （カNumericalvalueofP
Althoughthe same data are emtx）dded．thethe practicalpr00ffromthe measurementS．
numerical Ⅴalues of β of Webrlslaw
Mathematica山y we have seenit in the肝ig．4−5．β＝0．0554）and of Feclmer．slaw
Subsection3．4underthecondidonofβⅤ＝1．
（Fig．4・6．β＝0．0686）arenotthe same．This
IfwecompareFig．4−6and4・7．wenoticethat
discrepanCyCanbeconsideredduetoomitting
the numericalⅤalues ofβare di飴renteachotherforwhichwehavetoEX：rutinizeindetai1．

thedataof△Ⅴ＝0 inthecaseofⅥ屯tx）rlslaw．

butrightnowitcantx！PutOα1膿cau舘thismatterisnotcrucial．The（li飽rencemaycome

4．5．3（力ndidonofβⅤ＝1

fromaveragingpr∝e（1ures．

wastheconditionfbrtheⅥ屯l＊l／slawintra伍c

AsaremarkwecanSaythattherelationshipbetween speedVand distance・headway Xis

nowshownin（3．3）comesintotx）ing．Inthe
Otherword．we mayt光able to saythatthis

alwaysexpressedbyexponentialfunction．

conditionisthe鎌）1ution ofthe Wbbr、slaw ofβⅤ＝1．firstappearedin（3．6）inSection3．

6

（3．3）．
WbhavedehedVasVβ（＝1／β）・At／V＝Ⅴβ1

Ⅹ・Ⅴ（TohokuExp・血umヰ＝9．60521。（幻．20．151
4540

53thedistance・headwayX反側meS
X＝eX。＝2．718X。（4．7）

050ウ︼

50ウ︼

5ll

diinculttoagreethistheoreticalfbct．andwe︵ゼ∈︶AWb have to監rutini2X！What也．7）meanS．Ⅵねwonderthat（4．7）isindependentofspeedVIf
we assumeX。＝1Qmwhichwasmentionedin
3．5inSecdon3，alwaysXtxwmesalx）ut27m（atx）ut30m）independently of speed．Itis

3

becau諾eXP（βⅤβ）顎XP（1）苛（＝2・71828・−）・

0200

100

haveabigquestiononthisma仕erintheroad
tra瓜cnowanalysis．

300

Ⅹ（m）

Fig．4・9Ⅹ・VplotoflもhokuExpresswayAzuma
Makngclearthe above mendoned question
becametheessenceinourtheoreticalre鎗arCh
Fig．4−9isofcourselogarithmicexpressionof
the Wbtx）r・Feclmerlaw butthis丘gureo飽rs
thepersuasivedataforupperlmitof司）eed．In
theatx）VeCa舘、thehighestspeedwas150kph（bnnl）（about 42m／g）．Actual speedlimitis
legauysetwith100kph（＝28m／s）．

ontheroadtra伍cnow．ThiswⅧ＝貨mentioned
inthenextSection5．

4ゑ40pdond柁marks

VhsaidinthisSectionthatthecarspeedisa
艶nSationofthedriver．Theremaytx）argueSOnthismatter、butwecannota位）l・dbut′feelit、ifFinally the relationship between physicalstimuliandf3enSationcanbsaidas Sensation

weactual1yl黒thedrivers．Whenwedrivethe
CarS．WeWanttOrunaS払staspossible．butthis
cannotalwayst光Satisfiedl貯Cau艶Ofsa鈷tyor
exis血1g駐）regOingcar．Exactlywecan艶eintheis prol）Ortionaltothelogari止血10fphysicalStimulug．．Thisexpressionismorecommonin

general，butsincetheexponentialfunctionis

nextSession50fthedriverlshabittofo1lowthe

foregolngCaraSClo謝aSPOSSible．Insuchacase． COnVenient払r analyzingthe data we are
the driver almostal1time d（貯SnOtreCOgmZe COnSistentlyuslngit．
4．6（1）nClu由onof也isSec也on4
how払st，heisrumgwithaphysicalumit，Of

InthsSectionweconfirmedtherelationshipkm〟1．

An0ther thing we should consider the

betweendistance・headway（denotedbyX）canbeseenasthest血ulusandthespeed（denotedbyV）cantXさSeenaStheb心man艶nSa也onin

W抽即・Feclmerlawinroadtra伍cnowisthatJOf

thelimitofspeedVIntheV・Ⅹplott血g￡eldShownin fol・eXample Fig．4・6，Ⅹincrea能S Car driving system．Tb say more concretely
exponential1ywithrespecttolinearincrea求Of W抽即・Feclmer Law can well丘t tothe car

drivingphenomena．
Acknowledgments
The authorwishes toexpresshisthanksto
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