Low-Cost GPS/INS Integrated Land-Vehicular Navigation .

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DJournal of Traffic and Transportation Engineering 4 (2016) 23-33doi: ost GPS/INS Integrated Land-VehicularNavigation System for Harsh Environments UsingHybrid Mamdani AFIS/KF ModelNéda Navidi and René Jr LandryDepartment of Electrical Engineering, École de Technologie Supérieure (ETS), Montréal QC H3C 1K3, CanadaAbstract: Nowadays, GPS (global positioning system) receivers are aided by INS (inertial navigation systems) to achieve moreprecision and stability in land-vehicular navigation. KF (Kalman filter) is a conventional method which is used for the navigationsystem to estimate the navigational parameters, when INS measurements are fused with GPS data. However, new generation of INS,which relies on MEMS (micro-electro-mechanical systems) based low-cost IMUs (inertial measurement units) for the land navigationsystems, decreases the accuracy and the robustness of navigation system due to their inherent errors. This paper provides a new methodfor fusing the low-cost IMU and GPS measurements. The proposed method is based on KF aided by AFIS (adaptive fuzzy inferencesystems) as a promising solution to overcome the mentioned problems. The results of this study show the efficiency of the proposedmethod to reduce the navigation system errors in comparison with KF alone.Key words: GPS/INS integration, KF, AFIS.1. Introduction The last two decades have shown an increasing trendin the use of location based services in automotivevehicles applications including car tracking for theftprotection, fleet management services, automated carnavigation and emergency assistance. Most of theseapplications rely entirely on a GPS (global positioningsystem) receiver to provide ubiquitous navigationalsolution of the monitored vehicle. In order to providesuch a reliable solution, GPS requires an optimaloperating condition that is a clean line of sight to atleast four satellites. Unfortunately, this conditioncannot be fulfilled at all times, especially when drivingin severe urban environment where buildings, concretestructures, high passes and tunnels may attenuate,block or reflect incoming signals resulting in a poorlyaccurate navigation solution. Indeed, multipath is oneof the main sources of positioning errors for standaloneCorrespondingauthor: Néda Navidi, Ph.D. candidate,research fields: electrical engineering, navigation, positioningand tracking systems.GPS receivers used in severe urban environments.Multipath error on pseudo-range measurements canreach hundreds of meters [1], thus introducingsignificant errors in position computation.To reduce the impact of multipath, GPS is oftencombined to an INS (inertial navigation systems),which is a self-contained dead reckoning system basedon the integration of linear accelerations and angularrates. The procedure, in which raw inertialmeasurements are processed to compute the position,the velocity and the attitude of a mobile, is referred asthe INS mechanization process. Unfortunately, thelow-cost standalone INS positioning tend to divergeover time due to the significant inherent ical systems) based inertialsensors.To overcome shortcomings of standalone GPS orINS, both systems can be coupled together to form anintegrated navigation system. The INS/GPS integratedsystems exploit beneficial features of each individualsystem (i.e., short term precision of INS, and long-term

24Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF Modelstability of GPS), thus providing precise and ubiquitousnavigation. The INS/GPS integration is a mature topicthat is widely covered in the literature when it comes tohigh-end systems [2-5]. However, the recent studiesreported several shortcomings related to the use oflow-cost MEMS based IMUs (inertial measurementunits), particularly with regard to the use of linearizederror models.Various kinds of KF (Kalman filter) have beenextensively utilized for the integration of GPS withINS, and they can determine the optimal estimation ofthe system with minimum errors [6-9]. KF needs anadequate knowledge on the dynamic process of thesystem and measurement model. Another problem isthat the precision of MEMS-based IMUs issignificantly decreased by their drift and bias errors.This problem ruins the system performance, whenemploying a KF in car navigation [10]. Moreover, thecritical problem of the KF is the big degree ofcovariance divergence due to its modeling error [11].There are two different solutions to solve thecovariance divergence of KF: The first one is using theun-modeled states in KF. However, this solutionincreases the computational complexity of thesystem [11]. The second one is usage of the processnoise to develop the confidence, which can interceptthe KF to dismiss new values for estimating the statevector [10, 12, 13]. This paper uses Mamdani AFIS(adaptive fuzzy inference system) based on theemploying the process noise. The proposed model canbeused for tuning phase of the KF by tracking thecovariance values.The paper is organized as follows: Section 2 presentsthe GPS/INS integration methods; Section 3 describesthe proposed model to fuse the INS and GPSmeasurements in deeply; The results are provided inSection 4, whereas Section 5 draws the conclusion andfuture works.2. GPS/INS Integration Methods2.1 GPS/INS CouplingThere are different structures for the GPS/INSintegration, namely, uncoupled, loosely coupled,tightly coupled and ultra-tightly coupled. Table 1summarizes the advantages and the disadvantages ofthem. It is common to integrate the INS and GPSthrough loosely coupled structure. Because it not onlymaintains independency of stand-alone GPS and INSsolutions, but it can also provide more robustness forthe navigation solution [14]. The loosely coupledintegration is generally preferable in the GPS/INSintegration as being composed of three distinct entities,which are the stand-alone GPS solution, thestand-alone INS solution and the GPS/INS coupledsolution. This architecture, which is presented in Fig. 1,is shared by several authors [15-17].In general, the KF filters are recognized as the mostconventional method to estimate the navigationsystems (Fig. 1) [18]. The error model in the dynamicalmodel includes state errors of position, velocity andattitude, augmented by sensor state errors [19].2.2 KF (Kalman Filters)An optimal estimator is a state estimator whose gainis dynamically calculated to optimize a certain systemTable 1 A brief summary of commonly used INS/GPS integration architectures [14].TypeUn-coupledLoosely coupledTightly coupledUltra-tightly coupledAdvantageSimplicity of algorithm Separate INS and GPS KF; Small size of individual KF; Less computation complexity Optimal accuracy; Max 4 satellites requirement Reduce GPS dynamic stress; Less jamming errorsDisadvantageInstability in GPS outages Sub-optimal performance; Min four satellite requirement Large error size state model; More complex processingSpecial hardware requirement

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF Model25Acc bias/Gyr drift correctionsGPS receiverPositionVelocityGPS FilterGPS/INSFilterINSINS ngFig. 1 Loosely coupled GPS/INS integration.performance criteria. One of the widely usedoptimization criteria is the mean square error of theestimate. The optimal estimator based on thisoptimization criterion is called the Kalman filter. KFhas been accepted as a conventional method to estimatethe GPS/INS integration. The derivation of an errormodel which are applied in the KF can start with theconstruction of full scale true error models [18, 19].First, a definition of priori error in the estimated ofthe state vector and the associated covariance matrixcan be presented with:(1)(2)where,is priori error vector andis priori errorcovariance matrix.According to the equations of state estimator presentin the previous section, the estimated a posteriori statevector can be obtained by a linear combination of thevector of noisy measurements and estimated as a priori:(3)The error covariance matrix associated with thisestimate posteriori can be calculated from thefollowing expression:(4)(5)(6)The diagonal of the error covariance matrix containsthe variance of the estimation error of all system states.Thus, the trace of this matrix represents the sum of thevariance and it is thereby an unbiased indicator of themean square error of the estimate. The gain of theKalman filter, commonly called the Kalman gain, canbe selected to minimize the trace of the matrix. Thiscan be achieved by the following equality:022(7)(8)within Eq. (6) by the expressionSubstitutinggiven by Eq. (8), it is possible to simplify theexpression of the matrixas:(9)After repeating the previously presented equationsof the state estimator, spreading the error covariancematrix, and calculating the Kalman gain, the Kalmanfilter algorithm can be summarized by the equationspresented in Table 2. A system level block diagram ofthe discrete KF is shown in Fig. 2.The time update equations can be calculated thoughtthe estimation equations, while the measurementupdate equations can be presented thought thecorrection equations. Therefore, the final estimationalgorithm resembles anestimation-correctionalgorithm to solve the numerical problems as shown inFig. 3.The multiple dynamic models used in navigation are

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF Model26Table 2 Description of KF variables.ParameterΦDescriptionState transition matrix of a discrete linearState vector of a linear dynamic systemPrediction or a priori value of the estimated state vector of a linear dynamic systemCorrected or a posteriori value of the estimated state vector of a dynamic systemMeasurement vector or observation vectorKalman gain matrixMeasurement sensitivity matrix or observation matrix which defines the linear relationship between the state of thedynamic systems and measurements that can be madePredicted or priori value of estimated covariance of state estimation uncertainty in matrix formCorrected or posteriori value of the estimated covariance of state estimation uncertainty in matrix formProcess noiseMeasurement noiseZk Kk x̂ k yk( )Unitdelayxˆ k 1xˆk ( )Hk Φk k 11Fig. 2 Discrete-time KF diagram.New measurementMeasurement update orcorrectTime update or predictSystem modelFig. 3 Recursive process of prediction.non-linear, and, therefore, they can not be directlyexpressed in terms of the general form. Thelinearization of these systems around an optimal statevector is first requirement, so they can be expressedwith using this standard form. First, it is defined anon-linear system such that:·(10)(11)where,andare non-linear functions. It isis known andassumed that an optimal state vectorit is defined as follows:where,is optimal state trajectory and(12)is state oferror vector. Thus, Eqs. (8) and (9) can be rewritten as afunction of the optimal state vector and the error statevector as:

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF Model·(13)(14)It is assumed that the status error vector canbeneglected, and it is possible to approximate thenon-linear functions f and h using a Taylor series.Considering only the terms of first order, the result is:·(15)(16)By selecting an optimal state vector as equality, the following linearized general form isobtained:··(17)·(18)where:(19)(20)(21)2.3 AFISs (Adaptive Fuzzy Inference Systems)AFISs (adaptive fuzzy inference systems) is arule-based expert method for its ability to mimichuman thinking and the linguistic concepts rather thanthe typical logic systems [20]. The advantage of theAFIS appears when the algorithm of the estimationstates becomes unstable due to the system highcomplexity [21].AFIS architecture includes three parts: fuzzification,fuzzy inference and defuzzification. The first part isresponsible to convert the crisps input values to thefuzzy values, the second part formulates the mappingfrom the given inputs to an output, and the third partconverts the fuzzy operation into the new crisp values.The AFIS are able to convert the inaccurate data tonormalized fuzzy crisps which are represented by MF(membership functions) and the confidence-rate of theinputs. AFIS are capable to choose an optimal MFunder certain convenient criteria meaningful to aspecific application [22, 23].27Mamdani and Sugeno are the two practical AFIStypes which were used in several studies [24-27]. Themain difference between these two fuzzy algorithms isbased on the process complexity and the rule definition.Another important aspect to take into consideration isthat Mamdani needs more processing time than Sugeno.Sugeno type also provides less flexibility in the systemdesign compared to the Mamdani type. In general,Mamdani type is more efficient and accurate thanSugeno type [24, 25]. All those reasons have motivatedus to use the Mamdani type (Fig. 4) to design thefuzzy-part of the proposed GPS/INS integration model.3. Proposed AFIS/KF SystemThis paper employs IAE (innovation adaptiveestimation) concept in the fuzzy part of the proposedmodel [23, 28, 29]. The dynamic characteristics of thevehicle motion are based on the KF process. The AFIScan be exploited to increase the accuracy and the KF.Additionally, AFIS part can prevent the divergence inthe tuning phase of KF. Hence, Mamdani AFIS is usedas a structure for implementing the tuning of thenonlinear error model. Fig. 5 shows the proposedhybrid AFIS-KF model.MAMDANIMamdanitypeTYPEOUPUTMFsOutput MFsINPUTMFsInput MFsFig. 4General overview of AFIS using Mamdani type.VelocitypositionPositionIMUIMUGPSGPSKFKF Rk SkMamdaniAFISAFIS kFig. 5 The proposed hybrid Mamdani AFIS-KF model innavigation system.

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF rispValuesDeFuzzificationRule baseCorrectedCovarianceDatabaseFig. 6 The proposed Mamdani AFIS part of the hybrid AFIS-KF in navigation system.The proposed AFIS model is based on thecovariance matrix for the input of the Mamdani AFIS,as well as the difference of the actual and the estimatedcovariance matrices. Fig. 6 presents the proposed AFISoverview that is used in this paper. The estimatedcovariance matrix based on the innovation process iscomputed partly in the KF by:(22)where,andrepresent the estimated covarianceandareand the design matrices, respectively.previous covariance update and the measurement noisecovariance matrices. The actual covariance matrix ( )according to Ref. [30] can be presented by: (23)where,is the window size which is given by themoving window technique. Thus, the differencebetween the actual and estimated covariance matrices( ) can be presented by:(24)In fact, the value ofcan display the level ofdivergence between the actual and the estimatedcovariance matrices. When the value of is close tozero, the estimated and the actual covariance matriceswill be very similar and the absolute value of the canbe neglected. However, if the value of is not near tozero (smaller or greater than zero), an adaptation isconsidered in the algorithm and the value of Rk in Eq. (22)should be adjusted to compensate this difference.The proposed rules assessment according to thedifference between the actual and the estimatedcovariance matrices is described as three scenarios ofMF. If the value of is higher than zero, then the valueis turned down in accordance with the value ofofδR ; If the value of the is less than zero, then theis turned up in accordance with the valuevalue ofisof δR ; If the value of is close to zero, then.unchanged.where,Fig. 7a explains the grade of the membershipparameter in the five fuzzy sets. For example, ifisclose to zero, the grade of MF is shown with I2 in theAFIS. As the degree of MF in the fuzzy set is I1 or I4, itapproaches to minus one.The MF of the output for the proposed AFIS modelis presented in Fig. 7b. The output is the value ofwhich changes from –1 to 1. Moreover, it is fuzzified inseven levels from O1 to O7. These seven levels areimportant to differentiate between the likelihood levels.The next section presents the performance of theproposed AFIS/KF within simulation experiments.4. Simulation Experiments and AnalysisResults of the AFIS/KF method are compared withthose obtained from conventional KF to evaluate theperformance of the proposed model. A looselycoupled structure for GPS/INS integration isconsidered in this study. The navigation system isshown in Fig. 8. The error state related to system canbe presented as:(25)where,andare the north and the east positionandare the north and the easterrors; and

Loow-Cost GPS//INS Integrateed Land-Vehiicular Navigaation System forHarsh Envvironments Ussing Hybrid MamdaniMAFISS/KF Model299(a)(b)Fig. 7MF (mmembership fuunctions): (a) input;i(b) outpput.NoorthψaXaYEastFig. 8Naviggation system.velocity errrors, respecctively. ߜ߰ represents theheading erroor.The vehiccle trajectoryy was generatted in three mainmdifferent mootions (Fig. 9).9 Three maain time interrvalsare considerred for the straight and circlecmotionns ofthe trajectorry. The straigght was perfoormed 0 1,200 sand 2,401 3,600 s. Time segmments and thecounter-clocckwise circlee was done ini 1,201 2,4000 stome segmeent. The vehhicle moves with a consstantspeed (4.5 m/s).Two mainn points are coonsidered forr GPS availabbilitysignals in thhe scenario. First,Fthere arre different erroresouurces, like mulltipath errors for satellite constellations.It is called “normal“situuation” thatt maximummS unavailabiility is 5 s. Second, five obstructeddGPSsegmments weree consideredd in the GPSGsignalssduring the trajecctory. It is callled “bad situuation” whichhare shown withh the red pooints in Fig. 9. Table 3oint of eachhshows the duraation, start and stop pooutaage.Fig.F 10 showss the results inn normal situuation of GPSSwithhout GPS outtage more thaan 5 s. Fig. 11 depicts theeerroors of the KFF and AFIS-KKF model duuring the fiveeassuumed outagees. Fig. 12a shows a coomparison offRMMSE (Root Meean Square Errrors) betweeen KF and theepropposed AFIS-KKF model in the normal situation.Thisssfiguure also showws that the AFIS-KF atttenuates theeerroors by 30%, comparedcto the KF whenn the GPS issworrking in the normal situaation. Fig. 122b presents acommparison of RMSEsRbetween KF and the proposeddhybbrid AFIS-KKF model. This figuree obviouslyyillustrates the siignificant immprovement ofo navigationnution when ussing the propoosed AFIS-KKF model. It isssoluobserved that, afteraapplyinng AFIS-KF, errors wereedecreased by 75% compare too KF.

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System forHarsh Environments Using Hybrid Mamdani AFIS/KF Model3010,000North position t position (m)Fig. 9 Trajectory in three main time-segments due to the dynamics characteristic.20,000Table 3 Definition of the GPS outages based on their durations in the trajectory.Outages (No.)12345Period (s)6090928348Start-point (s)2968961,5102,1603,2300AFIS/KFNorth velocity error (m/s) East velocity error (m/s)50.20 0.2 50.10.20.3 0.40.5 0.6 0.7Time (h)0.80.9150 50.10.20.3 0.40.5 0.6 0.7Time (h)0.80.910.10.20.3 0.40.5 0.6 0.7Time (h)0.80.910.20 0.20.10.20.3 0.40.5 0.6 0.7Time (h)2Heading error ( )North position error (m)East position error (m)KFStop-point (s)3569861,6022,2433,2780.80.9110 1 20.10.20.3 0.40.5 0.6 0.7Time (h)Fig. 10 Position, velocity and heading errors in normal situation of GPS.0.80.91

AFIS/KKF0 10 2000.200.40.6Time (h)0.8150 500.200.40.6Time (h)543210 1 200.81North velocity error (m/s)

Low-Cost GPS/INS Integrated Land-Vehicular Navigation System for Harsh Environments Using Hybrid Mamdani AFIS/KF Model 24 stability of GPS), thus providing precise and ubiquitous navigation. The INS/GPS integration is a mature topic that is widely covered in the literature when it comes to high-end systems [2-5]. However, the recent studies

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