Multi-Mode Estimation for Small Fixed Wing Unmanned Aerial Vehicle Localization Based on a

19 Pages·2017·3.37 MB·English

sensors
Article
Multi-Mode Estimation for Small Fixed Wing
Unmanned Aerial Vehicle Localization Based
on a Linear Matrix Inequality Approach
MostafaElzoghby1,*,FuLi1,Ibrahim.I.Arafa2andUsmanArif1
1 SchoolofAutomationScienceandElectricalEngineering,BeihangUniversity,37XueyuanRoad,
HaidianDistrict,Beijing100191,China;[email protected](F.L.);[email protected](U.A.)
2 SchoolofControlandAutomation,MTC,Al-KhalifaAl-MaamoonStreetKobryElkobbah,
Cairo11331,Egypt;[email protected]
* Correspondence:[email protected];Tel.:[phone]
AcademicEditor:FelipeGonzalezToro
Received:21February2017;Accepted:14April2017;Published:18April2017
Abstract: Information fusion from multiple sensors ensures the accuracy and robustness of
anavigationsystem,especiallyintheabsenceofglobalpositioningsystem(GPS)datawhichgets
degradedinmanycases. Awaytodealwithmulti-modeestimationforasmallfixedwingunmanned
aerialvehicle(UAV)localizationframeworkisproposed,whichdependsonutilizingaLuenberger
observer-basedlinearmatrixinequality(LMI)approach.Theproposedestimationtechniquerelieson
theinteractionbetweenmultiplemeasurementmodesandacontinuousobserver.Thestateestimation
isperformedinaswitchingenvironmentbetweenmultipleactivesensorstoexploittheavailable
informationasmuchaspossible,especiallyinGPS-deniedenvironments. Luenbergerobserver-based
projection is implemented as a continuous observer to optimize the estimation performance.
TheobservergainmightbechosenbysolvingaLyapunovequationbymeansofaLMIalgorithm.
Convergenceisachievedbyutilizingthelinearmatrixinequality(LMI),basedonLyapunovstability
which keeps the dynamic estimation error bounded by selecting the observer gain matrix (L).
Simulation results are presented for a small UAV fixed wing localization problem. The results
obtainedusingtheproposedapproacharecomparedwithasinglemodeExtendedKalmanFilter
(EKF).Simulationresultsarepresentedtodemonstratetheviabilityoftheproposedstrategy.
Keywords: integratednavigationsystem;multi-modeestimation;sensordatafusion;Luenberger
observer;smallUAVlocalization
1. Introduction
The major issue in the field of autonomous Unmanned Aerial Vehicles (UAVs) and Micro
AerialVehicles(MAVs)istheirconsistentandpreciselocalization.Thisproblemisusuallysolvedby
combiningnavigationdataprovidedbydifferentsensors. Recently,theproblemhasattractedmany
researchersandalotofworkhasbeendoneintheautonomouslowcostUAVguidance,navigation
andcontrol(GNC)area,leadingtogreatsuccessduringthelastfewyears. Multiplesensorinformation
fusionforrobotnavigationincreasestheaccuracyandrobustnessofthenavigationinformation[1].
Differentmethodsofintegrationofnavigationdatafromdifferenttypesofsensorsandestimators
have been developed and investigated by researchers in the recent past few years. The focus has
mainlybeenontheintegrationofInertialNavigationSystem(INS)informationwithGPStoprovide
precise navigation information, but GPS data is occasionally unavailable due to outages. GPS is
asatellite-basednavigationsystemthatprovidesaccuratepositioninginformationanyplaceonthe
globe,butGPSisinsufficientformanynavigationapplicationsasastand-alonesystembecausewhile
Sensors2017,17,887;doi:10.3390/s17040887 www.mdpi.com/journal/sensors
Sensors2017,17,887 2of19
GPShassuperiorlong-termstabilityanderrorperformance,theaccuracyofGPSmeasurementscanbe
poorforshortperiodsoftimeduetoseveralerrorsrelatedwithitssignalacquisition. INS,ontheother
hand,isnotinfluencedbyexternalinterferences,andhashigherbandwidthwithgoodshort-term
accuracy,stabilityandnoisecharacteristics,butitpossessespoorlong-termreliabilityasthenavigation
accuracydegradeswithtimebecauseofbiasdriftandnoise,andthereforeitneedsexternalinformation
from GPS measurements for correction and initialization for long-term, high accuracy navigation.
GPS/INSintegratedsystemsarerobust, operateathigherbandwidth, providereliablenavigation
informationandhavebetternoisecharacteristicswiththelong-termstabilityofGPSandimproved
positioningcontinuitycomparedtoeitherstand-aloneGPSorINS[2–4].Nowadays,thetrendisto
increasethenumberofnavigationsensorstomitigatetheGPSdependence. Allsourcespositioning
navigation(ASPN)isanimportantconceptinaneweraofnavigationstudiestoachievelowcost,
effective,androbustnavigationsolutionsregardlessoftheavailabilityofaglobalpositioningsystem
(GPS)[5]. Alongwiththeimportanceofincreasingthenumberofnavigationinformationsources,the
designofintegratednavigationsystemusingestimatorsisimportanttoyieldmoreaccurate,robust
and reliable integrated systems. The key problem of any integrated navigation system is how to
achieveanoptimalestimationforposition,velocity,attitudesandstateofinterestparameters.
Multi-SensorDataFusion(MSDF)isdefinedastheacquisitionandprocessingofdatacollectedby
differentsensors.TherearedifferentMSDFtechniques,rangingfromKalmanfilteringmethodstothose
usedinsoftcomputingtechnology,whilesomeworkhasalsobeendoneonhybridarchitectures[6].
Theneedformultiplemodelarchitecturesemergesinnumerousapplicationswhereframework
changesoralternativemodelsaremoresuitabletodepictaparticularissue. Casesofsuchissuesare
faultdiagnosis,linearizationofnon-linearframeworkswiththeendgoalofcontrol,reconfiguration,
andsoon. Multi-modestateestimationframeworksturnouttobemoretroublesomethanstandard
single-modelestimationsandextensiveresearchhasfocusedonthismatter[7].
Multiple-modelestimationtechniquesforthemostpartexpectthattheactualmodelhasaplace
withinafinitearrangementofmodels.Apriorieventprobabilityforeverymodeisknownandwhat’s
more,thesystemequationsandmeasurementequationsaregenerallycorruptedbynoises. Inaddition,
the change starting with one framework mode then onto the next one is frequently depicted by
a Markov chain method. The deduction of the optimal estimator is troublesome regardless of the
possibilitythatthenoisesarethoughttobeindependentlyGaussiandistributed.
Asaresult,variousapproximatetechniqueshavebeendeveloped. Mosttechniquesdependon
theresidualgenerationexecutedbyabankoffilters(eachimplementedinaccordwithaparticular
model) and a hypothesis testing algorithm which is used to processes the residuals to decide on
aconditioned-modelestimateofstate. Theresidualsgiveanassessmentofhowneareachofthefilter
modelsistotheactualmodel.Suchtechniquesareforthemostpartalludedtoasmultiple-model(MM)
estimationapproaches[8].Upgradeshavebeenintroducedbyexchangingthecomputedstatistics
betweenthedifferentfiltersofthebank[9](thisisnormallydesignatedasanInteractingMultiple
Model(IMM)technique). IMMtechniquesarewidelyappliedintargettracking[10]. Numeroususes
ofthesetechniquesaresurveyedintheliterature(see,amongothers,[11]).
Thecomputationalcomplexitycostofanarrangementofparallelfiltersmaypreventonefrom
executingthemonlineforahighordersystemwithalargenumberofstatevariables,andalgorithms
havebeenproposedtodiminishtheamountofcalculationneeded.Aconceivablestrategytoaddress
thisdisadvantagecompriseschoosingthenumberofresidualgenerators(thusthecomputationalcost
isassigned)andafterthatanadaptivetuningofeachfiltercanbeintroduced[12]. Anotherstrategyis
alludedtoasmultiple-modelestimationwithvariablestructureanddependsontheusageofavariable
arrangementofadaptablemodels[13]. Expansionstoamorebroadclassofhybridframeworkhave
beenreported[14,15].
Sensors2017,17,887 3of19
Moreformalstrategiestomanageamultiple-modelestimationissuehavebeenintroducedas
oflateandprovideconvergenceresults. Theseestimationissuesmightbesolvedutilizinglimited
dimensionalfilters,aspointedoutin[16]. Variousalgorithmsareintroducedin[17]anddependon
differentBayesiancostequations. Astrategytomanagenon-Gaussiannoisesisintroducedin[18].
Heretheconsiderationisontheissueofreproducingthestateofadynamicframeworkbased
onaswitchingenvironmentingeneral. Thegoalistobuildupanestimatorreadytotrackthestate
ofthedynamicsystemframeworkbasedonalimitedamountofdataregardingthetechniquescited
above. Morespecifically,wesupposethatdifferentmeasurementinformationisavailablefromvarious
sensors(suchasGPS,IMU,altimeters...etc.) atdifferenttimeinstantsandinaddition,theGPSsignals
canfadeorbecomecorruptedinmanyavailabilityovertimecasesandthereforetheyaredegraded.
Furthermore,weassumenonoiseinfluencesinthemeasurementsanddynamicequationsandthus
anyuncertaintyisjustduetheinitialstateandswitchingbetweenmodes. Suchpresumptionsare
relaxedinthesimulationstogetamorepracticalevaluationcomparison,wherethesystemnoiseand
measurementnoisehavebeenintroduced. Theswitchingmodelisnotconsideredandtheprobability
informationisnotgiven. Thus,adecisionmakerdependsontheevaluationoftheresidualgeneration
that is performed based on the difference between the measured system output for each possible
operatingmodeandtheestimatedsystemoutputfromthecontinuousobserver. Theassessmentofthe
residualsappliesaresettothefiltermodelaccordingtothematchingoperatingmode. Asanoutcome,
thedynamicserroroftheeventualschemeactsasaswitchingsystem. Theprobleminanykindof
switchingobserveramongdifferentmodesistheobservergainselectiontoguaranteethestability
of the estimation dynamic errors and this problem remains difficult to solve and harder than the
design of the standard observer. The observer gain in case of the classical Luenberger observer is
chosenonthebasisofthepoleplacementtechniquewhereconvergenceoftheerrordynamicsmight
occur only if the poles of the dynamics error are placed in the strictly stable region, while in case
oftheobserverinaswitchingenvironmentthenthesolutionisbyseekingthecommonLyapunov
function.ThestabilityisguaranteedforsuchanerrordynamicsbythesolutionofacommonLyapunov
functionintheidealcasewhenthesystemisnotinfluencedbythenoiseandtherightdecisionfor
switchingistakenateverytimeinstant[19]. Linearmatrixinequality(LMI)techniquesareappropriate
todiscoverthissolutionandalsotopermitamoreeffectivedesignoftheobserverstoimprovethe
estimationperformance[20]. Inaddition,anenhancementhasbeenmadebyexecutingaprojection
strategy[21,22]ontheLuenbergerobserverbyincorporatingthelastmeasurementsintotheestimation
processtoupdatethecurrentestimate. Suchanenhancementresultsinhigherestimationperformance
byreducingtheestimationerror. Someofcurrentresearchworksfocusonobserver-basedcontroller
designschemeviausingtherelationofYounginequalityinajudiciousmannertocalculatetheobserver
and controller gains based on improved LMI conditions to ensure the stability of the closed-loop
system[23–25].
In our simulation we have selected the Extended Kalman Filter (EKF) to compare with the
proposedapproachbecauseEKFisasuccessfulandwidespreadestimatorfornonlinearsystemsand
is used in many applications [26] and it is well proven for inertial data fusion [27]. The Extended
KalmanFilteriswidelyusedforthepurposeofsensorfusioninUAVapplications,In[28–31]robust
navigationforfixedwingUnmannedAerialSystems(UASs)basedonintegrationofMSDFarchitecture
usinganExtendedKalmanFilterhasbeenintroduced.AnotherintegratednavigationsystemforUAV
localizationusingthemethodofnonlinearsmoothvariablestructurefilterispresentedandcompared
with an Extended Kalman Filter in [32]. Moreover, in [27,33] the authors evaluated the Extended
KalmanFilterforinertialdatafusion-basednavigationlocalizationformobilerobots. EKFhasbeen
mostwidelyusedfortheapplicationofmulti-sensorfusion,eitherinaself-sufficient,orinamultiple
modelmanner. ThemainreasonbehindthewidespreaduseofEKFisthelowercomputationalcost
andsimplicityindealingwithnonlinearsystems. Manyresearchershavemodifiedthemultiplemodel
approachusingEKFasperthesystemdesignandrequirements. In[34]amultiplemodelapproach
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usingEKFisdescribedforcollisionavoidanceandpositionestimation,andasimilarapproachfor
multi-targetstateestimationispresentedin[35].
Thearticleorganizationisasfollows: multi-modeestimationbasedonaswitchingobserverhas
beendescribedinSection2,Thenfeasiblesolutionsforthestabilityandperformancearediscussedin
Sections3and4,wherethestabilityanalysisoftheestimationerrorhasbeenaddressedinSection3
whiletheperformanceenhancement-basedprojectionconsiderationhasbeenaddressedinSection4.
Finally,inSection5,thesimulationframeworkandresults,discussionandacomparisonaregivento
showtheeffectoftheproposedmethod.
2. Multi-ModeEstimation-BasedSwitchingObserver
Letadiscretetimelinearsystembedescribedby:
x(k+1) = A x(k)+B u(k)+w ,
σ(k) σ(k) k
(1)
y(k) =C x(k)+v
σ(k) k
wherex(k)isthestatevector,u(k)istheinputvector,y(k)isthemeasurementvector,w isprocess
k
noise,v ismeasurementnoiseandσ(k)isafunctionofswitchingsignalwhichmapsbetweentheindex
k
ofthecurrenttimestepamongasetofindices{1,2,..,k}. Everyindexfromtheseindicesiscompatible
with various system and measurement equation configuration models, i.e., {A , B , C } ∈
σ(k) σ(k) σ(k)
{(A , B , C ),(A , B , C )... (A , B , C )}. Wesupposethepairs(A, C),i =1, 2,.., kareobservable,
1 1 1 2 2 2 k k k i i
andalsopresumethattheswitchingsequenceisarbitraryandnoaprioriinformationontheswitching
probabilityfromonemodeintoanotheroneisrequired. AswitchingLuenbergerobserverfor(1)is
asfollows:
xˆ(k+1) = A xˆ(k)+B u(k)+L r , k =1,2,... (2)
σ(k) σ(k) σ(k) σ(k)
whereL andr aretheobservergainmatrixandtheresidualatthetimek,respectively.
σ(k) σ(k)
We suppose that the different measurement information is available from various sensors at
differentdiscretetimeinstantsandinaddition,someofthesesensorsmaybenotavailableatdifferent
time instants, such as the GPS signals that can fade or become corrupted in many cases therefore
degradingtheiravailabilityandasaconsequence,weassumethatthemeasurementinformationfrom
differentsensorscanbegroupedintoamulti-measurementmodetoensuretheinformationavailability
fromdifferentsensorsintheabsenceofGPSinformationtokeepdescribingthesystemdynamicsall
thetime. Thisstageacquiresthemeasurementsateverytimestepandestimatesthesystemmode
based on the evaluation of the residuals. The system output is then generated by this most likely
systemmode. Theproposedhybridnavigationsystemcombinesallthemeasurementsfromallthe
sensorsthroughtwooperationalmodes,designatedasmode1and2.
Anessentialquestionistheassessmentoftheresiduals,i.e.,howtodecideachoiceontheworking
operatingmodeutilizingtheresiduals. Astheerrordynamicsaresupposedtobestable,apossible
assessment technique comprises choosing the mode that relates to the most reduced value of the
residuals. Foreachsystemmode,wecalculatetheresidualofthemeasurementwiththeoutputofthe
correspondingmode. Theresidualisthenusedtodecidetheoperatingmode;suchadecisionrelieson
thecriterionusedtodiscriminateamongthedifferentresiduals,theresultingdesignschemecomes
outtobehavelikeaswitchingsystem. Ageneraldebateonswitchingsystemframeworksisbeyond
thescopeofthisarticle,andforanintroductionreaderscanconsult[19]. Itisworthmentioninghere
thatwedon’tdependonaprobabilitydepictionofstochasticvariablesandeventsingeneral,and
thenatraditionalBayesianwaytodealwiththisissueisprevented. Inthiscontext,afeasiblesolution
comesouttobethatofchoosingthemodelwiththesmallestmatchingresiduals.
Generally,residualsaregeneratedfromthedifferencebetweenthemeasuredsystemoutputy(k)
foreachmodeandtheestimatedsystemoutputyˆ(k):
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r (k)y (k)−yˆ (k) = y (k)−Cxˆ(k),i =1, 2,..., N (3)
i i i i i
Theswitchingbetweenthetwomodes,dependingonthedecisionfunctioncanbeexpressedas:
(cid:40)
if RMS(cid:107)r (t)(cid:107) ≤ (cid:101)andt = t switchtomodeone
decisionfunction= 1 1 (4)
otherwiseRMS(cid:107)r (t)(cid:107) ≤ (cid:101)ort (cid:54)= t switchtomodetwo
2 1
where(cid:101)isarootmeansquareoftheresidualthresholdandt istheinstantatwhichallmeasurements
1
areavailableincludingtheGPS.ThedecisionfunctionnotonlyjudgestheavailabilityofGPS,but
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alsojudgestheaccuracyofthesensors’measurementswheretheswitchingbetweenthetwomodesis
basedonthemostreducedresidualwhichleadstothemostlikelymode. Thistechniqueacquiresthe
two modes is based on the most reduced residual which leads to the most likely mode. This technique
measurementsateverytimestepandestimatesthesystemmode. Asaconsequence,theswitching
acquires the measurements at every time step and estimates the system mode. As a consequence, the
observercanbedescribedbythefollowingexpression:
switching observer can be described by the following expression:
(cid:16) (cid:17)
(cid:1876)(cid:3548)((cid:1863)xˆ(+k+1)1)==(cid:1827)A(cid:3036) (cid:1876)i(cid:3548)x(ˆ(cid:1863)(k))++(cid:1828)B(cid:3036)i(cid:1873)u(((cid:1863)k))++L(cid:1838)i(cid:3036) (cid:3435)y(cid:1877)i(cid:3036)((k(cid:1863)))−−C(cid:1829)ixˆ(cid:3036)((cid:1876)(cid:3548)k()(cid:1863)),(cid:3439)i,(cid:1861)==1,12,,2.,.…..,., (5)(5)
where, (cid:1876)(cid:3548)(0) = (cid:1876)(0) is chosen a priori, (cid:1877) ((cid:1863)) is the measurement vector of the (cid:1861)th mode and (cid:1838) is
where, xˆ(0) = x(0)ischosenapriori,y(cid:3036) (k)isthemeasurementvectoroftheithmodeand L isthe(cid:3036)
i i
the related observer gain. The multi-measurement mode estimation-based switching observer
relatedobservergain. Themulti-measurementmodeestimation-basedswitchingobserverschemeis
schemdee pisi cdteedpiinctFeidg uinre F1i.gure 1.

Figure 1. Scheme of the multi-measurement mode estimation-based switching observer.
Figure1.Schemeofthemulti-measurementmodeestimation-basedswitchingobserver.
The essential problems emerging in the design of such an observer are related to stability,
The essential problems emerging in the design of such an observer are related to stability,
optimization, and performance. These problems will be addressed in the next sections and possible
optimization,andperformance. Theseproblemswillbeaddressedinthenextsectionsandpossible
solutiosonlsu twioinlls bwei lplrboepporosepdo.s ed.
3. Stability Analysis of the Estimation Error
The attention in the first place is placed on the stability of the estimation dynamics error (cid:1857)((cid:1863)) =
(cid:1876)((cid:1863))−(cid:1876)(cid:3548)((cid:1863)). Such error dynamics behave as a switching dynamic system, consequently the stability
is guaranteed for such an error dynamics case by the solution of a common Lyapunov function [36].
By taking into consideration the system in Equation (1) and with the assumption that the pairs ((cid:1827) ,(cid:1829))
(cid:3036) (cid:3036)
are observable, and if matrix (cid:1842) exists as a solution of the following Lyapunov inequality:
((cid:1827) −(cid:1838) (cid:1829))(cid:3021)(cid:1842)((cid:1827) −(cid:1838) (cid:1829))−(cid:1842) < 0,(cid:1861) = 1,2,…(cid:1840) (6)
(cid:3036) (cid:3036) (cid:3036) (cid:3036) (cid:3036) (cid:3036)
where, matrix (cid:1842) is a symmetric positive definite then the observer in Equation (2) includes a
dynamic estimation error that converges exponentially asymptotically to zero [36]. To prove this
result then, we recall the error dynamics equation as follows:
(cid:1857)((cid:1863)+1) = (cid:1876)((cid:1863)+1)−(cid:1876)(cid:3548)((cid:1863)+1) (7)
From the system Equation (1) and from the observer Equation (2) we can substitute in the error
dynamic Equation (7) then:
(cid:1857)((cid:1863)+1) = (cid:1827) (cid:1876)((cid:1863))+(cid:1828) (cid:1873)((cid:1863))−(cid:1827) (cid:1876)(cid:3548)((cid:1863))−(cid:1828) (cid:1873)((cid:1863))−(cid:1838) (cid:1870) (8)
(cid:3097)((cid:3038)) (cid:3097)((cid:3038)) (cid:3097)((cid:3038)) (cid:3097)((cid:3038)) (cid:3097)((cid:3038)) (cid:3097)((cid:3038))
where, the residual (cid:1870) = (cid:1877) ((cid:1863))− (cid:1829) (cid:1876)(cid:3548)((cid:1863)) then:
(cid:3097)((cid:3038)) (cid:3097)((cid:3038)) (cid:3097)((cid:3038))
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3. StabilityAnalysisoftheEstimationError
The attention in the first place is placed on the stability of the estimation dynamics error
e(k) = x(k)−xˆ(k). Such error dynamics behave as a switching dynamic system, consequently
thestabilityisguaranteedforsuchanerrordynamicscasebythesolutionofacommonLyapunov
function[36]. BytakingintoconsiderationthesysteminEquation(1)andwiththeassumptionthatthe
pairs(A,C)areobservable,andifmatrixPexistsasasolutionofthefollowingLyapunovinequality:
i i
(A −L C)TP(A −L C)−P <0, i =1,2,...N (6)
i i i i i i
where,matrixPisasymmetricpositivedefinitethentheobserverinEquation(2)includesadynamic
estimationerrorthatconvergesexponentiallyasymptoticallytozero[36]. Toprovethisresultthen,we
recalltheerrordynamicsequationasfollows:
e(k+1) = x(k+1)−xˆ(k+1) (7)
FromthesystemEquation(1)andfromtheobserverEquation(2)wecansubstituteintheerror
dynamicEquation(7)then:
e(k+1) = A x(k)+B u(k)−A xˆ(k)−B u(k)−L r (8)
σ(k) σ(k) σ(k) σ(k) σ(k) σ(k)
where,theresidualr = y (k)− C xˆ(k)then:
σ(k) σ(k) σ(k)
(cid:16) (cid:17)
e(k+1) = A x(k)−A xˆ(k)−L y (k)− C xˆ(k)
σ(k) σ(k) σ(k) σ(k) σ(k)
(cid:16) (cid:17)
e(k+1) = A (x(k)− xˆ(k))−L C x(k)− C xˆ(k)
σ(k) σ(k) σ(k) σ(k)
(cid:16) (cid:17)
e(k+1) = Aσ(k)(x(k)− xˆ(k))−Lσ(k)Cσ(k) x(k)− xˆ(k) (9)
(cid:16) (cid:17)(cid:16) (cid:17)
e(k+1) = A −L C x(k)− xˆ(k)
σ(k) σ(k) σ(k)
(cid:16) (cid:17)
e(k+1) = A −L C e(k)
σ(k) σ(k) σ(k)
WiththeconsiderationoftheLyapunovfunctionisVk = ekTPek,wegetVk+1 ≤ Vk,if:
(cid:16) (cid:17)T (cid:16) (cid:17)
A −L C P A −L C −P <0, (10)
σ(k) σ(k) σ(k) σ(k) σ(k) σ(k)
wherePisapositivesymmetricdefinitematrix,thentheEquation(6)canbederivedeasily.
Theassumptionof (A, C) areobservableisanimportantfeaturetoensuretheinequalityin
i i
Equation(6)allowsasolutionofasymmetricpositivematrixP,howeverthepresenceofsuchamatrix
Pisrequiredforachievingtheinequalitieswhichguaranteethestability. Therefore,thegoalistoget
thesolutionofthefollowingobservergainL problem,suchthatamatrixPexistsforthesolutionof
i
Lyapunovinequalities:
(A −L C)TP(A −L C)−P <0, i =1,2,...N (11)
i i i i i i
However,theaboveproblemisdifficulttosolveinthisform,so,itisappropriatetochoosethe
matrices L and Psimultaneously. Thereforethisproblemcanbechangedtoanothersimplerform
i
based on the relation of the Schur complement (for proof the reader is referred to [36]), thus the
Lyapunovinequalityintheaforementionedproblemisequivalentto[20]:
(cid:32) (cid:33)
P (PA − YC)
i i i >0, i =1,2,...N (12)
(PA − YC)T P
i i i
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Now, we can solve the Lyapunov inequality in this form by means of LMI techniques.
LMI techniques permit one to solve such problems using convex programming [20,37]. In order
toefficientlyperformsuchanoptimizationbylinearmatrixinequalities, wehaveusedtheCVX3
toolboxasaconvexprogrammingalgorithmtofindthematricesPandY andthenfindtheobserver
i
gainL asfollows:
i
L = P−1Y, i =1,2,...N (13)
i i
4. PerformanceEnhancementbyUsingLuenbergerObserverBasedProjection
TheclassicLuenbergerobserverinEquation(2)estimatesthestatevectorxˆ(k+1)attime(k+1)
byusingtheavailablemeasurementsy(k)attimek. Inthissectionweaimtodiscussanapproachto
computetheestimatestatevectorxˆ(k+1)attime(k+1)byusingtheavailablemeasurementvector
y(k+1) at time (k+1) just like a standard Kalman filter. We denote xˆ+(k) as the state estimates
at time k incorporating the output at time k as shown in Figure 2. Such an approach is based on
theprojectionmethodandtakesintoconsiderationthelastmeasurementtoenhancetheestimation
performance(forfurtherinformationaboutsuchanapproachthereaderisreferredto[36]). More
specifically,thisestimationtechniquecanbecharacterizedasfollows[36,38]:
(cid:16) (cid:17)
xˆ(k+1) = A xˆ+(k)+B u(k)+L y(k)− C xˆ+(k) (14)
σ(k) σ(k) σ(k) σ(k)
xˆ+(k+1) = xˆ(k+1)+P−1CT (C P−1CT )−1 (cid:16)y(k+1)− C xˆ(k+1)(cid:17) (15)
σ(k+1) σ(k+1) σ(k+1) σ(k+1)
where xˆ+(0) = x(0)ischosenapriori, L and Paretheobservergainandasymmetricpositive
σ(k)
definitematrixattimek,respectively.
Sensors 2017, 17, 887 7 of 18

Figure 2. State estimation based on the last measurement.
Figure2.Stateestimationbasedonthelastmeasurement.
The estimation error before incorporating (cid:1877)((cid:1863)+1) is as follows:
Theestimationerrorbeforeincorporatingy(k+1)isasfollows:
(cid:1857)((cid:1863)+1)= (cid:1876)((cid:1863)+1)−(cid:1876)(cid:3548)((cid:1863)+1)=(cid:16)((cid:1827)(cid:3097)((cid:3038))−(cid:1838)(cid:3097)((cid:3038))(cid:1829)(cid:3097)((cid:17)(cid:3038)))(cid:1857)(cid:2878)(k) (16)
e(k+1) = x(k+1)−xˆ(k+1) = A −L C e+(k) (16)
where (cid:1857) = (cid:1876)−(cid:1876)(cid:3548) and (cid:1857)(cid:2878) = (cid:1876)−(cid:1876)(cid:3548)(cid:2878), then the estimatioσn( k)errorσ (ak)fteσr( ki)ncorporating (cid:1877)((cid:1863)+1) is as
folwlohwesr:e e = x−xˆande+ = x−xˆ+,thentheestimationerrorafterincorporatingy(k+1)isasfollows:
e+(k+(cid:1857)(cid:2878)1((cid:1863)) =+1x)(k=+(cid:1876)(1(cid:1863))+−1xˆ)+−(k(cid:1876)(cid:3548)+(cid:2878)(1(cid:1863))+=1)(cid:104)I=−(cid:4670)(cid:1835)P−−1(cid:1842)C(cid:2879)T(cid:2869)(cid:1829)(cid:3097)(cid:3021)((cid:3038)(cid:2878)(cid:2869)()C(cid:3435)(cid:1829)(cid:3097)((cid:3038)(cid:2878)(cid:2869)P)−(cid:1842)1(cid:2879)C(cid:2869)(cid:1829)T(cid:3097)(cid:3021)((cid:3038)(cid:2878)(cid:2869)))(cid:3439)−(cid:2879)1(cid:2869)C(cid:1829)(cid:3097)((cid:3038)(cid:2878)(cid:2869))(cid:4671)(cid:105)(cid:1857)e(((cid:1863)k++11)) (1(71)7)
σ(k+1) σ(k+1) σ(k+1) σ(k+1)
Considering the Lyapunov functions are (cid:1848) = (cid:1857)(cid:2904)(cid:1842)(cid:1857) and (cid:1848)(cid:2878) =(cid:1857)(cid:2878)(cid:2904)(cid:1842)(cid:1857)(cid:2878), for stability we want
(cid:3038) (cid:3038) (cid:3038) (cid:3038) (cid:3038) (cid:3038)
to showC tohnasti d(cid:1857)e(cid:2878)ri(n(cid:1863)g) tchoenLvyearpguens otov zfuenroct aiosnysmapretoVtikc=allye TkbPye kparonvdinVgk+ =(cid:1848)(cid:3038)(cid:2878)e k+isT Pdeek+c,refoarsisntagb iinli ty (cid:1863)w. Tehwenan, tit is
suftfoicsiehnotw toth sahtoew+( kth)acto n (cid:1848)v(cid:3038)(cid:2878)e(cid:2878)r(cid:2869)ge<s t(cid:1848)o(cid:3038)(cid:2878)ze(cid:2869)r ,o sainsycem ipt tiost iocablvlyiobuys pfrroomvin EgqVua+tiiosnd e(1cr6e)a tshinagt i (cid:1848)n(cid:3038)k(cid:2878).(cid:2869)T<h e (cid:1848)n(cid:3038),(cid:2878)i tifis the
k
Lyasupfufincoievn tintoeqsuhaolwitiethsa atrVe k+s+a1ti<sfieVdk.+ L1e,ts’isn ccoenitsiidseorb: viousfromEquation(16)that Vk+1 < Vk+ ifthe
Lyapunovinequalitiesaresatisfied. Let’sconsider:
(cid:1848)(cid:2878) =(cid:1857)(cid:2878)(cid:3021) (cid:1842)(cid:1857)(cid:2878)
(cid:3038)(cid:2878)(cid:2869) (cid:3038)(cid:2878)(cid:2869) (cid:3038)(cid:2878)(cid:2869)
= (cid:4674)(cid:1857)((cid:1863)+1)−(cid:1842)(cid:2879)(cid:2869)(cid:1829)(cid:3021) (cid:3435)(cid:1829) (cid:1842)(cid:2879)(cid:2869)(cid:1829)(cid:3021) (cid:3439)(cid:2879)(cid:2869)(cid:1829) (cid:1857)((cid:1863)+1)(cid:4675)(cid:3021)(cid:1842)(cid:4674)(cid:1857)((cid:1863)+1)
(cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869))
−(cid:1842)(cid:2879)(cid:2869)(cid:1829)(cid:3021) (cid:3435)(cid:1829) (cid:1842)(cid:2879)(cid:2869)(cid:1829)(cid:3021) (cid:3439)(cid:2879)(cid:2869)(cid:1829) (cid:1857)((cid:1863)+1)(cid:4675)
(cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869))
(18)
= (cid:1848) − (cid:1857)(cid:3021)((cid:1863)+1) (cid:1829)(cid:3021) (cid:3435)(cid:1829) (cid:1842)(cid:2879)(cid:2869)(cid:1829)(cid:3021) (cid:3439)(cid:2879)(cid:2869)(cid:1829) (cid:1857)((cid:1863)+1)
(cid:3038)(cid:2878)(cid:2869) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869)) (cid:3097)((cid:3038)(cid:2878)(cid:2869))
< (cid:1848)
(cid:3038)(cid:2878)(cid:2869)
In the next section, the improvements of the estimation performance which were introduced via
the Luenberger observer-based projection consideration can appear clearly in the simulation results.
5. Simulation Framework and Results
In this section we present our simulation framework and results in order to validate the
proposed multi-mode estimation by using a Luenberger observer after a projection (LOAP)-based
LMI approach for the small fixed wing unmanned aerial vehicle localization problem.
The nonlinear state dynamics and measurement model which describe the dynamics of the body
of the small fixed wing unmanned aerial vehicle can be expressed by the following equations:
(cid:1876)(cid:4662)((cid:1872))=f((cid:1876)((cid:1872)),(cid:1873)((cid:1872)),(cid:1872)),
(19)
(cid:1877)((cid:1872))=ℎ((cid:1876)((cid:1872)),(cid:1873)((cid:1872)),(cid:1872))
where (cid:1876) is the state vector, which consists of attitude in terms of Euler angles, position and velocity
expressed in navigation frame, gyros and accelerometers biases, respectively, defined as follows:
(cid:1876) =(cid:4670)(cid:2038) (cid:2016) (cid:2032) (cid:1842)(cid:3041)(cid:1842)(cid:3032) ℎ(cid:1848)(cid:3041)(cid:1848)(cid:3032) (cid:1848)(cid:3031) (cid:1854)(cid:3104)(cid:3299) (cid:1854)(cid:3104)(cid:3300) (cid:1854)(cid:3104)(cid:3301) (cid:1854)(cid:3028)(cid:3299) (cid:1854)(cid:3028)(cid:3300) (cid:1854)(cid:3028)(cid:3301) (cid:4671) (20)
Furthermore, (cid:1873) is the system input which in our case is the IMU measurements, expressed as
follows:
(cid:1873) =(cid:4670)(cid:2033)(cid:3051)(cid:2033)(cid:3052)(cid:2033)(cid:3053)(cid:1858)(cid:3051)(cid:1858)(cid:3052)(cid:1858)(cid:3053)(cid:4671), (21)
Nonlinear navigation state model can be written as:
Sensors2017,17,887 8of19
V+ = e+T Pe+
k+1 k+1 k+1
= (cid:104)e(k+1)−P−1CT (C P−1CT )−1C e(k+1)(cid:105)TP(cid:104)e(k+1)
σ(k+1) σ(k+1) σ(k+1) σ(k+1)
−P−1CT (C P−1CT )−1C e(k+1)(cid:105) (18)
σ(k+1) σ(k+1) σ(k+1) σ(k+1)
= Vk+1− eT(k+1)CσT(k+1)(Cσ(k+1)P−1CσT(k+1))−1Cσ(k+1)e(k+1)
< V
k+1
Inthenextsection,theimprovementsoftheestimationperformancewhichwereintroducedvia
theLuenbergerobserver-basedprojectionconsiderationcanappearclearlyinthesimulationresults.
5. SimulationFrameworkandResults
Inthissectionwepresentoursimulationframeworkandresultsinordertovalidatetheproposed
multi-modeestimationbyusingaLuenbergerobserverafteraprojection(LOAP)-basedLMIapproach
forthesmallfixedwingunmannedaerialvehiclelocalizationproblem.
Thenonlinearstatedynamicsandmeasurementmodelwhichdescribethedynamicsofthebody
ofthesmallfixedwingunmannedaerialvehiclecanbeexpressedbythefollowingequations:
. (cid:16) (cid:17)
x(t) =f x(t),u(t),t ,
(cid:16) (cid:17) (19)
y(t) = h x(t),u(t),t
wherexisthestatevector,whichconsistsofattitudeintermsofEulerangles,positionandvelocity
expressedinnavigationframe,gyrosandaccelerometersbiases,respectively,definedasfollows:
(cid:104) (cid:105)
x = φθ ψP P hV V V b b b b b b (20)
n e n e d ωx ωy ωz ax ay az
Furthermore, u is the system input which in our case is the IMU measurements, expressed
asfollows:
(cid:2) (cid:3)
u = ω ω ω f f f , (21)
x y z x y z
Nonlinearnavigationstatemodelcanbewrittenas:
    
1 sinφtanθ cosφtanθ ωx,gyro−bωx
  0 cosθ −sinφ  ωy,gyro−bωy  
 0 sinφsecθ cosφsecθ ωz,gyro−bωz 
 
 Vn 
 
 V 
 e 
 −V 
 d 
   
 fx,accel−bax 
f(x,u) =  Cn f −b   (22)
 b y,accel ay  
 f −b 
 z,accel az 
 
 0 
 
 0 
 
 0 
 
 0 
 
 
 0 
0
Sensors 2017, 17, 887 8 of 18
(cid:1741) 1 sin(cid:2038)tan(cid:2016) cos(cid:2038)tan(cid:2016) (cid:2033)(cid:3051),(cid:3034)(cid:3052)(cid:3045)(cid:3042)−(cid:1854)(cid:3104)(cid:3299) (cid:1744)
(cid:1742)(cid:3429)0 cos(cid:2016) −sin(cid:2038) (cid:3433)(cid:4686)(cid:2033)(cid:3052),(cid:3034)(cid:3052)(cid:3045)(cid:3042)−(cid:1854)(cid:3104)(cid:3300)(cid:4687)(cid:1745)
(cid:1742) 0 sin(cid:2038)sec(cid:2016) cos(cid:2038)sec(cid:2016) (cid:2033) −(cid:1854) (cid:1745)
(cid:3053),(cid:3034)(cid:3052)(cid:3045)(cid:3042) (cid:3104)(cid:3301)
(cid:1742) (cid:1848) (cid:1745)
(cid:3041)
(cid:1742) (cid:1745)
(cid:1848)
(cid:3032)
(cid:1742) (cid:1745)
−(cid:1848)
(cid:1742) (cid:3031) (cid:1745)
(cid:1858) −(cid:1854)
f((cid:1876),(cid:1873))=(cid:1742) (cid:3051),(cid:3028)(cid:3030)(cid:3030)(cid:3032)(cid:3039) (cid:3028)(cid:3299) (cid:1745) (22)
(cid:1742) (cid:1829)(cid:3029)(cid:3041) (cid:4686)(cid:1858)(cid:3052),(cid:3028)(cid:3030)(cid:3030)(cid:3032)(cid:3039)−(cid:1854)(cid:3028)(cid:3300)(cid:4687) (cid:1745)
(cid:1742) (cid:1858) −(cid:1854) (cid:1745)
(cid:1742) (cid:3053),(cid:3028)(cid:3030)(cid:3030)(cid:3032)(cid:3039) (cid:3028)(cid:3301) (cid:1745)
0
(cid:1742) (cid:1745)
0
Sensors2017,17,887 (cid:1742) 0 (cid:1745) 9of19
(cid:1742) 0 (cid:1745)
(cid:1742) 0 (cid:1745)
whereCnisthetransformationma(cid:1743)trixwhichrepresents0arotationfrombodyto(cid:1746)navigationcoordinates
b
(cid:104) (cid:105)T
awnhdebree ca(cid:1829)u(cid:2924)s eisi ttihsea ntroarnthsofonromrmataiolnm amtraitxritxh ewnhoincehf rroemprietssecnhtas raac treoritsattiicosni sfCronm= boCdby two hneareviCgbatciaonn
(cid:2912) b n n
bcoeocrhdairnaactteesr iaznedd abse:cause it is an orthonormal matrix then one from its characteristics is (cid:1829)(cid:2924) =(cid:4670)(cid:1829)(cid:1854)(cid:4671)(cid:2904)
(cid:2912) (cid:1866)
where (cid:1829)(cid:1854) can be characterized as:
(cid:1866) 
cos(θ)cos(ψ) cos(θ)sin(ψ) −sin(θ)
Cb = sin(φ)sin(cθo)s(c(cid:2016)o)s(cψos)(−(cid:2032))sin(ψ)cos(φ) sin(φ)sinc(oθs)(s(cid:2016)i)ns(iψn)((cid:2032)+)cos(ψ)cos(φ) s−ins(inφ()(cid:2016)c)os(θ)  (23)
n  
(cid:1829)(cid:3041)(cid:3029) =(cid:4686)sisni(n(cid:2038)(θ))sicno(s(cid:2016)()φc)ocso(s(cid:2032)()ψ−)+sins(in(cid:2032)()ψc)ossi(n(cid:2038)()φ)sins(in(cid:2038)()ψs)ins(in(cid:2016)()θs)inc(o(cid:2032)s)(φ+)c−osc(o(cid:2032)s)(ψco)ss(i(cid:2038)n()φ)sinc(o(cid:2038)s)(φco)sc(o(cid:2016)s)((cid:4687)θ ) (23)
sin((cid:2016))cos((cid:2038))cos((cid:2032))+sin((cid:2032))sin((cid:2038)) sin((cid:2032))sin((cid:2016))cos((cid:2038))−cos((cid:2032))sin((cid:2038)) cos((cid:2038))cos((cid:2016))
TThhee bbaassiicc ffrraammee ffoorr tthhee iinneerrttiiaall sseennssoorrss uusseedd iinn tthhee ssiimmuullaattiioonn iiss tthhee bbooddyy--ffiixxeedd ccoooorrddiinnaattee ffrraammee
((bbooddyy xx––yy––zz ccoooorrddiinnaattee)) wwhheerree,, xx--aaxxiiss ppooiinntt iinn tthhee ffoorrwwaarrdd ddiirreeccttiioonn,, yy--aaxxiiss ppooiinntt ttoo rriigghhtt,, aanndd zz--aaxxiiss
ppooiinnttss ddoowwnnwwaarrddss hhoowweevveerr,,t thheel looccaallg geeooddeettiiccc coooorrddininaatteessa arreeN Noorrthth-E-Eaasst-tD-Doowwnn( N(NEEDD).).
IInn tthhee ssiimmuullaattiioonn wwee aassssuummee tthhaatt vvaarriioouuss mmeeaassuurreemmeenntt iinnffoorrmmaattiioonn ddaattaa iiss aavvaaiillaabbllee ffrroomm
ddiiffffeerreenntt sseennssoorrss aatt vvaarriioouuss ddiiffffeerreenntt ttiimmee iinnssttaannttss aanndd ffuurrtthheerrmmoorree,, ssoommee ooff tthheessee sseennssoorrss mmiigghhtt bbee
nnoott aavvaaiillaabbllee aatt vvaarriioouuss mmoommeennttss ooff ttiimmee,, ffoorr eexxaammppllee,, tthhee GGPPSS ssiiggnnaallss ccaann ffaaddee oorr bbee ccoorrrruupptteedd iinn
mmaannyy ssiittuuaattiioonnssc coonnsseeqquueenntltylyd degergardadinigngth tehea vaavilaaiblaibliitlyitayn adnads aasn aonu otcuotmcoemoef tohfi sthwise wsuep spuopspeothsea ttthhaet
mtheea smureeamsuernetminefnotr minaftoiormndataitoanf rodmatav afrrioomus sveanrsioourss issetonsboergs aitsh etroe dbien tgoattwheor-emdo dinetsot otwguoa-rmanotdeeest hteo
agvuaairlaabnitleitey tohfet haevmaielaabsuilriteym oenf ttdhae tamferoamsurveamrioenuts sdeantsao rfsroinmt hveaaribosuens cseeonfstohres iinnf otrhme aatibosnenfrcoem osfo mthee
oinfftohremmattoiokne ferpomde sscormibein ogf tthheemsy stote kmeedpy ndaemscircisbainllgt htheet ismyset.eFmig duyren3amshiocsw aslla tghrea ptihmicea. lFriegpurrees e3n sthaotiwons
fao grrsawpihtcichailn rgebperetwseenetnattiwono fmoro sdwesi.tching between two modes.

FFiigguurree 33.. GGrraapphhiiccaall rreepprreesseennttaattiioonn ffoorr sswwiittcchhiinngg bbeettwweeeenn ttwwoo mmooddeess..
In mode one, the observations for velocity and position are obtained from GPS, while in mode
Inmodeone,theobservationsforvelocityandpositionareobtainedfromGPS,whileinmode
two, the position and velocity in the north and east directions are acquired from the IMU and the
two,thepositionandvelocityinthenorthandeastdirectionsareacquiredfromtheIMUandtheheight
height from a barometric pressure sensor and finally the down velocity from the fusion between
fromabarometricpressuresensorandfinallythedownvelocityfromthefusionbetweenbarometer
barometer and vertical accelerometer measurements. The vertical velocity can be gathered on the
andverticalaccelerometermeasurements. Theverticalvelocitycanbegatheredonthebasisonthe
basis on the available vertical acceleration and altitude measurements [39].
availableverticalaccelerationandaltitudemeasurements[39].
The estimation of the continuous states is performed by utilizing a Luenberger observer based
TheestimationofthecontinuousstatesisperformedbyutilizingaLuenbergerobserverbasedon
on geometrical considerations, as portrayed in Section 4, and no a priori statistical information data
geometricalconsiderations,asportrayedinSection4,andnoaprioristatisticalinformationdataabout
about system and measurement noises and system uncertainties is required. Furthermore, we assume
systemandmeasurementnoisesandsystemuncertaintiesisrequired. Furthermore,weassumeno
noiseinfluencesonthemeasurementanddynamicequations,thusanyuncertaintyisjustbecauseof

theinitialstateandswitchingbetweenmodes. Suchpresumptionsarerelaxedinthesimulationstoget
afairevaluationcomparison,wherethesystemnoiseandmeasurementnoisehavebeenintroduced.
TheinitialstatesarechosenaprioriwhilethesystemprocesscovariancematrixPandtheobserver
gainmatrixLareselectedsimultaneouslybysolvingtheLyapunovinequalitybymeansoftheLMI
approach. NoaprioriinformationabouttheprocesscovariancematrixPisrequired,butthematrix
P as a positive symmetric definite matrix is chosen as a solution for the Lyapunov inequality by
means of the LMI technique. The existence of the positive definite matrix P as a solution of the
Lyapunovinequalityensuresthedynamicestimationerrorconvergesexponentiallytozero. Forsimple
denotationwecalltheproposedcontinuousobserveraLuenbergerobserverafterprojection(LOAP).
Wetestedtheproposedprojectionfilteronaclassofnonlinearsystemandcomparedtheresultswith
Sensors2017,17,887 10of19
theExtendedKalmanFilter(EKF).TheinitializationofthestateestimateforboththeLOAPandEKF
isthesameandbasedonutilizingthefirstmeasurementofGPStoprovideaninitialpositionand
velocity state estimate as follows 0.31 m, 2.23 m and 160.6 m for the initial north, east and height
positionsstateestimate,respectively,and20.8m/s,5.9m/sand−1.1m/sfortheinitialnorth,east
and down velocities state estimate, while the initialization of the other state estimates such as the
Eulerangles,gyrosbiasesandaccelerometersbiaseshavebeensettozero. Althoughweinitialized
thepositionandvelocitystateestimatefortheEKF,weassumetheEKFfilterstartedwithlargeinitial
uncertaintiescomparedtotheexpectedmagnitudesofthestatestobuildtheinitialprocesscovariance
matrix P withdeviations equal to0.5 rad, 100m and 10m/s forthe initialattitudes, position and
velocityuncertainties,respectively,alsoweuseaprioriinformationabouttheinitialuncertaintiesin
thegyroandaccelerometerbiasesequalto0.01rad/sand0.1m/s2,respectively. Also,wedesignthe
processnoisecovariancematrixforEKFbasedonhowmuchweexpectthedeviationofthereality
fromthemodelofthevehiclemotiontobe. Althoughnoaprioristatisticalinformationisrequired
fortheLOAP,weappliedsystemprocessnoisefortheLOAPthesameasEKFwithdeviationsequal
to0.002rad,0.5m,0.2m/sfortheattitude,positionandvelocityprocessnoises,respectively,and
assignedtheprocessnoisesofthegyroandaccelerometersensorsbiasesneartozerobecausewedo
notexpectthemtovaryveryquicklywithdeviationsequalto10−6rad/sand10−6m/s2forthegyro
andaccelerometerbiasesprocessnoise,respectively. Table1showsthetypeofthemeasureddataand
thesamplingrateusedforeachsensorwhileTable2showsacomparisonofthefactorsofthefilters
usedinthisstudy.
Table1.Themeasureddataandsamplingrateusedforeachsensor.
Sensor MeasuredData Frequency
GPS PositionandVelocity 1Hz
Gyro AngularSpeed 10Hz
Accelerometer Acceleration 10Hz
BarometricAltimeter Height 10Hz
Table2.Acomparisonfactorsofthefiltersusedinthisstudy.
APrioriStatistical Designofthe
Observer CovarianceMatrix(P) UpdateRate
InformationaboutNoises GainMatrix
Notrequired(but Selectedasasolution
Selectedbysolvingthe
introducedinthe oftheLyapunov
LOAP Lyapunovinequalityby 10Hz
simulationforfair inequalitybymeansof
meansofLMItechniques.
evaluationcomparison) LMItechniques.
Apriorichosen,then
EKF Required Usingacorrectionmatrix estimatedduringthe 10Hz
estimationprocess
Atrajectoryof100sisusedinthesimulationoftheaforementionedmodel. Figure4providesthe
3DpositiontrajectoryforthetruepositionsandGPSmeasurement.
TheresultsofourproposedapproacharecomparedwithsinglemodeExtendedKalmanFilter
techniqueandthesimulationisdividedintotwosections,thefirstisthecasewheretheGPSsignalis
availableatevery1Hz,whilethesecondisacasewheretheGPSsignalisabsentduringanintervalof
time. Thedataisrecordedandprocessedoffline.