Research Article A New Technique for Integrating...

Research ArticleA New Technique for Integrating MEMS-Based Low-Cost IMUand GPS in Vehicular Navigation

Neda Navidi1 Reneacute Jr Landry1 Jianhua Cheng2 and Denis Gingras3

1LASSENA Laboratory Ecole de Technologie Superieure 1100 Notre-Dame Street West Montreal QC Canada H3C 1K32Marine Navigation Research Institute College of Automation Harbin Engineering University Harbin 150001 China3LIV Laboratory Electrical and Computer Engineering Department Universite de Sherbrooke Sherbrooke QC Canada J1K 2R1

Correspondence should be addressed to Neda Navidi nedanav3gmailcom

Received 20 November 2015 Revised 4 April 2016 Accepted 28 April 2016

Academic Editor Maan E El Najjar

Copyright copy 2016 Neda Navidi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In providing acceptable navigational solutions Location-Based Services (LBS) in land navigation rely mostly on integration ofGlobal Positioning System (GPS) and Inertial Navigation System (INS) measurements for accuracy and robustness The GPSINSintegrated system can provide better land-navigation solutions than the ones any standalone system can provide Low-cost InertialMeasurement Units (IMUs) based on Microelectromechanical Systems (MEMS) technology revolutionized the land-navigationsystem by virtue of their low-cost miniaturization and widespread availability However their accuracy is strongly affected bytheir inherent systematic and stochastic errors which depend mainly on environmental conditions The environmental noise andnonlinearities prevent obtaining optimal localization estimates in LandVehicularNavigation (LVN)while using traditional KalmanFilters (KF) The main goal of this paper is to effectively eliminate stochastic errors of MEMS-based IMUs The proposed solutionis divided into two main components (1) improving noise cancellation using advanced stochastic error models in MEMS-basedIMUs based on combined Autoregressive Processes (ARP) and first-order Gauss-Markov Process (1GMP) and (2) modeling thelow-cost GPSINS integration using a hybrid Fuzzy Inference System (FIS) and Second-Order Extended Kalman Filter (SOEKF)The results obtained show that the proposed methods perform better than the traditional techniques do in different stochastic anddynamic situations

1 Introduction

Steadily increasing economical and environmental con-straints as well as stringent safety requirements have triggeredthe development and widespread use of low-cost LandVehicular Navigation (LVN) systems over the last decadeLVNs have many applications nonsafety applications suchas fleet management and traffic optimization or active safetyapplications such as Collision Mitigation Braking Systems(CMBS) and lane keeping systems LVNs can also be usedfor protecting vehicles from theft in vehicle tracking systemsor for reducing the greenhouse gas emissions in environ-mental monitoring systems Besides they can also be usedin autonomous car navigation systems and emergency assis-tance services Furthermore the need for recognizing driverrsquosbehavior in many applications requires low-cost and reliable

estimation of vehicular position and attitude Clearly univer-sal use of LVN in road transportation demands lower costwidespread availability and ever improving performance [1]

Advanced commercial Location-Based Services (LBS)must be able to solve difficult positioning problems especiallythose in indoor-parking urban canyons or dense foliagesituations by providing acceptable customer support when-ever the GPS signal is lost One issue that is recently inves-tigated in positioning-based systems relies on multi-GlobalNavigation Satellite Systems (GNSS) environment which isrealized due to the effective usage of the global constellationof GLONASS satellites of Galileo and Chinese BeiDouNavigation Satellite System (BDS) Multi-GNSS environmentoffers several advantages over a standalone GPS Indeedmulti-GNSS environment may increase the availability ofnavigation satellites where the received GPS signals are less

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 5365983 16 pageshttpdxdoiorg10115520165365983

2 Journal of Sensors

than 4 satellites in urban canyons Moreover it can improvethe accuracy of positioning especially when the navigationsystem considers the horizontal positioning accuracy [2]

However multi-GNSS-based techniques may increasethe intersystem interference particularly when multi-GNSSenvironments broadcast navigation signals in overlappedfrequency bands Moreover these techniques impose higherlevels of complexity because the analog-part of the receivershould deal with multiple-system multiple frequencies andconsiderable bandwidths [3] Finally the mentioned devel-oped techniques consist in postprocessing techniques thatrequire laboriousmanipulations and calculations whichmaynot be suitable for real-time commercial applications Amodern approach to solve the current positioning problems isto integrate the GPS signals with Low-cost Inertial Measure-ment Units (IMUs) data IMUs can estimate the position andthe attitude of the vehicle by employing an Inertial NavigationSystem (INS) mechanization process (accelerometers andgyroscopes) which is an integration of linear accelerationsand angular rates However MEMS-based low-cost IMUsperform accurately just for a limited time as their positioningand attitudinal errors increase steadily with time

MEMS-based IMUs are classified based on their massposition recognition their operation mode and their fab-rication procedure Piezoresistive Capacitive Piezoelectricand Resonant element sensors are different types of MEMSsensors which are placed in the classification of mass positionrecognition [4]These types of sensors transform the physicalwork caused by variation of movement into an electrical sig-nal Piezoresistive and Piezoelectric sensors are widely usedin MEMS-based IMUs because of their simplicity cost-effectiveness and wide dynamic range However they sufferfrom low precision and high sensitivity to temperature[4] Performance characteristics of MEMS-based sensorsinclude bandwidth noise floor misalignment drift linearitydynamic range and power consumption Thus the MEMS-based inertial sensor errors are generally composed ofstochastic errors which have an undesirable impact on thequality of the navigation solution [5]

The first part of the proposed methodology consists ofaugmenting the navigation solution with 1st-order Gauss-Markov Process (1GMP) for dynamically estimating andcompensating the stochastic errors of MEMS-based low-costIMUs Two different identification methods were exploitedin this study to identify the 1GMP parameters The first oneis a traditional method based on the analysis of Autocor-relation Function (ACF) and the second one is based onanalysis of 1st-order Autoregressive Process (ARP) modelACF does not fully match the random error characteristicsof low-cost MEMS-based IMUs What mainly causes thismismatch is the high nonlinearity of the low-cost inertialsensor errors leading to higher order GMPs (ie 2nd- or3rd-order GMPs) distributions instead of 1GMP [6] Thusthere is a need for the development of better model-basedmethods to accurately estimate the distribution of the 1GMPlow-cost inertial sensor errors This study proposes usingcombined 1GMP-ARP method as an alternative candidateto identify the 1GMPrsquos parameters distribution in the low-cost inertial sensors To estimate the ARPrsquos coefficients

three different algorithms were considered and implementednamely the Yule-Walker (YW) method the Burg methodand the modified-covariance method [7]

As these static errors have both high frequency (HF)term and low frequency (LF) term decreasing both of themis needed in order to develop the accuracy of the low-costinertial sensorsWavelet Denoising Techniques (WDTs) havebeen utilized in similar research studies due to their signif-icant role in removing the HF noises [8] So it is presenteda mixture of the WDT (with different levels) and 1GMP toevaluate the accuracy improvement of the low-cost inertialsensors when these methods are blended together

Traditionally GPSINS integration is carried out by usingKalman Filter (KF) techniques This approach has been val-idated in many studies where high-end IMUs were used [9]However recent studies report several shortcomings in KF-based INSGPS integration when using low-cost MEMS-based inertial sensors [6] Dynamic models that define thenavigation problem are originally nonlinear [10] but to allowthe use of simple linear filters such as the KF thesemodels aregenerally linearized under a small error assumptionWith theassumption made by the KF that the system is characterizedby a linear model driven by white Gaussian noise it is notpossible to describe the nonlinear error dynamics of MEMS-based INS [11]

The conventional 1st-order EKF is used extensively forintegration of high-end INS and GPS data which can achievea suboptimal estimation of Minimum Mean Square Error(MMSE) of the vehicle state vector [12] 1st-order ExtendedKalman Filter (EKF) is based on the assumption that thehigher order terms (2nd-order and up) of a Taylor seriesexpansion are small enough to be ignored This conditionmay hold for high-end navigation systems where the errorsare kept small and bounded but not for highly nonlinearsystems such asMEMS-based INS In addition to provide anacceptable position and attitude estimates the 1st-order EKFrequires proper nonlinear models of vehicle dynamics andmeasurement sensorsThefirst-order linearization especiallyin the presence of highly nonlinear dynamics often leads todivergence of the state covariance matrix [11] Thus usingsuch assumption for a low-cost navigation system may resultin unstable solution as demonstrated by several previouslypublished results [13]

Several research groups investigated the possibility ofusing sampling techniques to cope with highly non-Gaussianmultimodal error distributions in GPSINS integration andthus avoiding linearization of the dynamic models [13]Unscented Kalman Filter (UKF) and Particle Filter (PF) aretwo such alternative methods that can provide better per-formance than does classical KF or EKF but at high compu-tational costs [11 14] Some authors assert that the limitationsof the sampling and the Kalman-based techniques can beovercome by using Artificial Intelligence (AI) [15] Theintegration of AI algorithms with next generation embeddednavigation systems is made possible with the technologicaladvancement in computer technologies In the last decadevarious neural networks (NN) and Fuzzy Inference Systems-(FIS-) based integration scheme were introduced [16ndash18]AI algorithms require a training phase which is realized

Journal of Sensors 3

during GPS signal availability using a learning algorithmHowever AI-basedGPSINS integration schemes suffer fromtwo main drawbacks these methods generally require longtraining time that greatly limits their utilization in a real-timecommercial applicationmoreover they highly depend on theconsistency of the training data and show limited successif the error dynamics do not stay the same between theirtraining and prediction stages [11]

The second part of the proposed approach is to combineboth linearized navigation filters and AI techniques intoan intelligent cognitive navigator SOEKF as the linearizednavigation filter helps in reducing the linearization errorduring the filtering process and FIS-based approach as an AItechnique predicting the error states of the SOEKF based ona Covariance Matching Estimation Technique (COMET) Infact the dynamic characteristics of vehicle motion in bodyframe and the navigation frame form based on the SOEKFprocessThe FIS can be exploited to increase the accuracy andthe robustness of the SOEKF and to prevent its divergencein the tuning phase of SOEKF The performances of the pro-posed methods were evaluated by a road test using MatlabThe results show that the proposed method is effective inreducing the root mean square error (RMSE) of position by45 compared with the SOEKF

The remainder of the paper is organized as followsSection 2 describes the main methods of this study Section 3deals with the proposed methods and the contributions ofthis paper for GPSINS integration and the noise modelingThe method employed here for the GPSINS integration isbased on the proposed hybrid FIS-SOEKF model Addi-tionally the proposed noise modeling of low-cost inertialsensor utilizes the combined 1GMP-ARP Section 4 dealswith the road test and the detailed analysis of experimentaldata followed by a discussion on the performance obtainedFinally Section 5 concludes this paper by summarizing theresults achieved and by offering suggestions for future workto improve the proposed methods

2 Background

This section presents details on background of techniquesand ideas used in this study More specifically two maintopics will be studied in this section Second-Order ExtendedKalman Filter (SOEKF) as a data-fusion technique for auto-motive navigation and stochastic error modeling techniquein Inertial Navigation Systems

21 Second-Order Extended Kalman Filter (SOEKF) Linearstate estimators have been widely used in the early literaturefor guidance navigation and control of various types ofvehicles including aircraft space shuttles ships submarinesand automotive land vehicles [9] As mentioned in Intro-duction in order to perform optimal estimation using theclassical Kalman Filter-based technique on the nonlinear andstochastic INS dynamic model the model must be linearizedaround assumptions First-order and second-order ExtendedKalman Filter (EKF) are two Kalman Filter-based techniqueswhich linearized based on the assumption of first and secondorders of a Taylor series expansion

In this study SOEKF was addressed for proposed data-fusion of low-cost INS and GPS aided with FIS Thus asummary of SOEKF performance is presented in this sectionThe SOEKF is an estimator method which can solve theproblem of estimating the state of a controlled process Themodel of SOEKF is

119909119896 = 119891 (119909119896minus1 119896 minus 1) + 119902119896minus1

119910119896= ℎ (119909119896 119896) + 119903119896

(1)

where 119909119896 isin R119898 is the state 119910119896

isin R119898 is the measurement119902119896minus1 sim 119873(0 119876119896) is the process noise 119903119896 sim 119873(0 119877119896) is meas-urement noise and 119891 and ℎ are the dynamic and measure-ment model functions The prediction and correction stepsof SOEKF are described by [19ndash21]

minus

119896= 119891 (119909119896minus1 119896 minus 1) +

1

2sum119894

119890119894tr 119865(119894)

119909119909(119909119896minus1 119896 minus 1)

sdot 119875119896minus1

119875minus

119896= 119865119909 (119909119896minus1 119896 minus 1) 119875119896minus1119865119909

119879(119896minus1 119896 minus 1) +

1

2sum119894

119890119894

sdot 119890119894119879tr 119865(119894)

119909119909(119909119896minus1 119896 minus 1) 119875119896minus1119865

(1198941015840)


+ 120578119896

(2)

where minus

119896and 119875minus

119896are respectively the time-propagated state

estimates and covariance 119865119896 and 119867119896 are the state transitionand the design matrices 119909119896 is the error state vector and 119911119896the measurement vector

V119896 = 119911119896 minus ℎ (119909minus

119896 119896) +

1

2sum119894

119890119894tr 119867(119894)

119909119909(119909119896 119896) 119875119896 (3)

COV119896 = 119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896

+ 120583119896

(4)

119870119896 = 119875minus

119896119867119909119879(119909minus

119896 119896) + COVminus1

119896 (5)

119909119896 = 119909minus

119896+ 119870119896V119896 (6)

119875119896 = 119875minus

119896minus 119870119896119878119896119870119896

119879 (7)

where 119865119909(119909 119896 minus 1) and 119867119909(119909 119896) are Jacobian matrices of 119891119894and ℎ119894 which are expressed by

[119865119909 (119909 119896 minus 1)]1198941198941015840

=120597119891119894 (119909 119896 minus 1)

1205971199091198941015840

[119867119909 (119909 119896)]1198941198941015840

=120597ℎ119894 (119909 119896)

1205971199091198941015840

(8)


119865(119894)119909119909

(119909 119896 minus 1) and119867(119894)119909119909

(119909 119896) are the Hessian matrices of 119891119894 andℎ119894 which are defined by

[119865(119894)

119909119909(119909 119896 minus 1)]

1198941198941015840=

1205972119891119894 (119909 119896 minus 1)

1205971199091198941205971199091198941015840

[119867(119894)

119909119909(119909 119896)]

1198941198941015840=

1205972ℎ119894 (119909 119896)

1205971199091198941205971199091198941015840

(9)

and 119890119894 is the unit vector in direction of the coordinate axis 119894

22 Stochastic ErrorModeling Technique in Inertial NavigationSystems Techniques developed for the correction of inertialsensor errors can generally be divided into two categoriesthat are (1) deterministic error calibration procedures and (2)stochastic error estimation models Only the stochastic errorestimation models are considered in this work

In general stochastic errors can be separated into twocategories that is white noise and colored (or correlated)noise White noise is a random process characterized by anAutocorrelation Function (ACF) of the form of a Dirac Onthe other hand colored noise may exhibit different ACFsdepending on the nature of the noise (pink red grey etc)

White noise can generally be mitigated using adequatelow-pass filter orWavelet Denoising Technique (WDT)whilecolored noise must be precisely modeled in order to beestimated and compensated Traditionally colored noise ofIMUs is modeled as a first-order Gauss-Markov Process(1GMP) with an exponentially decaying ACF This sectionpresents details on stochastic processes to model the colorednoise as well as Wavelet Denoising Technique (WDT) toeliminate the white noise in IMUs

221 Stochastic Processes Themost important errors to con-sider are stochastic errors since these errors if not dealt withwill have a significant negative impact on the accuracy ofvehicular tracking output [5] Several research groups startedinvestigating the possibility of using Mont-Carlo simulationtechnique for modeling the stochastic errors of MEMSdevices due to multicomponent character of the sensorsHowever it increases the number of degrees of freedom inthe dynamic model [22] So the alternative methods likegeneralized stochastic perturbation technique StochasticFinite Element Method (SFEM) and Second-Order SecondMoment (SOSM) approach are considered in [23ndash25]

Although there are several stochastic processes thissection addresses only those that were considered for thisstudy namely first-order Gauss-Markov Process (1GMP)and Autoregressive Processes (ARP) The relation betweenthese two stochastic processes and proposed stochastic errormodeling will be detailed in Section 31

First-Order Gauss-Markov Process (1GMP) Traditionally thenoises of IMUs aremodeled as 1st-order Gauss-Markov Proc-ess (1GMP) [8 26ndash28]The continuous-time and the discrete-time models of GMP can be given respectively by (10) and(11) Consider

(119905) = minus119860 (119905)

119879119888+ 119908 (119905) (10)

1

e1205902

1205902

minus120591 120591

minus1

120573+1

120573

R(120591)

Figure 1 Autocorrelation Function (ACF) for 1GMP

119860119896+1 = 119890minusΔ119905119879

119888 sdot 119860119896 + 119908 (119899) (11)

where 119860119896 and 119908(119899) are the 1GMP and the white noise (WN)and Δ119905 and 119879119888 are the sampling and the correlation timesrespectively The noise covariance can be expressed by

1205902

119908119896= 1205902

119909119896(1 minus 119890

minus2Δ119905119896119879119888) (12)

Classically 1GMPrsquos parameters can be estimated by Auto-correlation Function (ACF) Several studies investigated theuse of ACF in analyzing the stochastic error of the inertialsensors and also to acquire 1GMPrsquos parameters It can bemodeled with an exponentially decaying ACF as shown inFigure 1 The essential parameters required to model thisprocess are 120590

2 and 120573 which are given by

119877 (120591) = 1205902119890120573|120591|

120573 =1

119879119888 (13)

where 1205902 120591 and 120573 are the noise covariance the time lag andthe reciprocal of the process correlation time respectively Asshown in Figure 1 this process limits the uncertainty of thesignal and this forms the distinctive feature of this processAnd so 119877(120591) at any correlation time is less than or equal to119877(0) However the accuracy of ACF results depends on thelength of the recorded data

Autoregressive Processes (ARP) ACFmight be matched to thehigher order of GM instead of the 1st-order one for MEMSinertial sensors as stated by many workers [29] Thus theother method to model the errors which is presented in [29]is Autoregressive Processes (ARP)ARP is one type ofAutore-gressive Moving Average (ARMA) process which is pro-duced by the combination of past values and is expressed by

119860119896 = minus

119887

sum119899=1

120572119899 sdot 119860119896minus119899 + 1205730120596119896 (14)

where 119860119896 119887 120572119899 1205730 and 120596119896 are the ARP output order ofprocess model parameters standard deviation and whitenoise respectively Given the value of 119887 minimizing the rootmean square error (RMSE) between the original signal andthe signal estimated by ARP can help us to identify the ARPrsquoscoefficients This error is presented by

120598119896 = sum119896

[119860119896 +

119887

sum119899=1

120572119899119860119896minus119899]

2

(15)


1 LOD 2 LODs 3 LODs

A3

A2

A1

D1

D2

D3

X(n)

Low-pass

High-pass

Low-pass

High-pass

Low-pass

High-pass

Figure 2 Application of Wavelet Denoising Technique (WDT) with different Levels of Decomposition (LODs)

There are several identification algorithms to estimate theARPrsquo coefficients For this study three methods namelyYule-Walker (YW) Burgersquos and modified-covariance(MCOV) algorithms will be considered for implementationThe method that gives the minimum RMSE will be the bestone to estimate the ARPrsquos coefficients So an investigation ofthe best method to estimate the ARPrsquo coefficients is con-sidered in this study

222 Wavelet Denoising Technique (WDT) Wavelet Denois-ing Technique (WDT) is based on the wavelet filteringmethod which can eliminate the stochastic error in themeas-urements of the inertial sensors at High Frequencies (HFs)without altering important information contained in thesignal [30 31] A Multiresolution Analysis (MRA) algorithmwas performed on the WDT which can decompose a signalinto different subbands with various time and frequency res-olutions Conceptually in WDT signal 119909(119899) is filtered by onehigh-pass filter and one low-pass filter with downsampling bytwo (darr2) in the continuous Levels of Decomposition (LODs)as shown in Figure 2

The maximum amount of information about the originalsignal 119909(119899) was contained in the approximation coefficients(119860119896) in each subband Hence the minority information of

119909(119899) was contained in HF noise components which wereidentified by detailed coefficients (119863119896) This technique canbe combined with the different noise modeling process like1GMP and ARP to investigate the stochastic error of inertialsensors After applying the technique the coefficients of1GMP were determined from the enduring noises Morerelated explanations are presented in Sections 3 and 4

3 Proposed Methods

This paper proposes combining two independent and com-plementary solutions into a global integrated navigationsystem to provide stable low-cost ubiquitous automotive nav-igation in severe urban environments The first proposed

solution consists of augmenting the navigation solutionwith combined 1GMP-ARP for dynamically estimating andcompensating the static errors of low-cost inertial Thesecond proposed solution is to combine both linearized nav-igation filters and AI techniques into an intelligent cognitivenavigator SOEKF as the linearized navigation filter helps inreducing the linearization error during the filtering processand FIS-based approach as an AI technique helps in predict-ing the error states of the SOEKFThis section presents detailson these two proposed solutions

31 Proposed Stochastic Error Modeling of Low-Cost IMUAutocorrelation Function (ACF) which is presented in Sec-tion 2 does not fully match the random error characteristicsof low-cost MEMS-based IMUs What mainly causes thismismatch is the high nonlinearity of the low-cost inertialsensor errors leading to other distributions instead of 1GMP[6] Thus there is a need for the development of bettermodel-based methods to accurately estimate the distributionof the 1GMP low-cost inertial sensor errors This study pro-poses using a new method named ldquocombined 1GMP-ARPmethodrdquo as an alternative candidate to identify the 1GMPrsquosparameters distribution in the low-cost inertial sensors Thefundamental concept of the model is 1GMP However its 120590

2

and 120573 are calculated by 1st-order ARP rather than by ACFSubstituting 119887 = 1 within (14) gives the first-order ARP asfollows

119860119896 = 1205721 sdot 119860119896minus1 + 1205730120596119896 (16)

Combining (11) and (16) the value of 120573 in 1GMP can berelated to 1205721 in ARP as

120573 = minus ln 1205721

Δ119905 (17)

In this study we call this process ldquocombined 1GMP-ARPrdquomodelThemain advantage of themodel is that the inaccurateACF does not have a detrimental effect on the estimated


IMU

Acc measurements

Gyro measurements

Rotation angular rate of Earth Kinematic

equation of gravitational

velocity Attitudevelocitypositionof IMU

Position vector in the Earth

frame

Gravity calculation

Mag measurements

IMU mechanization

GPS

Kinematic equation of velocity

SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus

Figure 3 Proposed hybrid FIS-SOEKF model in navigation system

correlation time As it has been mentioned before using theidentification algorithms (ie YW Burgersquos and MCOV algo-rithms) to estimate the ARPrsquos coefficients of the combined1GMP-ARP model is the goal of this section Furthermorein order to compare the combined 1GMP-ARP model withclassical 1GMP model first the 1GMP parameters are experi-mentally estimated by ACF then the parameters are obtainedby ARP using different identification algorithms

Furthermore Wavelet Denoising Technique (WDT) canbe combined with the ACF (as the experimental method toestimate the classical 1GMP parameters) or with proposedcombined 1GMP-ARP (as an alternative method to estimatethe parameters of 1GMP) to investigate the stochastic error ofinertial sensors After applying the technique the coefficientsof 1GMP obtained by ARP or ACF were determined fromthe enduring noises These combinations are augmented inGPSINS integration system to improve the navigationalsolution and their results are presented in Section 4

32 Proposed Hybrid FIS-SOEKF Model for GPSINS Integra-tion Two different approaches of adaptive Kalman filteringnamely Innovation Adaptive Estimation (IAE) and MultipleModel Adaptive Estimation (MMAE) are considered by sev-eral research groups [32 33]These two approaches are basedon Covariance Matching Estimation Technique (COMET)This study adopted the IAE concept in the fuzzy part of theproposed model

The dynamic characteristics of vehicle motion in bodyframe and the navigation frame form the basis for the SOEKFprocessThe FIS can be exploited to increase the accuracy andthe robustness of the SOEKF and to prevent its divergence inthe tuning phase of SOEKFHence FISwas used as a structurefor identifying the dynamic variations and for implementingthe real-time tuning of the nonlinear error model It canprovide a good estimation in maintaining the accuracy and

the tracking-capability of the system Figure 3 depicts how theproposed hybrid FIS-SOEKF model performs as the ubiqui-tous navigation system

Fuzzy Inference System (FIS) is a rule-based expertmethod that can mimic human thinking and understandlinguistic concepts rather than the typical logic systems [34]The advantage of the FIS is realized when the algorithm ofthe estimation states becomes unstable because of the highcomplexity of the system FIS are also used for the knowledgeinduction process because they can serve as estimators forgeneral purpose [35]

FIS architecture performs three types of operations fuzzi-fication fuzzy inference and defuzzification Fuzzificationconverts the crisp input values into fuzzy values fuzzy infer-encemaps the given inputs into an output anddefuzzificationconverts the fuzzy operation into the new crisp values TheFIS can convert the inaccurate data to normalized fuzzycrisps which are represented by the ranges of possible valuesMembership Functions (MFs) and the confidence-rate of theinputs In addition the FIS are capable of choosing an optimalMF under certain specific criteria applicable to a specificapplication The deterministic output of FIS and its perfor-mance depend on the effective fuzzy rules the considereddefuzzification process and the reliability of the MF values

The proposed FIS model is based on the innovationprocess of the covariancematrix as the input of the FIS as wellas the difference between the actual covariancematrix and thetheoretical covariance matrix Figure 4 shows the proposedFIS overview which was used in this study The theoreticalcovariance matrix based on the innovation process wascomputed partly in SOEKF by (4) The actual covariancematrix according to [36] presented is proposed by

120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment

Rules builderFIS type

Member functionsof inputs

Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance

Analyticalcovariance

Input

New

matrix

Output

covariance

Figure 4 Proposed FIS part for hybrid FIS-SOEKF in navigationsystem

where 119863 is the window size (it is designated by moving win-dow techniques and experimentally chosen as119863 = 20) and V119894is [V1 V2 sdot sdot sdot V119898]

119879 So the difference is presented by

120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)

In fact 120591 can display the Degree of Incompatibility(DOI) between the actual and the theoretical covariancematrices When 120591 is near zero it means that these two valuesnearly match and the absolute value of the difference can benegligible However if 120591 is smaller or greater than zero itmeans that the theoretical value (COV119896) is greater or smallerthan the actual value (120572) To correct this difference thediagonal element of 120583119896 in (4) should be adjusted relying onthe DOI The proposed rules of assessment according to thedifference between the actual and the theoretical covariancematrices are described as three scenarios of FMs

(1) If 120591 is greater than 0 then 120583119896 will decrease due to 120575120583119896

(2) If 120591 is less than 0 then 120583119896 will increase due to 120575120583119896

(3) If 120591 cong 0 then 120583119896 will remain unchanged

Consider 120575120583119896

= 120583119896 minus 120583119896minus1 The related FMs are shown inFigure 5(a) wherein ldquoDrdquo ldquoBrdquo and ldquoIrdquo denote ldquoDecreaserdquoldquoBalancerdquo and ldquoIncreaserdquo respectively In addition (2) and(5) show that if 120583119896 is perfectly observed variation in 120578119896 canmake changes in COV119896 directly So with the observationof the incompatibility between the actual and the theoret-ical covariance matrices augmentation or diminishmentbecomes essential for 120578119896 The proposed rules of assessmentfor this purpose are described as two MFs of FIS

(1) If 120591119896 minus 120591119896minus1 = 0 then 120578119896 will remain unchanged(2) If 120591119896 minus 120591119896minus1 = 0 then 120578119896 will change due to 120575120578

119896

Table 1 Technical specification of the angular rate and the acceler-ation sensors in micro-iBB

Accelerometer GyroscopeData rate 200Hz Data rate 200HzTemperaturerange minus40sim+85∘C Temperature

range minus40sim+85∘C

Input range plusmn2 g Input range plusmn250∘secLinearsensitivity 2mgLSB Sensitivity m∘secdigit

Accelerationnoise density 220 120583gradicHz

Rate noisedensity 003∘secradicHz

Nonlinearity 02∘sec

Consider 120575120578119896

= 120578119896 minus 120578119896minus1 The related FMs are shown inFigure 5(b) wherein ldquoLZrdquo ldquoZrdquo and ldquoGZrdquo denote ldquolower thanzerordquo ldquozerordquo and ldquogreater than zerordquo respectively

4 Experimental AnalysisResults and Discussion

The static raw data for analysis of the stochastic errorwas obtained from the wireless Micro-intelligent Black Box(Micro-iBB) which was designed and assembled by theVTADS team of the LASSENA Lab in the ETS Micro-iBBconsists of a triad of accelerometers gyroscopes magne-tometers and a temperature sensor The serial numbers ofthe accelerometer the gyroscope and the demonstrationboard are LSM303DLHC L3GD20 and STEVAL-MKI119V1respectivelyThe selected aiding devices inMicro-iBB consistof a GNSS receiver (u-blox 7) Table 1 and Figure 6 presentrespectively the details of the Micro-iBB and its technicalspecification which were used in this paper

41 Stochastic Error Modeling Technique This section ana-lyzes the stochastic error to determine the effect of randomerrors on gyroscopes and accelerometers of the Micro-iBBFirst the ACFs were considered after recording the raw dataWDT was applied before data processing to weaken high fre-quency noises and remove preliminary biases in the sensorsBy using WDT uncorrelated and colored noises in the signalcould be reduced Figure 7 depicts the effect of different LODsin eliminating the systemrsquos noiseThen the related parametersof ACF for 1GMP namely 1205902

119908119896and 120573 should be extracted

Figure 8(a) shows the ACF for accelerometers after usingWDT with six LODs This figure confirms the existence ofresidual noise along 119910-axis and 119911-axis of the accelerometerseven after applying the WDT This noise is the uncorrelatedterm of the noise which cannot be attenuated by the ACF in1GMPTheACFof the119909-axis of the accelerometer implies thatthe 119909-axis is more correlated than are the other axes

Figure 8(b) shows the ACF of gyroscopes after usingWDT with 6 LODs This figure shows that although the119910-axis and 119911-axis gyroscopes were influenced principally byHF and white noises the 119909-axis of the gyroscope shows morecorrelated components than the other axes do The com-parison between Figures 1 and 8 clearly shows that


IBD

05040302010minus01minus02minus03minus04minus050

05

1

(a)

LZ GZZ

025015005minus005minus015minus0250

05

1

(b)

Figure 5 Membership Functions (MFs) for (a) 120575120583119896and (b) 120575120578119896

Figure 6 Wireless Micro-intelligent Black Box (Micro-iBB)

1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)

Figure 7 WDT with different LODs for (a) 119909-axis of accelerometers and (b) 119910-axis of gyroscopes in Micro-iBB in stationary situation

the ACF cannotmodel the low-costMEMS IMUsrsquo errors with1GMP perfectlyTherefore for this study it was performed tomodel 1GMP by utilizing ARPrsquos parameters as an alternativesolution to overcome this problem in the analysis of stochasticerrors The experimental results of ACF for all sensors aresummarized in Table 2 under ldquoACF-basedrdquo columnFollowing this MCOV YW and Burgrsquos algorithms have alsobeen applied to the denoised measurements (with differentlevels of decomposition) in order to estimate the parametersof combined 1GMP-ARP Results for these algorithms also aresummarized in Table 2 under ldquoYW-ARP-basedrdquo ldquoMCOV-ARP-basedrdquo and ldquoBURGrsquos-ARP-basedrdquo columns for 6 levelsof decomposition These results present a considerabledifference between the parameters obtained with each of thefour identifications methods

42 GPSINS Integration To evaluate the performance ofthe proposed models of GPSINS integration their resultsare compared with those of the conventional methods ofGPSINS integration For demonstration the loosely coupledGPSINS integrationwas utilized in a three-dimensional nav-igation system In the loosely coupled GPSINS integrationposition and velocity observed by GPS are the auxiliarymeasurements So in the SOEKFmeasurementmodel can beas follows

120575119911 = 119867120575119909 + 120578 (20)

where 119867 is [11986711987703times303times2

03times311986711987703times2

] and 119867119877 = 1198683times3 Measurement noisevector 120578 is [120578119877 120578119881]

119879 denoting the position and velocitynoise Vector 120578 is defined by measurement noise model 119877 in


times105

times10minus4

ACCx

ACCy

ACCz

minus05 0 05 1minus1

Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)

Figure 8 ACF of 1GMP for (a) accelerometers and (b) gyroscopes in Micro-iBB after employing WDT with 6 LODs

Table 2 The parameters needed to model 1GMP with different methods for ARP after 6 LODs of WTD

Models ACF-based YW-ARP-based MCOV-ARP-based BURGrsquos-ARP-based1205902

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573

Acc119909 32 times 10minus12 12600 34 times 10

minus10 1380 14 times 10minus11 1740 16 times 10

minus11 11100Acc119910 29 times 10


minus11 11100 31 times 10minus11 11500

Acc119911 71 times 10minus12 11100 54 times 10minus10 1320 46 times 10minus11 1630 51 times 10minus11 1900Gyr119909 25 times 10minus12 11700 41 times 10minus10 160 39 times 10minus11 1520 62 times 10minus11 11600Gyr119910 18 times 10



Gyr119911 37 times 10minus12 11500 43 times 10


minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled

according to the GPS position and velocity uncertainty withzero mean and covariance matrix The error states vectorrelating to the navigation system of this study is as follows

120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)

where 120575119877 and 120575119881 are respectively [120575119877119873 120575119877119864 120575119877119880] and[120575119881119873 120575119881119864 120575119881119880] denoting position and velocity residualerrors in EastNorthUp (ENU) frame 120575120595 is [120575120595119873 120575120595119864 120575120595119880]

and is the attitude error 120575119887119892 and 120575119887119886 are respectively[120575119887119892

119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-

chastic error states of accelerometers and gyroscopes alongthe three axes In fact 120575119887119892 and 120575119887119886 are included in the statevector of error propagation model to dynamically estimatethe six stochastic error states From the definition of 1GMP

in (10) the dynamic equations related to stochastic errorstates of accelerometers and gyroscopes can be shown by

[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)

Table 2 presents the 1GMPrsquos parameters which are estimatedby ACF and the combined 1GMP-ARP (using different algo-rithms) for 6 LODs ofWTD So the 120573119892 120573119886119908119892 and119908119886 in (21)can be initialized by the values in Table 2 for 6 LODs ofWTD

The proposed land-navigation solution was implementedwith an experimental setup for a road test The tests wereperformed by Micro-iBB whose specifications are detailedunder Section 41 The results were evaluated against a ref-erence solution provided by Novatel SPAN technology Itconsists ofNovatel receiver andHoneywell tactical-grad IMU


minus7366 minus7364 minus7362 minus736 minus7358 minus7356

4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)

Figure 9 The freeway trajectory showing the five outages (a) on google map and (b) on Matlab

to validate the proposed method and to assess the overallperformance during the GPS outages

The raw data of the route-test was gathered from differentsensors of Micro-iBB The selected trajectories satisfy theoverall quality and reliability requirements of the proposedmodels of this study in real environments of various condi-tionsThe first trajectory was figured out in an urban freewaywhich allowed the authors to drive at high speedThe secondtrajectory was in downtown area that contains numerousprominent skyscrapers The test along this trajectory wasperformed at slow speed with frequent stops because of hightraffic and crowded road intersections

The goal is to investigate the efficiency of the proposedhybrid FIS-SOEKF using different LODs of WDT and usingvarious algorithms to estimate ARPrsquos coefficients for 1GMPThe WDTs considered for this purpose are 2 LODs 6 LODsand 8 LODs and the algorithms are MCOV YW and Burgrsquosalgorithm The proposed combinations are compared withtraditional SOEKF and two hybrid FIS-SOEKFs which usedACF to estimate the 1GMP coefficient with different LODsof WDT All the hybrid FIS-SOEKF solutions followed theidea of updating stochastic error states to the gyroscope andaccelerometers measurements RMSEs were estimated in allnavigational solutions with Novatel SPAN as the referencesolution

421 First Scenario Freeway The first route-test trajectoryselected for this study starts from freeway 5 near VigerEast Avenue (45509023 minus73557511) and ends at freeway 71near Lebeau Avenue (45524299 minus73652472) in MontrealQC Canada by boulevard Ville-Mariehighway 720 Westand highway 15 North (see Figure 9) This trajectory wasdone for nearly 11min of uninterrupted car navigation overa distance of 144 kmThe freeway trajectory was chosen withfive natural GPS outages to examine the systemrsquos performanceduring short and long outages that exist in urban canyonsFigure 9(a) depicts the route-test with the reference solutionon google map Figure 9(b) presents the position of the GPSoutages onMatlabwith red lines along the selected trajectory

Figure 10 presents the number of available satellites andthe time-duration of each GPS outage The first GPS outagein tunnel area occurred for 30 sec with less than number of7 available satellites There was no satellite for about two-thirds of the duration of this outage (see Figure 10(a)) Thesecond and third outages occurred along the two skyways for180 sec and 60 sec durations (see Figures 10(b)-10(c)) Figures

10(d)-10(e) present the last two outages in urban canyon areafor durations of 50 and 30 sec Generally higher movingspeed increases horizontal positioning errors Therefore thisscenario of high speed driving with several outages can testthe validity and robustness of the proposed models

Figure 11 presents the root mean square and the maxi-mum errors in horizontal positioning during the five GPSoutages in the first scenario for the twelve compared solu-tions The advantage of using the proposed method with 6LODs for updating stochastic error states to gyroscope andaccelerometer measurements can be seen by comparing itwith the traditional SOEKF and hybrid FIS-SOEKFs whichare used in ARP or ACF to estimate the 1GMP coefficientutilizing different LODs of WDT Figure 11 illustrates theresults obtained by (i) the traditional SOEKF without updat-ing stochastic error states shown as SOEKF and (ii) thoseobtained by the proposed hybrid FIS-SOEKF using ACF-based 1GMP after 2-LOD denoising shown as ldquoFS(ACF2)rdquoand after 6-LOD denoising shown as ldquoFS(ACF6)rdquo Therest of the solutions used the proposed hybrid FIS-SOEKFand are shown as ldquoFSrdquo in this figure (iii) ldquoMCOV-ARPrdquoldquoYW-ARPrdquo and ldquoBURGrsquos-ARPrdquo represent the solutions thatmodeled 1GMP by using ARPrsquos coefficients under ldquomodified-covariancerdquo ldquoYule-Walkerrdquo and ldquoBurgrsquosrdquo algorithms respec-tively The values which follow each solution denote thespecific LOD of WTD that was used in that solution

From the foregoing data it can be seen that the proposedhybrid FIS-SOEKF which used ACF-based 1GMP after 6-LOD denoising performed significantly better than didthe traditional SOEKF without updating stochastic errorstates This is because the proposed hybrid FIS-SOEKF candeal with the nonlinearity of systems and models howeverSOEKF exploits second-order linearized models to estimatethe state of systemsThe results also show that ldquoBURGrsquos-ARPrdquooutperformed ldquoSOEKFrdquo and also the proposed hybrid FIS-SOEKF that used ACF-based 1GMP after denoising

Furthermore in terms of the ARP-based 1GMP after 6-LOD denoising ldquoBURGrsquos-ARP6rdquo shows a major improve-ment in positioning accuracy as compared with the perfor-mance of ldquoYW-ARP6rdquo and ldquoMCOV-ARP6rdquo This is becauseBurgrsquos algorithm provides the highest accuracy when appliedto low-cost MEMS-based INS as compared to the otheralgorithms that determine the ARPrsquos coefficients ideallyDuring the second outage of 180 sec duration it can be seenthat the performance of all solutions degraded considerablybecause at least four satellites are needed in loosely coupled


0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)

100 15050Duration (s)

02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)

Figure 10 Available satellites in each outage for freeway trajectory (a) outage 1 (b) outage 2 (c) outage 3 (d) outage 4 and (e) outage5

510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)

53 421Outage (number)

SOEKFFS(ACF2)FS(ACF6)FS(MCOV-ARP2)FS(MCOV-ARP6)FS(MCOV-ARP8)

FS(YW-ARP2)FS(YW-ARP6)FS(YW-ARP8)FS(BURGrsquoS-ARP2)FS(BURGrsquoS-ARP6)FS(BURGrsquoS-ARP8)

(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)

2 3 4 51Outage (number)

(b)

Figure 11 (a) Root mean square error and (b) maximum errors in horizontal positioning during five GPS outages in the first scenario(freeway)


ReferenceSOEKFFS(ACF2)FS(ACF6)FS(MCOV-ARP2)FS(MCOV-ARP6)FS(MCOV-ARP8)


45476

45478

4548

45482

45484

45486

45488

4549

45492

minus73585 minus73575minus73595East position (∘)

Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785

minus73618 minus73616minus7362



East position (∘)

Nor

th p

ositi

on (∘

)

(b)

Figure 12 Different navigation solutions for the freeway trajectory (a) outage 2 and (b) outage 4

45495455

455054551

45515

minus73575 minus7357 minus73565 minus7356 minus73555minus7358East (∘)

Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)

Figure 13 Second (downtown) trajectory showing the five outages (a) on google map and (b) on Matlab

GPSINS integration Against this requirement the numberof satellites available was less than four for more than halfof the outage duration besides the duration itself beingquite long In addition using the low-cost MEMS-based INSworsened this situation So the second outage gave the worstresult with the maximum horizontal error being 630m andmean horizontal error 187m these cannot be shown in thefigure as the values are high

Figure 12 presents two sections of the freeway trajectoryduringGPS outages 2 and 4This figure shows the referenceand all the navigation solutions considered in this paperThese results confirm that the performance of the proposedhybrid FIS-SOEKF used with ARP-based 1GMP after 6-LOD denoising is acceptable as a new navigational solution

in freeways where the duration of GPS outages can be amaximum of 60 sec

422 Second Scenario Downtown Thesecond route-test tra-jectory selected starts from McGill Street near La CommuneOuest Avenue (45498718 minus73552831) and ends up at PinsOuest Avenue (45501466 minus73584548) in Montreal QCCanada by Boulevard St-LaurentShebrooke Ouest Streetand Peel Street (see Figure 13) Testing along this trajectorywas carried out throughmedium and high traffic volumes fornearly 50min of car navigation covering a distance of 105 kmThis downtown trajectory which runs through skyscraperswas chosen because of its five natural GPS outages to examinethe systemrsquos performance during short and long outages


2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)

Figure 14 Available number of satellites during each outage for downtown trajectory (a) outage 1 (b) outage 2 (c) outage 3 (d) outage4 and (e) outage 5

Figure 13(a) depicts route-test with the reference solution ongooglemap Figure 13(b) presents the details of the position ofthe GPS outages on Matlab with red lines along the selectedtrajectory

Figure 14 shows the number of available satellites and thetime-duration of each GPS outage for downtown trajectoryof Montreal To examine the proposed navigation solutionduring longer GPS outages the near outages were clubbedinto bigger outages The number of the bigger outages in thisscenario thus became 5 Figure 15 shows the RMSE and themaximumerror in horizontal positioning during the fiveGPSoutages for the twelve compared solutions The results of thisscenario confirm the accuracy obtained in the results of theprevious scenario

Figure 16 illustrates the results obtained for outages 2and 4 by (i) the traditional SOEKF without updating sto-chastic error states shown as ldquoSOEKFrdquo and (ii) thoseobtained by the proposed hybrid FIS-SOEKF using ACF-based 1GMP after 2-LOD denoising shown as ldquoFS(ACF2)rdquoand after 6-LOD denoising shown as ldquoFS(ACF6)rdquo (iii) Therest of the solutions used the proposed hybrid FIS-SOEKFwhich are shown as ldquoFSrdquo ldquoMCOV-ARPrdquo ldquoYW-ARPrdquo andldquoBURGrsquos-ARPrdquo represent the solutions that modeled 1GMPby using ARPrsquos coefficients under ldquomodified-covariancerdquoldquoYule-Walkerrdquo and ldquoBurgrsquosrdquo algorithms respectively Thevalues that follow each solution denote the specific LOD ofWTD that was used in that solution

From the data presented it can be seen that the proposedhybrid FIS-SOEKF which used ACF-based 1GMP after 2-LOD and 6-LOD denoising ldquoFS(ACF2)rdquo and ldquoFS(ACF6)rdquoperformed significantly better than did the traditionalSOEKF without updating stochastic error states The resultsalso show that ldquoFS(BURGrsquos-ARP)rdquo among all the ldquoSF (ARP)rdquonavigation solutions outperformed ldquoSOEKFrdquo ldquoFS(ACF2)rdquoand ldquoFS(ACF6)rdquo Furthermore in terms of the ARP-based1GMP after 6-LOD denoising ldquoFS(BURGrsquos-ARP6)rdquo shows amajor improvement in positioning accuracy in the down-town scenario as compared with the accuracy obtained byldquoFS(YW-ARP6)rdquo and ldquoFS(MCOV-ARP6)rdquo This is becauseBurgrsquos algorithm provides the highest accuracy when appliedto low-cost MEMS-based INS as compared to the otheralgorithms that determine the ARPrsquos coefficients perfectly

As regards the number of satellites and the duration ofeach GPS outage it can be seen that there are less than 4satellites for two-thirds of the duration and less than 5 sat-ellites during the remaining part of the duration for outages2 and 4 However RMSE in outage 4 is much more thanthat in outage 2 because the duration of outage 4 (180 sec)is longer than that of outage 2 (110 sec) The results alsoshow that outage 5 presents in spite of its longest duration(310 sec) lesser RMSE than that of outages 3 and 4 This isbecause 6ndash8 satellites existed for less than half of the durationof outage 4 and less than 4 satellites for three-quarters of theduration Outage 2 shows the lowest error (2m) because ofits short outage duration


0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)

Figure 15 Horizontal positioning errors during the five GPS outages in the downtown scenario (a) rootmean square error and (b)maximumerrors

455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075

minus73571minus73572minus73573minus73574



East position (∘)

Nor

th p

ositi

on (∘

)

(b)

Figure 16 Different navigation solutions for downtown trajectory (a) outage 2 and (b) outage 4


5 Conclusions

In this paper two novel auxiliary methods are proposedto improve the performance of ultralow-cost MEMS-basedIMU in a vehicular navigation system The proposed meth-ods can cover two areas modeling stochastic errors andperforming GPSINS integration The proposed 1GMP-ARPconsists of a complete stochastic error modeling of the noisecomponents in MEMS low-cost inertial sensors This papershows a big cooperation between 1GMP ARP and WDT tocharacterize the noise components Different levels of decom-position in denosing technique as well as various methodsto identify the ARPrsquo parameters were considered to clarifythe best one for stochastic error modeling in MEMS low-cost inertial sensors Result presents that Burgrsquos methodafter applying the six levels of decomposition in denosingtechnique performs much better than modified-covarianceand Yule-Walker methods with different levels of decom-position The second proposed method consists of an inte-gration of SOEKFFIS to enhance the performance of low-cost GPSINS integration for navigation data fusion The FISpart is exploited for dynamic adjustment of the process noisecovariance by observing the innovation process which isutilized in SOEKF-part to maintain further enhancement inthe estimation of the accuracy The results confirm that theproposed hybrid FIS-SOEKF can provide further improve-ment in the overall performance compared to SOEKF inharsh environment

Future research work related to this study will focus onmore complicated GPSINS integration structures such astightly and ultratightly coupled and noise modeling withhigher order stochastic error to reduce the errors in thepositing of the navigation system In addition GPSINS inte-gration will expand to GNSSBDSlow-cost-INS integrationto utilize the advantages of the multi-GNSS environment

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is part of the project entitled VTADS VehicleTracking and Accident Diagnostic System It is supportedby the Natural Sciences and Engineering Research Councilof Canada (NSERC) and Ecole de Technologie Superieure(LASSENA Lab) in collaboration with two industrial part-ners namely iMetrik Global Inc and Future Electronics

References

[1] J W Zellner R M van Auken D P Chiang P C Broen JKelly and Y Sugimoto ldquoExtension of the Honda-DRI lsquoSafetyImpact Methodologyrsquo (SIM) for the NHTSA Advanced CrashAvoidance Technology (ACAT) program and application to aprototype advanced collision mitigation braking systemrdquo SAEInternational Journal of Passenger CarsmdashMechanical Systemsvol 2 no 1 pp 875ndash894 2009

[2] T Hai Ta D Minh Truong T Thanh Thi H Nguyen T DinhNguyen andG Belforte ldquoMulti-GNSS positioning campaign inSouth-East Asiardquo Coordinates vol 9 pp 11ndash20 2013

[3] D M Truong and T H Ta ldquoDevelopment of real multi-GNSSpositioning solutions and performance analysesrdquo in Proceedingsof the International Conference on Advanced Technologies forCommunications (ATC rsquo13) pp 158ndash163 Ho Chi Minh Viet-nam October 2013

[4] K-P Schwarz and N El-Sheimy ldquoFuture positioning andnavigation technologiesrdquo in Study Performed under the ScientificServices Agreement with Batelle Columbus Division and Topo-graphic Engineering Center Fort Belvoir VA USA 1999

[5] M S Grewal L R Weill and A P Andrews Global PositioningSystems Inertial Navigation and Integration John Wiley ampSons New York NY USA 2007

[6] J Georgy U Iqbal andANoureldin ldquoQuantitative comparisonbetween kalman filter and particle filter for low cost INSGPSintegrationrdquo in Proceedings of the 6th International Symposiumon Mechatronics and its Applications (ISMA rsquo09) pp 1ndash7Sharjah United Arab Emirates March 2009

[7] A Oppenheim and R Schafer Discrete-Time Signal ProcessingPearson New International Edition Pearson Education UpperSaddle River NJ USA 2013

[8] A Noureldin T B Karamat M D Eberts and A El-ShafieldquoPerformance enhancement of MEMS-based INSGPS integra-tion for low-cost navigation applicationsrdquo IEEE Transactions onVehicular Technology vol 58 no 3 pp 1077ndash1096 2009

[9] T B Karamat Implementation of tightly coupled INSGPS inte-gration for land vehicle navigation [MS thesis] Electricalengineering Royal Military College of Canada 2009

[10] J Pusa Strapdown inertial navigation system aiding with non-holonomic constraints using indirect Kalman filtering [MSthesis] Department of Mathematics Tampere University ofTechnology 2009

[11] Z Shen J Georgy M J Korenberg and A Noureldin ldquoLowcost two dimension navigation using an augmented KalmanfilterFast Orthogonal Search module for the integration ofreduced inertial sensor system and Global Positioning SystemrdquoTransportation Research Part C Emerging Technologies vol 19no 6 pp 1111ndash1132 2011

[12] L Zhao H Qiu and Y Feng ldquoStudy of robust 119867infin filteringapplication in loosely coupled INSGPS systemrdquoMathematicalProblems in Engineering vol 2014 Article ID 904062 10 pages2014

[13] P Aggarwal Z Syed and N El-Sheimy ldquoHybrid Extended Par-ticle Filter (HEPF) for integrated civilian navigation systemrdquo inProceedings of the IEEEION Position Location and NavigationSymposium (PLANS rsquo08) pp 984ndash992 Monterey Calif USAMay 2008

[14] J Cheng D Chen R Landry Jr L Zhao and D Guan ldquoAnadaptive unscented kalman filtering algorithm for MEMSGPSintegrated navigation systemsrdquo Journal of Applied Mathematicsvol 2014 Article ID 451939 8 pages 2014

[15] K-W Chiang and Y-W Huang ldquoAn intelligent navigator forseamless INSGPS integrated land vehicle navigation applica-tionsrdquoApplied Soft Computing Journal vol 8 no 1 pp 722ndash7332008

[16] A Noureldin A El-Shafie andM R Taha ldquoOptimizing neuro-fuzzy modules for data fusion of vehicular navigation systemsusing temporal cross-validationrdquo Engineering Applications ofArtificial Intelligence vol 20 no 1 pp 49ndash61 2007


[17] L Semeniuk and A Noureldin ldquoModeling of INS position andvelocity errors using radial basis function neural networksfor INSGPS integrationrdquo in Proceedings of the ION NationalTechnical Meeting pp 18ndash20 Monterey Calif USA January2006

[18] A Noureldin A El-Shafie and M Bayoumi ldquoGPSINS inte-gration utilizing dynamic neural networks for vehicular navi-gationrdquo Information Fusion vol 12 no 1 pp 48ndash57 2011

[19] J Skaloud Optimizing georeferencing of airborne survey systemsby INSDGPS [PhD thesis] Department of Geomatics Engi-neering Alberta Canada 1999

[20] Y Bar-Shalom X R Li and T Kirubarajan Estimation withApplications to Tracking and Navigation Theory Algorithms andSoftware John Wiley amp Sons New York NY USA 2004

[21] B Sadeghi and B Moshiri ldquoSecond-order EKF and unscentedKalman filter fusion for tracking maneuvering targetsrdquo in Pro-ceedings of the IEEE International Conference on InformationReuse and Integration (IEEE IRI rsquo07) pp 514ndash519 IEEE LasVegas Nev USA August 2007

[22] R Ardito C Comi A Corigliano and A Frangi ldquoSolid damp-ing inmicro electromechanical systemsrdquoMeccanica vol 43 no4 pp 419ndash428 2008

[23] A Kumar and D Chakraborty ldquoEffective properties of thermo-electro-mechanically coupled piezoelectric fiber reinforcedcompositesrdquo Materials amp Design vol 30 no 4 pp 1216ndash12222009

[24] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationWiley-Interscience 1992

[25] M Kaminski and A Corigliano ldquoSensitivity probabilistic andstochastic analysis of the thermo-piezoelectric phenomena insolids by the stochastic perturbation techniquerdquoMeccanica vol47 no 4 pp 877ndash891 2012

[26] M El-Diasty and S Pagiatakis ldquoA rigorous temperature-dependent stochastic modelling and testing for MEMS-basedinertial sensor errorsrdquo Sensors vol 9 no 11 pp 8473ndash84892009

[27] P Lavoie Systeme de Navigation Hybride GPSINS a Faible Coutpour la Navigation Robuste en Environnement Urbain M IngGenie Electrique Ecole de Technologie Superieure MontrealCanada 2012

[28] S Nassar K-P Schwarz N El-Sheimy and A NoureldinldquoModeling inertial sensor errors using autoregressive (AR)modelsrdquo Navigation vol 51 no 4 pp 259ndash268 2004

[29] S Nassar K-P Schwarz N El-Sheimy and A NoureldinldquoModeling inertial sensor errors using autoregressive (AR)modelsrdquo Navigation Journal of the Institute of Navigation vol51 no 4 pp 259ndash268 2004

[30] C H Kang S Y Kim and C G Park ldquoImprovement of alow cost MEMS inertial-GPS integrated system using waveletdenoising techniquesrdquo International Journal of Aeronautical andSpace Sciences vol 12 no 4 pp 371ndash378 2011

[31] N El-Sheimy S Nassar and A Noureldin ldquoWavelet de-noisingfor IMU alignmentrdquo IEEE Aerospace and Electronic SystemsMagazine vol 19 no 10 pp 32ndash39 2004

[32] F Jiancheng and Y Sheng ldquoStudy on innovation adaptive EKFfor in-flight alignment of airborne POSrdquo IEEE Transactions onInstrumentation andMeasurement vol 60 no 4 pp 1378ndash13882011

[33] J Ma S Ni W Xie and W Dong ldquoAn improved strong track-ing multiple-model adaptive estimation a fast diagnosis algo-rithm for aircraft actuator faultrdquo Transactions of the Institute ofMeasurement and Control 2015

[34] D Bhatt P Aggarwal V Devabhaktuni and P BhattacharyaldquoA new source difference artificial neural network for enhancedpositioning accuracyrdquo Measurement Science and Technologyvol 23 no 10 Article ID 105101 2012

[35] L M Bergasa J Nuevo M A Sotelo R Barea and M ELopez ldquoReal-time system formonitoring driver vigilancerdquo IEEETransactions on Intelligent Transportation Systems vol 7 no 1pp 63ndash77 2006

[36] R Vershynin ldquoHow close is the sample covariance matrix tothe actual covariancematrixrdquo Journal ofTheoretical Probabilityvol 25 no 3 pp 655ndash686 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



than 4 satellites in urban canyons Moreover it can improvethe accuracy of positioning especially when the navigationsystem considers the horizontal positioning accuracy [2]

However multi-GNSS-based techniques may increasethe intersystem interference particularly when multi-GNSSenvironments broadcast navigation signals in overlappedfrequency bands Moreover these techniques impose higherlevels of complexity because the analog-part of the receivershould deal with multiple-system multiple frequencies andconsiderable bandwidths [3] Finally the mentioned devel-oped techniques consist in postprocessing techniques thatrequire laboriousmanipulations and calculations whichmaynot be suitable for real-time commercial applications Amodern approach to solve the current positioning problems isto integrate the GPS signals with Low-cost Inertial Measure-ment Units (IMUs) data IMUs can estimate the position andthe attitude of the vehicle by employing an Inertial NavigationSystem (INS) mechanization process (accelerometers andgyroscopes) which is an integration of linear accelerationsand angular rates However MEMS-based low-cost IMUsperform accurately just for a limited time as their positioningand attitudinal errors increase steadily with time

MEMS-based IMUs are classified based on their massposition recognition their operation mode and their fab-rication procedure Piezoresistive Capacitive Piezoelectricand Resonant element sensors are different types of MEMSsensors which are placed in the classification of mass positionrecognition [4]These types of sensors transform the physicalwork caused by variation of movement into an electrical sig-nal Piezoresistive and Piezoelectric sensors are widely usedin MEMS-based IMUs because of their simplicity cost-effectiveness and wide dynamic range However they sufferfrom low precision and high sensitivity to temperature[4] Performance characteristics of MEMS-based sensorsinclude bandwidth noise floor misalignment drift linearitydynamic range and power consumption Thus the MEMS-based inertial sensor errors are generally composed ofstochastic errors which have an undesirable impact on thequality of the navigation solution [5]

The first part of the proposed methodology consists ofaugmenting the navigation solution with 1st-order Gauss-Markov Process (1GMP) for dynamically estimating andcompensating the stochastic errors of MEMS-based low-costIMUs Two different identification methods were exploitedin this study to identify the 1GMP parameters The first oneis a traditional method based on the analysis of Autocor-relation Function (ACF) and the second one is based onanalysis of 1st-order Autoregressive Process (ARP) modelACF does not fully match the random error characteristicsof low-cost MEMS-based IMUs What mainly causes thismismatch is the high nonlinearity of the low-cost inertialsensor errors leading to higher order GMPs (ie 2nd- or3rd-order GMPs) distributions instead of 1GMP [6] Thusthere is a need for the development of better model-basedmethods to accurately estimate the distribution of the 1GMPlow-cost inertial sensor errors This study proposes usingcombined 1GMP-ARP method as an alternative candidateto identify the 1GMPrsquos parameters distribution in the low-cost inertial sensors To estimate the ARPrsquos coefficients

three different algorithms were considered and implementednamely the Yule-Walker (YW) method the Burg methodand the modified-covariance method [7]

As these static errors have both high frequency (HF)term and low frequency (LF) term decreasing both of themis needed in order to develop the accuracy of the low-costinertial sensorsWavelet Denoising Techniques (WDTs) havebeen utilized in similar research studies due to their signif-icant role in removing the HF noises [8] So it is presenteda mixture of the WDT (with different levels) and 1GMP toevaluate the accuracy improvement of the low-cost inertialsensors when these methods are blended together

Traditionally GPSINS integration is carried out by usingKalman Filter (KF) techniques This approach has been val-idated in many studies where high-end IMUs were used [9]However recent studies report several shortcomings in KF-based INSGPS integration when using low-cost MEMS-based inertial sensors [6] Dynamic models that define thenavigation problem are originally nonlinear [10] but to allowthe use of simple linear filters such as the KF thesemodels aregenerally linearized under a small error assumptionWith theassumption made by the KF that the system is characterizedby a linear model driven by white Gaussian noise it is notpossible to describe the nonlinear error dynamics of MEMS-based INS [11]

The conventional 1st-order EKF is used extensively forintegration of high-end INS and GPS data which can achievea suboptimal estimation of Minimum Mean Square Error(MMSE) of the vehicle state vector [12] 1st-order ExtendedKalman Filter (EKF) is based on the assumption that thehigher order terms (2nd-order and up) of a Taylor seriesexpansion are small enough to be ignored This conditionmay hold for high-end navigation systems where the errorsare kept small and bounded but not for highly nonlinearsystems such asMEMS-based INS In addition to provide anacceptable position and attitude estimates the 1st-order EKFrequires proper nonlinear models of vehicle dynamics andmeasurement sensorsThefirst-order linearization especiallyin the presence of highly nonlinear dynamics often leads todivergence of the state covariance matrix [11] Thus usingsuch assumption for a low-cost navigation system may resultin unstable solution as demonstrated by several previouslypublished results [13]

Several research groups investigated the possibility ofusing sampling techniques to cope with highly non-Gaussianmultimodal error distributions in GPSINS integration andthus avoiding linearization of the dynamic models [13]Unscented Kalman Filter (UKF) and Particle Filter (PF) aretwo such alternative methods that can provide better per-formance than does classical KF or EKF but at high compu-tational costs [11 14] Some authors assert that the limitationsof the sampling and the Kalman-based techniques can beovercome by using Artificial Intelligence (AI) [15] Theintegration of AI algorithms with next generation embeddednavigation systems is made possible with the technologicaladvancement in computer technologies In the last decadevarious neural networks (NN) and Fuzzy Inference Systems-(FIS-) based integration scheme were introduced [16ndash18]AI algorithms require a training phase which is realized





2 Background





119910119896= ℎ (119909119896 119896) + 119903119896

(1)



minus

119896= 119891 (119909119896minus1 119896 minus 1) +

1

2sum119894

119890119894tr 119865(119894)

119909119909(119909119896minus1 119896 minus 1)

sdot 119875119896minus1

119875minus


119879(119896minus1 119896 minus 1) +

1

2sum119894

119890119894

sdot 119890119894119879tr 119865(119894)


(1198941015840)


+ 120578119896

(2)

where minus

119896and 119875minus



V119896 = 119911119896 minus ℎ (119909minus

119896 119896) +

1

2sum119894

119890119894tr 119867(119894)

119909119909(119909119896 119896) 119875119896 (3)

COV119896 = 119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896

+ 120583119896

(4)

119870119896 = 119875minus

119896119867119909119879(119909minus

119896 119896) + COVminus1

119896 (5)

119909119896 = 119909minus

119896+ 119870119896V119896 (6)

119875119896 = 119875minus

119896minus 119870119896119878119896119870119896

119879 (7)


[119865119909 (119909 119896 minus 1)]1198941198941015840

=120597119891119894 (119909 119896 minus 1)

1205971199091198941015840

[119867119909 (119909 119896)]1198941198941015840

=120597ℎ119894 (119909 119896)

1205971199091198941015840

(8)


119865(119894)119909119909

(119909 119896 minus 1) and119867(119894)119909119909


[119865(119894)

119909119909(119909 119896 minus 1)]

1198941198941015840=

1205972119891119894 (119909 119896 minus 1)

1205971199091198941205971199091198941015840

[119867(119894)

119909119909(119909 119896)]

1198941198941015840=

1205972ℎ119894 (119909 119896)

1205971199091198941205971199091198941015840

(9)








(119905) = minus119860 (119905)

119879119888+ 119908 (119905) (10)

1

e1205902

1205902

minus120591 120591

minus1

120573+1

120573

R(120591)


119860119896+1 = 119890minusΔ119905119879

119888 sdot 119860119896 + 119908 (119899) (11)


1205902

119908119896= 1205902

119909119896(1 minus 119890

minus2Δ119905119896119879119888) (12)



119877 (120591) = 1205902119890120573|120591|

120573 =1

119879119888 (13)



119860119896 = minus

119887

sum119899=1

120572119899 sdot 119860119896minus119899 + 1205730120596119896 (14)


120598119896 = sum119896

[119860119896 +

119887

sum119899=1

120572119899119860119896minus119899]

2

(15)


1 LOD 2 LODs 3 LODs

A3

A2

A1

D1

D2

D3

X(n)

Low-pass

High-pass

Low-pass

High-pass

Low-pass

High-pass






3 Proposed Methods




2


119860119896 = 1205721 sdot 119860119896minus1 + 1205730120596119896 (16)


120573 = minus ln 1205721

Δ119905 (17)



IMU

Acc measurements

Gyro measurements





frame

Gravity calculation

Mag measurements

IMU mechanization

GPS


SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus










120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













2 Background





119910119896= ℎ (119909119896 119896) + 119903119896

(1)



minus

119896= 119891 (119909119896minus1 119896 minus 1) +

1

2sum119894

119890119894tr 119865(119894)

119909119909(119909119896minus1 119896 minus 1)

sdot 119875119896minus1

119875minus


119879(119896minus1 119896 minus 1) +

1

2sum119894

119890119894

sdot 119890119894119879tr 119865(119894)


(1198941015840)


+ 120578119896

(2)

where minus

119896and 119875minus



V119896 = 119911119896 minus ℎ (119909minus

119896 119896) +

1

2sum119894

119890119894tr 119867(119894)

119909119909(119909119896 119896) 119875119896 (3)

COV119896 = 119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896

+ 120583119896

(4)

119870119896 = 119875minus

119896119867119909119879(119909minus

119896 119896) + COVminus1

119896 (5)

119909119896 = 119909minus

119896+ 119870119896V119896 (6)

119875119896 = 119875minus

119896minus 119870119896119878119896119870119896

119879 (7)


[119865119909 (119909 119896 minus 1)]1198941198941015840

=120597119891119894 (119909 119896 minus 1)

1205971199091198941015840

[119867119909 (119909 119896)]1198941198941015840

=120597ℎ119894 (119909 119896)

1205971199091198941015840

(8)


119865(119894)119909119909

(119909 119896 minus 1) and119867(119894)119909119909


[119865(119894)

119909119909(119909 119896 minus 1)]

1198941198941015840=

1205972119891119894 (119909 119896 minus 1)

1205971199091198941205971199091198941015840

[119867(119894)

119909119909(119909 119896)]

1198941198941015840=

1205972ℎ119894 (119909 119896)

1205971199091198941205971199091198941015840

(9)








(119905) = minus119860 (119905)

119879119888+ 119908 (119905) (10)

1

e1205902

1205902

minus120591 120591

minus1

120573+1

120573

R(120591)


119860119896+1 = 119890minusΔ119905119879

119888 sdot 119860119896 + 119908 (119899) (11)


1205902

119908119896= 1205902

119909119896(1 minus 119890

minus2Δ119905119896119879119888) (12)



119877 (120591) = 1205902119890120573|120591|

120573 =1

119879119888 (13)



119860119896 = minus

119887

sum119899=1

120572119899 sdot 119860119896minus119899 + 1205730120596119896 (14)


120598119896 = sum119896

[119860119896 +

119887

sum119899=1

120572119899119860119896minus119899]

2

(15)


1 LOD 2 LODs 3 LODs

A3

A2

A1

D1

D2

D3

X(n)

Low-pass

High-pass

Low-pass

High-pass

Low-pass

High-pass






3 Proposed Methods




2


119860119896 = 1205721 sdot 119860119896minus1 + 1205730120596119896 (16)


120573 = minus ln 1205721

Δ119905 (17)



IMU

Acc measurements

Gyro measurements





frame

Gravity calculation

Mag measurements

IMU mechanization

GPS


SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus










120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119865(119894)119909119909

(119909 119896 minus 1) and119867(119894)119909119909


[119865(119894)

119909119909(119909 119896 minus 1)]

1198941198941015840=

1205972119891119894 (119909 119896 minus 1)

1205971199091198941205971199091198941015840

[119867(119894)

119909119909(119909 119896)]

1198941198941015840=

1205972ℎ119894 (119909 119896)

1205971199091198941205971199091198941015840

(9)








(119905) = minus119860 (119905)

119879119888+ 119908 (119905) (10)

1

e1205902

1205902

minus120591 120591

minus1

120573+1

120573

R(120591)


119860119896+1 = 119890minusΔ119905119879

119888 sdot 119860119896 + 119908 (119899) (11)


1205902

119908119896= 1205902

119909119896(1 minus 119890

minus2Δ119905119896119879119888) (12)



119877 (120591) = 1205902119890120573|120591|

120573 =1

119879119888 (13)



119860119896 = minus

119887

sum119899=1

120572119899 sdot 119860119896minus119899 + 1205730120596119896 (14)


120598119896 = sum119896

[119860119896 +

119887

sum119899=1

120572119899119860119896minus119899]

2

(15)


1 LOD 2 LODs 3 LODs

A3

A2

A1

D1

D2

D3

X(n)

Low-pass

High-pass

Low-pass

High-pass

Low-pass

High-pass






3 Proposed Methods




2


119860119896 = 1205721 sdot 119860119896minus1 + 1205730120596119896 (16)


120573 = minus ln 1205721

Δ119905 (17)



IMU

Acc measurements

Gyro measurements





frame

Gravity calculation

Mag measurements

IMU mechanization

GPS


SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus










120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 LOD 2 LODs 3 LODs

A3

A2

A1

D1

D2

D3

X(n)

Low-pass

High-pass

Low-pass

High-pass

Low-pass

High-pass






3 Proposed Methods




2


119860119896 = 1205721 sdot 119860119896minus1 + 1205730120596119896 (16)


120573 = minus ln 1205721

Δ119905 (17)



IMU

Acc measurements

Gyro measurements





frame

Gravity calculation

Mag measurements

IMU mechanization

GPS


SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus










120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










IMU

Acc measurements

Gyro measurements





frame

Gravity calculation

Mag measurements

IMU mechanization

GPS


SOEKF

FIS

Attitude

velocity

position

rnGPS nGPS

+

minus

120575Rn 120575Vn

120572

COVk

120591k

Δ120578k Δ120583k

rnINS nINS 120595

nINS

+minus

+minus










120572 =1

119863

119896

sum119894=119896minus119863+1

V119894V119894119879 (18)


Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Rule assessment



Fuzz

ifica

tion

Def

uzzi

ficat

ion

Fuzzy controller

Observed covariance


Input

New

matrix

Output

covariance




120591 = COV119896 minus 120572 =1

119863

sdot

119896

sum119894=119896minus119863+1

119867119909 (119909minus

119896 119896) 119875minus

119896119867119909119879(119909minus

119896 119896)

+1

2sum

1198941198941015840

1198901198941198901198941015840119879tr 119867(119894)

119909119909(119909minus

119896 119896) 119875minus

119896119867(1198941015840)

119909119909119909minus

119896 119896119875119896 + 120583119896

minus V119896V119896119879

(19)





Consider 120575120583119896



119896








Consider 120575120578119896









IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










IBD


05

1

(a)

LZ GZZ


05

1

(b)



1000 2000 3000 4000 5000 6000minus001

0001002003004

Time (s)

Ang

ular

velo

city

(∘s

)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

(a)

1000 2000 3000 4000 5000 6000minus1

minus050

051

Time (s)

Raw data2-LOD WDT

6-LOD WDT8-LOD WDT

Spec

ific f

orce

(ms

2 )

(b)




120575119911 = 119867120575119909 + 120578 (20)


03times311986711987703times2




times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times105

times10minus4

ACCx

ACCy

ACCz


Time lag (s)

10

8

6

4

R(120591)

2

0

minus2

(a)

GyrxGyryGyrz

times105

times10minus7


Time lag (s)

20

15

10

R(120591)

5

0

minus5

(b)




119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573 120590

2

119908119896120573



minus11 11100Acc119910 29 times 10








minus11 11200

SOEKF where 119877 = diag [1205752119877

1205752119877

1205752119877

1205752119881

1205752119881

1205752119881] is modeled


120575119909 = [120575119877 120575119881 120575120595 120575119887119892 120575119887119886]119879

(21)



119909

120575119887119892119910

120575119887119892119911] and [120575119887119886

119909

120575119887119886119910

120575119887119886119911] denoting the sto-



[120575 119887119892

120575 119887119886

] = [120573119892 0

0 120573119886][

120575119887119892

120575119887119886

] + [119908119892

119908119886] (22)





4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











4548455

45524554

East (∘)

Nor

th (∘

)

(a)


45545514552 Start

1

2

34

5

End

East (∘)

Nor

th (∘

)

(b)












0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

2

4

6Sa

telli

te (

)

10 15 20 25 305Duration (s)

(a)


02468

Sate

llite

()

(b)

02468

Sate

llite

()

20 30 40 50 6010Duration (s)

(c)

0

2

4

6

Sate

llite

()

10 20 30 40 500Duration (s)

(d)

2

4

6

8

Sate

llite

()

5 10 15 20 25 300Duration (s)

(e)


510152025303540455055

RMS

horiz

onta

l pos

ition

erro

r (m

)




(a)



50

100

150

200

250

MA

X ho

rizon

tal p

ositi

on er

ror (

m)


(b)





45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












45476

45478

4548

45482

45484

45486

45488

4549

45492


Nor

th p

ositi

on (∘

)

(a)

45476

454765

45477

454775

45478

454785




East position (∘)

Nor

th p

ositi

on (∘

)

(b)


45495455

455054551

45515


Nor

th (∘

)

(a)

End Start

34

5

2

1

45498455

45502455044550645508

455145512


Nor

th (∘

)

(b)







2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2

4

6Sa

telli

te (

)

40 60 80 10020Duration (s)

(a)

3

4

5

Sate

llite

()

40 60 80 10020Duration (s)

(b)

2

4

6

8

Sate

llite

()

40 60 80 100 12020Duration (s)

(c)

40 60 80 100 120 140 16020Duration (s)

12345

Sate

llite

()

(d)

100 150 200 250 30050Duration (s)

2468

Sate

llite

()

(e)








0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70RM

S ho

rizon

tal p

ositi

on er

ror (

m)




(a)

0

20

40

60

80

100

120

140

160

180

200

MA

X ho

rizon

tal p

ositi

on er

ror (

m)




(b)


455036

455038

45504

455042

455044

455046

455048

45505

455052

455054




East position (∘)

Nor

th p

ositi

on (∘

)

(a)

455055

45506

455065

45507

455075




East position (∘)

Nor

th p

ositi

on (∘

)

(b)



5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 Conclusions



Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A New Technique for Integrating...

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