[PDF] BAROMETRIC AND GPS ALTITUDE SENSOR FUSION Vadim

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BAROMETRICANDGPSALTITUDESENSORFUSION

VadimZalivaandFr anzFr anchetti

CarnegieMellonUniversity

IndexTerms - sensorfusion,GPS, barometric,altitude

ABSTRACT

Thealtitudeof amoving vehicleas reportedbyGPS suffers frominterm ittenterrorscausedbytemporary obstructionof thesatellitesby buildings,mountains, etc.Additionally, it isaffected bysystematicerrorscausedbymultipath effects, ionosphericandtropospheric effects, andotherhardw arede- signlimitationsand naturalfactors.

Atmosphericpressure, measuredbya portablebaromet-

ricsensor, couldalsobeusedto determinealtitude,is not susceptibletoproblems causedbyobstruction ofsatellites, andcanpro videreliablemeasurements outdoorseveninur - banandmountainous regions.

Inthispaper ,wepropose analgorithmwhichimproves

accuracyandprovidestighter confidencebounds ofaltitude measurementsfroma mobilephone(or anyde viceequipped withGPSand barometricsensors)by meansofsensor fus ion techniqueswithoutthe needfor calibration.

Ourexperiments haveshown thattheproposedalgorithm

providesmoreaccuratemeasurementswith tightercon fidence boundscomparedto usingeitherof thetwo sensors,baromet- ricorGPS, alone.

1.INTRODUCTION

Sensorfusiontechniques areusedto combineinformation frommultiplesources (sensors)withthe goalofimpro ving accuracyorreliabilityov erthe resultsfroman individual source.Agood introductionintothe subjectofsensor fusion isgiv enin[1],andmoredetailsonits mathematicalmethods canbefound in[2]. Inthispaper ,willconsider arealisticscenariowhere ave- hicle(ora pedestrian)ismo vingoutdoorson theearthss ur- face.Themovement trajectoryisassumed tobesmoothin altitudeandnot tofeaturesudden altitudejumps. Thevehicl e mustbeequipped withGPSand atmosphericpressuresen- sors,suchas canbefound inmany modernmobilephones, likeSamsungGalaxyS4 andGoogleNexus4. Thismaterialis basedonresearch sponsoredby DARPA underagree- mentnumberF A8750-12-2-0291.TheU.S. Governmentisauthorizedto re- produceand distributereprints forGovernmentalpurposesnotwithstanding anycopyrightnotationthereon. Wewillaapplysensor fusionapproachto thisparticular scenario.Ouralgorithm willuse knowledgeof howthese two particularsensorsbeha ve.Some ofassumptionsaboutalti- tudesensorbeha viouraresho wninthefollowingtable:

FeatureGPSBarometric

AffectedbyobstructionsYes No

AlwaysavailableNoYes

BiasedNo Yes

Biasdriftswith weatherNoY es

Typicalaccuracy5m[3]0.17-0.5m[4]

Accuracyreportedbysensor YesNo

Ourapproachis basedonthe premisethatbarometric

altitudemeasurementspro videmoreaccurate information onrelativ ealtitudechangescomparedtoGPS.Itdoesha ve measurementnoise, butthe magnitudeofthisnoiseistyp- icallysignificantlysmaller thanthatof GPSmeasurements. onmeasurementsof barometricpressure requirescalibration. Suchcalibrationis valid foronlylimited timeperiodsdue tonaturalchanges intheatmospheric pressurecausedby weather. barometricmeasurementsto reflecttrueabsolute altitude. Usingstatisticalproperties ofGPSand barometricmea- surementsaswel lassome assumptionsabouttheeffectsof weather,wederiv econfidenceinterv alsforfusedmeasure- ments.Then,using theseintervals, wedefinea costfunction whichisused tooptimizethe temporalwindo wsizeused in fusion.

Inrelatedw orkYao[5] integratedbarometricaltitude

measurementsasa virtualsatelliteinto GPSpositioncalcu- lations.Ourapproach doesnotreplace GPSpositioncal- culationandcould beusedon topofother GPSaccuracy improvementmodels[6],aswellaswith otherGNSSsystems likeGLONASS.Inother relatedworkGebre-Egziabher[7] presentsanempirical barometricaltitudeconfidence bound basedonhistorical meteorologicaldatawithout attempting to estimatethealtitude.

Weclaimthatouralgorithm achieves thefollowing:

1.Resultingfused altitudemeasurements willt ypically

havetightertoleranceintervals,compared toGPSsen- soralone.

2.Resultingfused altitudemeasurementswill typically

havemoreaccurateabsolutevalues,compared tobaro- metricsensoralone.

3.Thealgorithm providesnot onlyfusedabsolute altitude

values,butalsostatistical confidenceintervals. Additionallybenefits ofourapproach arethatthe altitude couldbeestimated accuratelyev enwhenGPS signalistem- porarilylost(dead zones).This isanonline algorithmwhich couldbeused inreal-timeand thecalibrationstep isnotre- quired.

2.THEALGORITHM

SensorFusion

Letthe true(unknown) altitudeattime tbea(t).W ewill useg(t)todenoteGPS altitudemeasurements attimet.Ac- cordingtoAndroid APIdocumentation[8]the yarenormall y distributed:g⇠N(a, 2 g Withoutlossofgenerali ty,we willassumeGPS measure- mentsaresampled atfixed intervalswith thesamplenumber ibeinganaltitude g[i]andst andarddeviation g [i].Thesam- plesarenumbered inre verseorder withi=1beingthemost recent.The algorithmcouldbe easilyadoptedfor variable ratesampling. Similarly,wewillmodelthe barometricaltitudeas b⇠ N(a+, 2 b ).Here,denotesthe unknownbias ofbaro- metricaltitudemeasurements. Unlike GPS,thev ariance 2 b is notprovided bythesensorandis unknown. Wewillstartby estimatingtheparameters oftheg(t)dis- tributionusingMLEbased onthelast nsamples.Thepooled varianceandthe samplemean (biased)are: s 2 g 1 n n X i=1 2 g [i](1) g 1 n n X i=1 g[i](2)

Themeanhas standarderrorof themean(SEM)

sg p n andisnormally distributed aroundthetrue altitude:µ g N(a, s 2 g n Next,wewillestimatethe parametersof theb(t)distribu- tionusingMLE basedon thelastmsamples.

Here,mandndenotethenumber ofGPSand barometric

samples,respecti vely,foragiventimeinterval.Generally speaking,mn,becausethe barometricmeasurements are alwaysavailable,even whenGPSmeasurementsaremissing.

Samplemean andsamplev ariance(biased)are

b 1 n n X i=1 b[i] s 2 b 1 n n X i=1 (b[i]µ b 2 (3)

UsingSEM,we describethe distribution forµ

b N(a+ s 2 b n

Now,estimatethevalueof as:

b g .This estimatedvalue isthesumoftw onormallydistrib utedran- domvariables. Suchasumwillbealso distributednormally: ⇠N(, s 2 b m s 2 g n Weobtainthecorrectedaltitude forsampleias:ˆa[i]= b[i]+ .Thisv alueas thesumoftwonormally distributed randomvariables willhavedistribution:

ˆa[i]⇠N(µ

b ,s 2 b s 2 b m s 2 g n )(4)

OrdinaryLeastSquar esVariance Correction

Thesamplev arianceestimationin Equation(3)estimatesthe spreadofbvaluesrelative tothehorizontallineb=µ b which correspondstoa vehicle movingon ahorizontalplane.How- ever,ifthevehicleascendsordescends(e.g. carclimbinga hill),thismethod isnotsuitable forestimationof Gaussian noiseinb.Intuiti vely,thecorrectapproachwouldbetoes- timatethev arianceorthogonalto thedirectionofthemov e- ment.

First,wefit astraight linethroughthe {b

1 ...b m }using theordinaryleastsquares method.Accordingto theGauss- Markovtheorem,thisisthebest linearunbiasedestimator (BLUE).Thisw ouldgiv eus↵andparametersofa linear model b. Underthe normalityassumption,thefirst andsecondmo- mentsofthe distributioncouldbeestimated as[9]: =(5) s s 1 m2 P m i=1 P m i=1 (t[i]µ t 2 (6) toprojectthem verticallyto thefittedline asshowninFig- ure1: b p 1+ 2 (7) b .To doso,letus setk()= p 1+ 2 ,sowe canwrite: b b uctis[10]: b 2 var(k)+[E(k)] 2 (9) usingsamplestatistics as: 1 m m X i=1 1 m m X i=1 (b[i]t[i]↵)(10) s v u u t 1 m m X i=1 2 v u u t 1 m m X i=1 (b[i]t[i]↵µ 2 (11)

Treatingkasafunction ofrandom variable,itis pos-

sibletoestimate thefirstand thesecondmoments, E(k)and var(k)respectively,usingfirstorderTaylor'se xpansions[11]:

E[k()]⇡k(µ

k 00 2 2

2(µ

3 )+s 2

2(µ

2 +1) 3/2 (12) var[k()]⇡(k 0 2 2 2 s 2 2 +1 (13) Substituting(12,13)into (9)andusing valuesobtained by b )whichwe willbeusing insteadofs b in(4): b 2 s 2 2 2 +1 2 s 2 s 2 +1 s (2(µ 3 )+s 2quotesdbs_dbs29.pdfusesText_35

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