Performance Improvement of Inertial Navigation System by Using Magnetometer with Vehicle ...

12 Pages·2015·2.53 MB·English

Hindawi Publishing Corporation
Journal of Sensors
Volume 2015, Article ID 435062, 11 pages
http://dx.doi.org/10.1155/2015/435062
Research Article
Performance Improvement of Inertial Navigation System by
Using Magnetometer with Vehicle Dynamic Constraints
DaeheeWon,1JongsunAhn,2SangkyungSung,2MoonbeomHeo,3
Sung-HyuckIm,3andYoungJaeLee2
1DepartmentofAerospaceEngineeringScience,UniversityofColoradoatBoulder,Boulder,CO80309,USA
2DepartmentofAerospaceInformationEngineering,KonkukUniversity,Seoul143-701,RepublicofKorea
3DivisionofSatelliteNavigation,KoreaAerospaceResearchInstitute,Daejeon305-806,RepublicofKorea
CorrespondenceshouldbeaddressedtoYoungJaeLee;[email protected]
Received10March2015;Revised23July2015;Accepted27July2015
AcademicEditor:JianhuaTong
Copyright©2015DaeheeWonetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
A navigation algorithm is proposed to increase the inertial navigation performance of a ground vehicle using magnetic
measurementsanddynamicconstraints.Thenavigationsolutionsareestimatedbasedoninertialmeasurementssuchasacceleration
andangularvelocitymeasurements.Toimprovetheinertialnavigationperformance,athree-axismagnetometerisusedtoprovide
theheadingangle,andnonholonomicconstraints(NHCs)areintroducedtoincreasethecorrelationbetweenthevelocityand
theattitudeequation.TheNHCsprovideavelocityfeedbacktotheattitude,whichmakesthenavigationsolutionmorerobust.
Additionally,anacceleration-basedrollandpitchestimationisappliedtodecreasethedriftwhentheaccelerationiswithincertain
boundaries.ThemagnetometerandNHCsarecombinedwithanextendedKalmanfilter.Anexperimentaltestwasconductedto
verifytheproposedmethod,andacomprehensiveanalysisoftheperformanceintermsoftheposition,velocity,andattitudeshowed
thatthenavigationperformancecouldbeimprovedbyusingthemagnetometerandNHCs.Moreover,theproposedmethodcould
improvetheestimationperformancefortheposition,velocity,andattitudewithoutanyadditionalhardwareexceptaninertial
sensorandmagnetometer.Therefore,thismethodwouldbeeffectiveforgroundvehicles,indoornavigation,mobilerobots,vehicle
navigationinurbancanyons,ornavigationinanyglobalnavigationsatellitesystem-deniedenvironment.
1.Introduction to using an inertial navigation system, various applications
haveadoptedamagnetometertocomputeheadings,includ-
Navigationinformationrepresentssomeofthemostimpor- ing mobile robots, unmanned aerial vehicles (UAVs), and
tant data for autonomous applications. Inertial navigation missiles[5–8].
systems(INSs)havebeenusedtoprovidestandalonenaviga- Magnetometers have been used with inertial sensors
tionsolutionsforalongtime,includingposition,velocity,and (accelerometersandgyroscopes)toimprovetheattitudeesti-
attitude information. Almost all autonomous applications mation performance. In addition to improving the attitude
adapt INSs as the base system [1–4]. These can estimate a accuracy,amagnetometeralsoprovidesaslightimprovement
solutionwithoutanyexternaldata,buttheaccuracydecreases in the position accuracy. A navigation equation for the
with an increase in the operating time. In particular, the positionandvelocityincludestheattitudeelements,andthe
headingestimationisrelativelypoorcomparedtotheother accuracy of the position depends on the attitude. However,
attitude elements (roll and pitch angle), and it has poor theimprovementisnotsignificantbecausethepositionand
observability. In terms of the attitude accuracy, the earth’s velocityarehighlydependentonthemeasuredacceleration
magnetic field is very helpful when estimating the heading ratherthantheattitude.Moreover,informationisonlytrans-
angle.Becausethismagneticfielddoesnothavedrift,itcan ferred from the attitude to velocity and not in the opposite
be used as a reference for the heading angle. In addition directionforfeedback.Therefore,itisdifficulttoimprovethe
2 JournalofSensors
positionandattitudeatthesametime.Inthispaper,wepro- Theheadingiscomputedbyusingcalibratedmagneticmea-
poseamethodtoimprovetheINSperformanceforaground surement.TheNHCsareappliedtoincreasethecorrelationof
vehicleusingamagnetometeranddynamicconstraints. thevelocityandattitude.Thismethodcanprovideavelocity
It is well known that the navigation solution provided feedbacktotheattitude,whichmakesthenavigationsolution
by a conventional INS contains drift error [1, 2, 4, 9]. The more robust than a conventional navigation equation that
drift in the roll and pitch can be reduced or corrected onlytransfersinformationfromtheattitudetovelocity.The
using a zero velocity update or acceleration-based attitude magneticheadingandNHCsarecombinedwithanextended
estimation.However,becausethesemethodsarenoteffective Kalman filter (EKF). An experimental test showed that the
for heading estimation (yaw), a magnetometer has been proposedmethodsignificantlyimprovestheaccuracyofthe
used to resolve the problem. Various calibration methods position, velocity, and attitude estimations. As a result, the
[10–13] have been proposed for the accurate estimation of method can increase the survival time of dead reckoning
the heading angle. The sensor has been rotated to measure in GNSS-denied environments such as indoors or in urban
∘
the data in all directions (360 ). The scale factor and offset canyons.Therefore,theproposedmethodcouldbeeffective
havebeencomputed,afterwhichcompensatedmagneticdata forgroundvehicles,indoornavigation,mobilerobots,vehicle
wereusedtocalculatetheheadingangle.SokenandHajiyev navigation in urban canyons, or navigation in any GNSS-
[14]proposedonlinemagnetometercalibration,andAlonso deniedenvironment.
and Shuster [15] and Springmann and Cutler [16] showed InSection2ofthispaper,thebasicnavigationequation
an attitude estimation method without an inertial sensor. used to compute the navigation solution is introduced. The
However,thesepreviousstudiesonlyfocusedonimproving NHCs are described in Section3, and then the estimation
theattitudeestimation,withoutimprovingthepositionand filter used to combine the magnetic heading and NHCs
velocityaccuracy. is discussed in Section4. The simulation and experimental
Global navigation satellite systems (GNSSs) have been test results are presented in Section5. The conclusions are
integrated with inertial sensors to obtain accurate position presentedinSection6.
andvelocitysolutions.Loosely,tightly,andultratightlycou-
pled integration[2, 17, 18] are famoustechniques forGNSS
2.BasicNavigationEquation
integration.However,GNSSintegrationalwaysrequiresvisi-
blesatellites,andthenavigationsolutionalsodivergesifthere Inertialmeasurementsareusedtocomputebasicnavigation
is no visible satellite or sometimes even with less than four solutions. In addition to this, magnetic field measurement
satellites.Recently,avisionsensorwasintroducedtoinertial improvestheheadingaccuracy.Inthissection,anavigation
navigation to cope with the visible satellite problem [3, 19– equation is introduced for both the inertial sensor and
21]. The tracked visual point method provides more robust magnetometer.
navigation solutions than conventional INS. However, the
visionsensoristoosensitivetotheamountoflight,andsome-
2.1. Inertial Navigation System (INS). An inertial measure-
timestherearenoavailablepointstouse.Moreover,itcannot
ment unit (IMU) provides the acceleration and angular
remove the drift error without other absolute information
velocity. These measurements can be used to compute a
[21, 22]. Both the GNSS and vision integration approaches
navigation solution, including the position, velocity, and
needadditionalhardware,whichincreasethecost.Moreover,
attitudeinformation.Thistechniqueiscalledinertialnaviga-
theyarehighlydependentonthesurroundingenvironment.
tion.INSequationscanbeformulatedinvariouscoordinate
Vehicledynamicconstraintssuchasnonholonomiccon-
systems, including north-east-down (NED) frame; Earth-
straints (NHCs) have been introduced to improve the INS
Centered,Earth-Fixed(ECEF)frame;andlongitude-latitude-
performance and could decrease the drift rate [9, 23–26].
height (LLH) frame. In this paper, we use equations based
The basic assumption for NHCs is that a ground vehicle is
ontheNEDframeandthetargetIMUisalow-costmicro-
governedbytwoNHCsifitdoesnotjumpofforslideonthe
electromechanical systems (MEMS) IMU. The NED frame
ground.Inreality,theseconstraintsareacceptableforgeneral
issimplerandmoreintuitivethanothercoordinatesystems.
vehicleoperation,eventhoughtheyaresometimesviolated
Thenavigationsolutioniscomputedbyusingtheacceleration
as a result of the vibrations of the vehicle’s engine and sus-
andangularvelocity[1,2,4].Thediscretenavigationequation
pensionorbyside-slipduringcornering[24–27].However,
isasfollows:
this method still has no feedback from the attitude and it
is not accurate without external information. Only relative p𝑛(𝑡+1) p𝑛(𝑡)+k𝑛(𝑡)Δ𝑡
ncuarvrigenattiomneiassuproesmsibenletsb.asedonthepreviousinformationand [[k𝑛(𝑡+1)]]=[[[k𝑛(𝑡)+[𝐶𝑏𝑛(𝑡)f𝑏(𝑡)+g𝑛]Δ𝑡]]]
Fortheaccurateestimationofnavigationsolutions,pre- [𝜓𝑛(𝑡+1)] [ 𝜓𝑛(𝑡)+𝐸𝑛(𝑡)𝜔𝑏(𝑡)Δ𝑡 ] (1)
𝑏
vious researchers only focused on improving the attitude
usingamagnetometer,ortheyusedadditionalhardwareand +w(𝑡),
had operating environment limitations. In this paper, we
propose a method to improve the estimation performance wherep𝑛istheposition,k𝑛isthevelocity,𝜓𝑛istheattitude,
fortheposition,velocity,andattitudewithoutanyadditional 𝐶𝑏𝑛 is the direction cosine matrix, 𝐸𝑏𝑛 is the rotation rate
hardware except an inertial sensor and magnetometer. For transformationmatrixfromthebodyframetothenavigation
𝑏 𝑛
robust estimation, we use a magnetic heading with NHCs. frame,f isthespecificforceinthebodyframe,g isgravity,
JournalofSensors 3
𝜔𝑏 is the angular rate in the body frame, Δ𝑡 is the time x
interval,andwistheprocessnoiseoftheINS.Thenavigation
solution is computed and updated by using (1) recursively. Magnetic heading mxh True heading
without declination Magnetic north
The IMU should be calibrated, and its initial biases should correction Declination angle
becompensatedpriortocomputingthenavigationsolution.
True north
Fortunately, most commercial IMUs are calibrated in the mh y
fH(a1)co.twoZerevyr,eoro,vrtheletohicenirtieytiauelxpsidtsatatteeavv(aaZliulUaebPslTere)mcaaanliidbnrugatynirokonncoopwmrnopcateossspsinersogc[ae1rs]es. Magnetic vector y
well-knownmethodsforinitializinganINS[1,9,23]. z
WithonlyanIMU,anINSdoesnothaveanycorrection
process via external information,which makes the solution Figure 1: Magnetic vector presented on horizontal plane with
inaccurate. In this paper, we use the magnetic heading and declinationangle.
NHCstoimprovetheaccuracyofthenavigationsolution.The
magneticheadingispresentedinthesubsequentsection,and
thenthedetailsoftheNHCsareintroducedinSection3. OceanicandAtmosphereAdministration(NOAA)provides
magnetic declinations all over the world (http://www.ngdc
2.2. Magnetic Heading. A magnetometer measures and .noaa.gov/geomag/declination.shtml). After the declination
detectsboththesignandthemagnitudeoftheearth’smag- is determined, the compensation parameters can be com-
neticfield.Then,thehorizontalcomponentsofthemagnetic puted by rotating magnetometer to measure all direction
field are used to determine magnetic north, which differs magneticfields.
fromtruenorth.Thedifferenceiscalledthedeclinationangle, Inreality,amagneticdistortionexistsbecausethemagne-
whichshouldbecorrectedforaccurateheadingestimation. tometerismountedonavehiclethatincludesferrousmate-
Moreover,themagneticfieldisdistortedbytheeffectofthe rials nearby. The ferrous materials will distort, or bend, the
vehicle or surrounding environment. Thus, three processes magneticfield.Inpreviousresearch[10–13],thesemagnetic
shouldbeperformedinadvanceofdeterminingtheheading: distortionsweredividedintomoredetailedfactorsforhigher
(1) tilt compensation to obtain the horizontal magnetic accuracy. In thispaper, the distortionsare divided intotwo
measurements,(2)declinationanglecompensationtoadjust categories: hard- and soft-iron effects. First, the hard-iron
to true north, (3) and ferrous distortion compensation by effectcomesfromanynearbyironorsteelonthevehicle.This
adjusting the scale factor and offset of the measurements. effectproducesaconstantmagnitudeerrorinthemagnetic
Theseprocessesarecalledmagnetometercalibration. readingalongeachaxis.Thesoft-ironeffectcomesfromthe
First,amagnetometerattachedtoavehicleoritssurface interactionoftheearth’smagneticfieldandanymagnetically
isnotalwayshorizontaltotheearth’slocalplane.Thereexists soft material. The amount of distortion from the soft-iron
tiltangle(rollandpitch)withrespecttothelocalhorizontal dependsonthesensororientation[10,11].
plane,whichmakesitdifficulttocomputetheheading.Com- Those distortions can be accounted for and should be
monsensorstomeasurethistiltangleincludeaccelerometers, removed from the magnetic readings before computing the
gyroscopes, gimbaled structures, and fluid sensors. In this magneticheading.Thehorizontalmagneticreadingsshould
∘
paper,weuseathree-axisaccelerometerandgyroscopeinside make a circle for the 360 rotation without any distortions.
the IMU. Once the magnetic readings (𝑚𝑏,𝑚𝑏, and 𝑚𝑏) Inreality,itshowsanellipsoidpattern,aswellasbiaseswith
𝑥 𝑦 𝑧
respecttotheoriginasshowninFigure2.Wecancompensate
are measured on the body frame, they are converted into a forthesedistortionsbydeterminingthescalefactor(𝑠𝑥,𝑠𝑦)
h(𝐶oℎri)zownitthaltphleanroellus(𝜙in)gaandcopoitrcdhin(a𝜃t)eatnragnlessfoirnmFaitgiounrem1.aThtrixe andoffset(𝑜𝑥,𝑜𝑦):
𝑏
transformationequationisshownbelow: 𝑚ℎ 𝑠 ⋅𝑚ℎ 𝑜
[ 𝑥]=[ 𝑥 𝑥]+[ 𝑥]. (3)
𝑚𝑏 𝑚ℎ 𝑠 ⋅𝑚ℎ 𝑜
𝑚ℎ [ 𝑥] [ 𝑦] [ 𝑦 𝑦] 𝑦
[ 𝑥]=𝐶ℎ[𝑚𝑏],
𝑚ℎ 𝑏[ 𝑦] Thescalefactorchangestheellipsoidtoacircle,andtheoffset
[ 𝑦] [𝑚𝑧𝑏] (2) moves the center of circle to (0,0). These two factors are
calculated from the maximum and minimum values of the
𝐶ℎ =[cos𝜃 sin𝜙sin𝜃 cos𝜙sin𝜃], magnetic readings for the 360∘ rotation. The scale factor is
𝑏 0 cos𝜙 −sin𝜙 determinedbyusingthefollowingequations:
wherethe“𝑏”and“ℎ”superscriptsrepresentthebodyframe max(𝑚𝑦ℎ)−min(𝑚𝑦ℎ)
andhorizontalplane,respectively,andsubscripts𝑥,𝑦,and𝑧 𝑠𝑥 =max(1, max(𝑚ℎ)−min(𝑚ℎ)),
𝑥 𝑥
denotetheaxes.
(4)
Next, we need to compensate for the declination angle. max(𝑚ℎ)−min(𝑚ℎ)
Thisiseasilydeterminedandcompensatedforusingalookup 𝑠𝑦 =max(1, 𝑥 𝑥 ).
tablebasedonthegeographiclocation[10–13].TheNational max(𝑚𝑦ℎ)−min(𝑚𝑦ℎ)
4 JournalofSensors
x
0.1
After calibration→
(cid:4)b
x
0.05
a)
G 0
xis ( (cid:4)yb ≈0
y-a (cid:4)zb≈0
−0.05 z y
Figure3:VelocitypresentedonbodyframewithNHCs.
−0.1
←Raw measurement general vehicle operation, even though they are sometimes
violatedasaresultofthevibrationsofthevehicle’sengineand
−0.1 −0.05 0 0.05 0.1
suspensionandbyside-slipduringcornering[9,23–27].The
x-axis (Ga)
constraintscanberegardedasazerovelocityupdate(ZUPT)
for the velocity of the vehicle in the plane perpendicular to
Figure 2: Magnetic readings on horizontal plane: blue circle
represents raw measurements and red circle shows values after theforwarddirection(𝑥-axis).Inotherwords,thevelocities
compensationbycalibrationparameters. on the lateral and vertical axes (𝑦 and 𝑧 velocities on body
frame)inFigure3arealmostzero,asshownbelow:
V𝑏 ≈0,
Andtheoffsetisdeterminedbyusingthescalefactormin- 𝑦
(7)
imumandmaximumreadingsasfollows: V𝑏 ≈0.
𝑧
𝑜𝑥 =(max(𝑚𝑥ℎ)−min(𝑚𝑥ℎ)−max(𝑚𝑥ℎ))⋅𝑠𝑥, Acoordinatetransformationmatrix(𝐶𝑛𝑏)isusedtoconvert
2 thevelocityfromthenavigationframetothebodyframe,and
(5) itisexpressedasfollows:
max(𝑚ℎ)−min(𝑚ℎ)
𝑜𝑦 =( 𝑦 2 𝑦 −max(𝑚𝑦ℎ))⋅𝑠𝑦. k̂𝑏 =𝐶𝑛𝑏k̂𝑛. (8)
The 𝑦- and 𝑧-axis velocities computed using (8) should be
Once the magnetic readings are compensated by using the zero because of the NHCs in (7). However, they are not
tilt angle, declination, and distortions, then the magnetic exactly zero because of the velocity and attitude estimation
headingcanbecomputedbyusinganarctangentfunctionas errorsoftheINS.Theseerrorscanbeestimatedandcorrected
follows: by using a Kalman filter. Two velocity residuals from the
NHCs are considered as measurements for the navigation
𝜓 =tan−1(𝑚ℎ𝑦). (6) filter.Theequationfortheresiduals(𝑟)isexpressedasfollows:
mag 𝑚ℎ
𝑥 𝛿V𝑏 ̂V𝑏 −V𝑏 ̂V𝑏
𝑟=[ 𝑦]=[ 𝑦 𝑦]=[ 𝑦]. (9)
Finally, we can obtain the magnetic heading, which will be 𝛿V𝑏 ̂V𝑏−V𝑏 ̂V𝑏
𝑧 𝑧 𝑧 𝑧
combinedwiththeNHCsusinganEKF.
Alinearizedobservationmodelcanbeobtainedbyapplying
perturbationin(8),whichcanbeexpressedasfollows:
3.Nonholonomic Constraints(NHCs)
𝛿V𝑏 𝛿k𝑛
An INS has inherent drift error in its navigation solutions [ 𝑦]=[𝐻 𝐻 ][ ],
unless correction information is provided by an external 𝛿V𝑏 NHC,V NHC,𝜓 𝛿𝜓𝑛
𝑧
system.Thereareseveraltechniquestoreducethiserror,such
𝐶 𝐶 𝐶
asimprovednavigationalgorithms[24,25,28],adenoising 𝐻 =[ 12 22 32],
IMU[29],andtheuseofvehiclekinematicconstraints[9,25– NHC,V 𝐶 𝐶 𝐶 (10)
13 23 33
27]. This paper focuses on vehicle kinematic constraints,
𝐻
whicharecalledNHCs. NHC,𝜓
ThebasicassumptionforNHCsisthatagroundvehicle
−V 𝐶 +V 𝐶 V 𝐶 −V 𝐶 −V 𝐶 +V 𝐶
is governed by 2 NHCs if it does not jump off or slide on =[ 𝑑 22 𝑒 32 𝑑 12 𝑛 32 𝑒 12 𝑛 22],
the ground. In reality, these constraints are acceptable for −V 𝐶 +V 𝐶 V 𝐶 −V 𝐶 −V 𝐶 +V 𝐶
𝑑 23 𝑒 33 𝑑 13 𝑛 33 𝑒 13 𝑛 23
JournalofSensors 5
where 𝐶𝑖𝑗 is the 𝑖th row and 𝑗th column of 𝐶𝑏𝑛 and V𝑛, and NHCs. In addition to these equations, the acceleration
V𝑒, and V𝑑 denote the north, east, and down velocities, readings can be used as measurements to predict the roll
respectively. and pitch angles if the accelerations are within reliable
The NHCs can improve the velocity estimation, which boundaries. The accelerometer can be regarded as a gravity
also improves the position estimation. However, because sensorwhentheaccelerationsaresolowthatwecanconsider
external and absolute information is still not available, it themtobeequaltozero[34,35].Theaccelerometerreading
is only possible to obtain a relative navigation solution. should be ‖f𝑏‖ ≈ 𝑔 under low accelerations. Then, the roll
Moreover, this technique is vulnerable to errors in the yaw andpitchanglescanbeestimatedbyanalyzingthemagnitude
estimation. of the acceleration reading on each axis. The acceleration
rangeisdeterminedasshownin[34],anditissetto0.9𝑔 ≤
4.EstimationFilter ‖f𝑏‖ ≤ 1.1𝑔 in this paper. The relationship between the
accelerometerreadingandtheattitudeiswrittenasfollows:
The equations for the magnetic heading and NHCs were
presented in the previous section. The magnetic heading 𝑔sin𝜃
is computed from the corrected magnetic readings using f𝑏 =𝐶𝑏f𝑛 =[[−𝑔sin𝜙cos𝜃]], (13)
an arc-tangent function. We can also obtain two additional 𝑛
measurementequationsfromthevelocityconstraintsofthe [−𝑔cos𝜙cos𝜃]
NHCs. This section will address the details of the filter
−𝑓𝑏
structure used to implement the navigation algorithm with 𝜙=tan−1( 𝑦),
anIMUandamagnetometer. −𝑓𝑏
𝑧
An EKF is used for the navigation filter. EKF has two
distinct steps: predict and update. In predict step, the state (14)
vector is propagated from the current epoch into the next 𝑓𝑏
epochbyusingasystemdynamicmodel.Then,wecanpredict 𝜃=tan−1( 𝑥 ).
the state for the next time step. After the prediction, the √(𝑓𝑏)2+(𝑓𝑏)2
𝑦 𝑧
predicted state vector is refined using a measurement, and
thestatevectorisestimated,whichminimizesthesquareof Finally, we can determine the magnetic heading (𝜓 ), 𝑦
mag
the error [30–33]. To implement the EKF, the state vector and 𝑧 velocities on body frame (V𝑏,V𝑏), and roll and pitch
andsystemmodelshouldbedefinedforthepredictionand 𝑦 𝑧
angles based on the acceleration (𝜙 ,𝜃 ). The measure-
observationmodelfortheupdate. acc acc
mentorderisslightlychanged,andtheobservationmodecan
beexpressedasfollows:
4.1. System Model. First, the state vector is defined on the
navigationframeusingtheposition,velocity,andattitude.It 𝑧=[V𝑏 V𝑏 𝜙 𝜃 𝜓 ]𝑇,
isexpressedasfollows: 𝑦 𝑧 acc acc mag
x=[p𝑛 k𝑛 𝜓𝑛]𝑇. (11) z𝑘 =𝐻𝑘𝛿x𝑘+𝜐𝑘, (15)
The system model for EKF should be linear because of the 𝐻 =[02×3 𝐻NHC,V 𝐻NHC,𝜓],
EKFconstraints.Equation(1)canbelinearizedandexpressed 𝑘 03×3 03×3 𝐼3×3
withrespecttotheerrorofthestatevectorin(11).Thefinal
systemmodelwiththeerrorstatevector(𝛿x)canbewritten where the measurement noise (𝜐𝑘) is assumed to be white
noise. This model is only reliable under low accelerations.
asfollows:
Therefore, we need additional observation model for high
𝛿x𝑘+1 =𝐹𝑘𝛿x𝑘+𝑤𝑘, accelerations.Therollandpitchmeasurementsarenolonger
reliable under a high acceleration. Thus, the observation
[𝐼3×3 Δ𝑡𝐼3×3 03×3 ] (12) modelshouldbechangedasfollows:
𝐹𝑘 =[03×3 𝐼3×3 Δ𝑡[f𝑛×]], 𝑧=[V𝑏 V𝑏 𝜓 ]𝑇, (16)
[03×3 03×3 𝐼3×3 ] 𝑦 𝑧 mag
swkheewr-es𝐼ymismtheetriidcemntaittryixmoaftfr𝑛ix,t,h0eissuabzscerriopmt𝑘adtreinxo,t[efs𝑛×th]eistitmhee 𝐻𝑘 =[0021××33 𝐻0N1H×C3,V [𝐻0N0HC,1𝜓]]. (17)
stamp,andthesubscript𝑎×𝑏showsthenumberofmatrix
rows(𝑎)andcolumns(𝑏).Theprocessnoise(𝑤𝑘)isassumed 4.3.FilterStructure. Weobtainthesystemmodelandobser-
to be white noise. Equation (12) propagates the state vector vationmodeltoimplementtheEKF.Figure4showsablock
intothenexttimestep(𝑘+1). diagramoftheproposedmethodforestimatingthenaviga-
tion solution using a magnetometer and an IMU. Basically,
4.2. Observation Model. From (6) and (10), we can obtain thenavigationsolutioniscomputedusingtheINSblock,and
the three observation equations for the magnetic heading theoutputisnotcompensateduntiltheEKFresultisfedback
6 JournalofSensors
Calibration data
mh,mh 𝜓
x y mag
Mag Heading
Roll, pitch
fb,𝜔b pn,vn,𝜓n Navigation
IMU INS EKF
solution
vn,𝜓n
NHCs
b
f
Roll, pitch
Error corrections
Figure4:FilterstructureusedtoestimatenavigationsolutionwithmagnetometerandIMU.
Table1:Sensorsusedintest. Table2:Magnetometercalibrationresults.
Sensor Model
Axis Parameter Value
Targetsensor
𝑋 Scalefactor 1.284
IMU AnalogDevicesADIS16364
Offset 0.032
Magnetometer HoneywellHMC1053
Referencesystem 𝑌 Scalefactor 1.000
Rover Offset 0.029
IMU HoneywellHG1700-AG11
GPS NovAtelProPak-3Receiver
Referencestation NovAtelDL-V3Receiver
Software NovAtelInertialExplorer
Then, the reference position, velocity, and attitude were
computed by using commercial software (NovAtel Inertial
Explorer).
recursively.Theheadingmeasurementiscalculatedusingthe
In the test, the vehicle remained stationary for the first
tilt-compensatedmagneticreadingandcalibrationdata,and
200s to obtain the zero velocity data. The initial attitude
then it is sent to the EKF as a measurement. In addition,
wasobtainedbyusingZUPTwiththisstationaryIMUdata
the INS output is used to generate the NHCs with (7) and
(gyrocompassingwasnotusedbecauseoflowresolutionof
(8), whereas the roll and pitch are computed based on the
the target IMU). After that, the vehicle made three small
acceleration using (14). The estimation filter is constructed
circles to calibrate the magnetometer. Then, the vehicle
from(11)to(17),andthenthenavigationsolutionisestimated
followedaneight-shapedroadthreetimesasshownFigure5.
recursively.
Figure6showsthevelocityandaccelerationduringthetest.
In this section, the navigation filter was formulated to Theproposedmethod(INS+NHC+Mag)wasanalyzedin
implementtheEKF.Thenextsectionwilldiscusstheresults
comparisontothreeconventionalapproaches,aswellasthe
ofanexperimentaltestoftheproposedmethod.
referencedatafromNovAtelSPAN.Thesethreeapproaches
included an INS, INS with magnetic heading information
5.TestandAnalysis (INS+Mag),andINSwithNHCs(INS+NHC).
All four navigation filters, including the proposed
5.1. Test Settings. An experimental test with a car was method, were implemented after the ZUPT, which was
conducted to verify the proposed method. The data were conducted by using the Kalman filter with a zero velocity
collectedatKonkukUniversityinKoreaonAugust13,2013, condition.Asaresult,allofthemethodsstartedwiththesame
and postprocessed using MATLAB. The target sensor was initialvaluesafter200s.Themagnetometerwasassumedto
anAnalogDevicesADIS16364,whichincludedathree-axis beprecalibratedpriortoestimatingthenavigationsolutions.
accelerometerandgyroscope.Thetargetmagnetometerwas The calibration was conducted by using the data collected
aHoneywellHMC1053.ThereferencesensorwasaNovAtel fromthecirculartrack.Thedeclinationanglewasdetermined
∘
SPAN, which consisted of a ProPak-V3 GPS Receiver and fromtheNOAAlookuptabletobe8.05 atthetestsite(Seoul,
Honeywell HG1700-AG11 IMU. A NovAtel DL-V3 Receiver Korea). Finally, the calibration parameters were obtained
was used for the reference station to compute precise as presented in Table2. The compensated magnetometer
and accurate solutions. The sensors are shown in Table1. readingsafterthecalibrationareshowninFigure2.
JournalofSensors 7
Table3:Attitudeerroranalysiswithrespecttomean,STD,andRMS.
Mean STD RMS
Roll Pitch Yaw Roll Pitch Yaw Roll Pitch Yaw
INS 0.14 0.38 −48.60 1.00 1.06 32.02 1.01 1.12 58.19
INS+Mag 0.13 0.45 2.39 1.00 1.19 5.85 1.00 1.27 6.32
INS+NHC 0.08 −0.12 −47.70 0.26 0.47 30.80 0.27 0.48 56.77
INS+NHC+Mag 0.07 −0.23 −1.06 0.26 0.50 4.73 0.28 0.55 4.84
“INS+NHC”and“INS+NHC+Mag”werenotdiverse.This
wasbecausetheNHCsusedvelocityconstraints,whichcould
3 “8” shapedtrack×3 prevent the velocity from drifting. As a result, the NHCs
improvedthepositionestimations.
The attitude estimation results are shown in Figure10
and listed in Table3. The results were analyzed for each
axis.Threeparameters(mean,standarddeviation(STD),and
root mean square (RMS)) were introduced for a statistical
analysis. In terms of the roll and pitch angles, the NHCs
improvedtheestimationperformancebyapproximatelytwo
times. For the yaw angle, the magnetometer definitely kept
1 Stop (200s) the error bound lower, whereas in other cases without the
2 Circletrack×3 magnetometer, it continuously increased. “INS + NHC”
did not have reliable performance for the yaw estimation,
although it could estimate the roll and pitch angles well.
Figure5:Testtrajectorytocollectthedata. Therefore, it was determined that the proposed method
showed thebest attitudeestimationperformanceforallthe
axes.
In this section, the estimation performance for a nav-
5.2.TestResults. Theestimatedpositionresultsarepresented
igation solution was analyzed. A comprehensive analysis
in Figure7, and the position errors with respect to the
of the performance in terms of the position, velocity, and
referenceareshowninFigure8.WithoutNHCs,theposition
attitude showed that the proposed method had the best
resultswerediverseinshorttime.Whenthemagnetometer
performanceforallofthenavigationinformationbecauseof
wasintegratedwiththeINS,thedriftratewasslightlyslower
themagnetometerandNHCs.
than that in the INS-only case. However, in both cases
without NHCs (INS and “INS + Mag”), the results were Thepositionsolutionstillhadalittledrifterrorbecause
of the absence of external information such as that from
meaningless because they had significant errors and were
a GPS. It would be possible to use a GPS to bound the
difficult to use for position estimation. The position error
for “INS + NHC” was decreased dramatically compared to position solution, even though the update rate is fairly low.
Therefore, the proposed method could be used to obtain
theerrorbeforetheintegrationofNHCs.Thetrendsofthe
reliableestimationsolutionsforgroundvehiclenavigation.
estimatedtrajectoryshowedpatternssimilartothereference,
even though they had drift error. The proposed method
(INS+NHC+Mag),whichintegratedthemagnetometerin 6.Conclusion
addition to the NHCs, showed the best position estimation
andwasclosetothereferencetrajectory.Theerrorwasfour Anavigationalgorithmwasproposedtoimprovetheinertial
timesbetterthanthatfor“INS+NHC.”The“INS+NHC” navigationperformanceofagroundvehicleusingmagnetic
resultsmovedinthewrongdirectionaftertheinitialcircular measurements and dynamic constraints. To improve the
turn,andthedriftratewashigherthanthatoftheproposed estimationperformance,athree-axismagnetometerwasused
method. to estimate the heading angle and NHCs were introduced
Figure9 shows the estimated velocity error. Each error toincreasethecorrelationbetweenthevelocityandtheatti-
comesfromtakingthenormofthethree-axisvelocityerrors. tude equation. The NHCs could provide feedback from the
The navigation solution was not estimated until 200s had velocitytotheattitude,whichmadethenavigationsolution
passedbecauseoftheZUPTinitialization.TheNHCscould more robust compared to conventional inertial navigation
keeptheerrorfromdiverging,whereasbothresultswithout equations that only transfer information in one direction,
NHCs had significant errors. Their maximum errors were fromtheattitudetovelocity.Additionally,acceleration-based
around 60 and 90m/s (more than 200km/s), even though rollandpitchestimationswereappliedtodecreasethedrift
the speed did not exceed 7m/s during the test. Thus, these when the acceleration was within certain boundary. The
errors were not reasonable. In addition, both results from magnetometer and NHCs were combined with an EKF. An
8 JournalofSensors
x-axis x-axis
m/s) 105 2m/s) 21
Velocity ( −−1005 celeration ( −−012
0 200 400 600 800 Ac 0 200 400 600 800
Time (s) Time (s)
y-axis y-axis
y (m/s) 1050 2on (m/s) 210
Velocit −−105 celerati −−12
0 200 400 600 800 Ac 0 200 400 600 800
Time (s) Time (s)
y (m/s) 1050 z-axis 2on (m/s) 210 z-axis
Velocit −−105 celerati −−12
0 200 400 600 800 Ac 0 200 400 600 800
Time (s) Time (s)
(a) (b)
Figure6:Referencevelocity(a)andacceleration(b)duringthetest.
×104
2.5 200
150
2
100
1.5
m) m) 50
h ( 1 h (
Nort Nort 0
0.5
−50
0
−100
−0.5 −150
−15000 −10000 −5000 0 5000 −100 −50 0 50 100 150 200 250
East (m) East (m)
Ref INS+Mag Ref INS+Mag
INS INS
Figure7:Estimatedtrajectoryonhorizontalplane(eastandnorthaxes).
experimental test was conducted to verify the proposed The dynamic constraints could dramatically decrease the
method.Thepositionandvelocityerrorwasdecreaseddra- drift of the position and velocity errors, and the roll and
maticallywhentheNHCswereintegrated,whereastheatti- pitcherrorsweredecreasedbyapproximately50%compared
tudeerrorsweredecreasedbythemagnetometerintegration. with the cases without NHCs. However, the yaw angle still
JournalofSensors 9
×104
3 350
300
2.5
250
2
m) m)
or ( or ( 200
err 1.5 err
n n
ositio ositio 150
P P
1
100
0.5
50
0 0
0 200 400 600 800 1000 0 200 400 600 800
Time (s) Time (s)
INS INS+NHC
INS+Mag INS+NHC+Mag
Figure8:Estimatedpositionerror(norm)withrespecttoreference.
100 Roll
or 2
or (m/s) 6800 Attitude err(deg.)−20
err 0 100 200 300 400 500 600 700 800
city 40 Time (s)
o
Vel Pitch
20
ror 2
0 de ereg.) 0
0 100 200 300 Ti4m0e0 (s)500 600 700 800 Attitu(d −2
0 100 200 300 400 500 600 700 800
INS INS+NHC Time (s)
INS+Mag INS+NHC+Mag
Yaw
Figure9:Estimatedvelocityerror(norm)withrespecttoreference. or 50
r
ude erdeg.) −500
had drift. We conclude that the proposed method could Attit( −100
0 100 200 300 400 500 600 700 800
improve all of the navigation performances as a result of
Time (s)
usingthemagnetometeranddynamicconstraints,basedon
acomprehensiveanalysisoftheperformanceintermsofthe INS INS+NHC
position,velocity,andattitude. INS+Mag INS+NHC+Mag
The proposed method could improve the estimation
Figure 10: Estimated attitude error on each axis with respect to
performancesfortheposition,velocity,andattitudewithout
reference.
additional hardware, except an inertial sensor and a mag-
netometer.Additionally,themethodincreasedtheaccuracy
compared with a conventional INS and provided accurate
informationforalongertime.Asaresult,themethodcould theproposedmethodcouldbeeffectiveforgroundvehicles,
increasethesurvivaltimeofdeadreckoninginGNSS-denied indoornavigation,mobilerobots,vehiclenavigationinurban
environmentssuchasindoorsorinurbancanyons.Therefore, canyons,andnavigationinanyGNSS-deniedenvironment.
10 JournalofSensors
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