Merits and limitations of using fuzzy inference system for temporal integration of INS/GPS in vehicular navigation

Soft Comput (2007) 11:889–900DOI 10.1007/s00500-006-0140-0

ORIGINAL PAPER

Merits and limitations of using fuzzy inference system for temporalintegration of INS/GPS in vehicular navigation

Rashad Sharaf · Mahmoud Reda Taha ·Mohammed Tarbouchi · Aboelmagd Noureldin

Published online: 24 October 2006© Springer-Verlag 2006

Abstract Most of the present vehicular navigationsystems rely on global positioning system (GPS) com-bined with inertial navigation system (INS) for reliabledetermination of the vehicle position and heading. Inte-grating both systems provide several advantages andeliminate their individual shortcomings. Kalman filter(KF) has been widely used to fuse data from both sys-tems. However, KF-based integration techniques sufferfrom several limitations related to its immunity to noise,observability and the necessity of accurate stochasticmodels of sensor random errors. This article investigatesthe potential use of adaptive neuro-fuzzy inference sys-tem (ANFIS) for temporal integration of INS/GPS invehicular navigation. An ANFIS-based module named“P–δP” is designed, developed, implemented and testedfor fusing INS and GPS position information. The fu-sion process aims at providing continuous correction ofINS position to prevent its long-term growth using GPSposition updates. In addition, it provides reliable pre-diction of the vehicle position during GPS outages. TheP–δP module was examined using real navigation sys-tem data compromising an Ashtech Z12 GPS receiverand a Honeywell LRF-III INS. The proposed moduleproved to be successful as a modeless and platform inde-pendent module that does not require a priori knowl-edge of the navigation equipment utilized. Limitationsof the ANFIS module are also discussed.

R. Sharaf · M. Tarbouchi · A. NoureldinDepartment of Electrical and Computer Engineering,Royal Military College of Canada, Kingston, ON, Canada

M. Reda Taha (B)Department of Civil Engineering, MSC01 10701,The University of New Mexico, Albuquerque,NM 87131-0001, USAe-mail: [email protected]

Keywords Fuzzy systems · Inertial navigation ·Data fusion · Positioning systems

1 Introduction

The last two decades have shown an increasing trendin the use of positioning and navigation (POS/NAV)technologies in land vehicle (mainly for car navigation)applications (Lobo et al. 1995; El-Sheimy and Schwarz1994). Application areas of POS/NAV technologies inland transportation are numerous including automatedcar navigation, emergency assistance, fleet management,asset tracking, collision avoidance, environment moni-toring, and automotive assistance. The convergence oflocation, information management and communicationtechnologies have created a rapidly emerging marketknown as location-based service (LBS). LBS is acritical enabling technology using location as a filterto extract relevant information to provide value-addedservice such as location-aware billing, automated adver-tising services and other location-based informationsought by the user based on his or her own location.Because of the importance of location information, reli-able in-vehicle navigation and guidance systems havebecome necessary, providing not only location informa-tion but also route-guidance and location-sensitive ser-vices. Most of the current vehicular navigation systemsrely on global positioning system (GPS) that is capa-ble of providing accurate position and velocity infor-mation, especially when operating in differential mode(usually called DGPS). To be able to provide such accu-rate measurements, GPS needs at least four satelliteswith good geometry. In addition, there must be directline of sight between the GPS antenna and those

890 R. Sharaf et al.

satellites. Unfortunately this is not the case all the timesince a GPS signal is usually lost when driving aroundobstacles (downtown area, high passes or tunnels onhighways, tree lined streets,. . .), or with poor weatherconditions (Farrell 1998). The satellite signal blockageresults in deterioration of the overall position accuracy.On the other hand, inertial navigation system (INS)is a self-contained system mounted inside the vehicleand does not rely on external references like GPS. INSincorporates three orthogonal accelerometers and threeorthogonal gyroscopes, which measure three linearaccelerations and three angular rates respectively. Thosesensors, when mounted on the moving platform,point to three mutually orthogonal directions calledthe body frame (b-frame), which is defined along theforward, transverse and vertical directions of the vehi-cle. The position of the vehicle (determined by GPSor INS) is provided at the local level frame (l-frame).The l-frame axes are defined along the East, the Northand the vertical directions. Transformation betweenboth frames is possible through a 3 × 3 transformationmatrix.

A set of mathematical transformations and integra-tion with respect to time are applied to the rawmeasurements from the INS sensors (Fig. 1). Thesetransformations are known as the mechanization equa-tions (El-Sheimy and Schwarz 1994; Salychev 1998).Usually, the mechanization equation is referenced tothe l-frame (Salychev 1998). When measurements pro-vided by the INS accelerometers are transformed fromthe b-frame to the l-frame and then integrated withrespect to time, they result in three velocity componentsalong the East, North and vertical directions. Further-more, a second integration with respect to time yieldsthe position along the East, North and vertical direc-tions. Meanwhile, integration with respect to time of thegyroscope measurements, also in the l-frame, providesthree attitude components. These are the roll, pitch andthe azimuth (heading angle). The pitch is the angle thatthe vehicle makes with the horizontal surface; the rollis the angle that corresponds to rotations along the vehi-cle axle while the azimuth is the angle between the for-ward axis of the vehicle and the North direction in thehorizontal plane.

During the mechanization procedure, the accuracy ofINS position components deteriorate with time due tothe inherit sensor errors that exhibit considerable long-term growth (Mynbaev 1994; Noureldin et al. 2002).Those errors include white noise, correlated randomnoise, bias instability and angle random walk (IEEEStd.#647, 1995). Those errors, usually modeled usingstochastic processes, can cause a significant degradationin INS performance over long-time operation. To over-

come the shortcomings of stand alone GPS or INS, bothsystems might be integrated into a single navigation sys-tem as shown in Fig. 2. Data fusion of the integrated sys-tem has been carried out for many years using Kalmanfiltering (KF) (Hostetler and Andreas 1983; Hargrave1989), which utilizes a dynamic model of INS position,velocity and attitude errors as well as stochastic modelof sensor errors.

The major inadequacy related to the utilization ofKF for INS/GPS integration is the necessity to haveaccurate stochastic model for each of the sensorerrors (Scherzinger and Reid 1994). If the input datadoes not fit the model, KF may not provide reliableestimates. While navigational grade and high-end tac-tical grade systems have stochastic errors that can beappropriately modeled using KF, it is always challeng-ing to model the errors of low-end tactical grade andMEMS-based IMUs. Moreover, relatively poor predic-tions of vehicle position were reported when KF wasused with stand alone INS during GPS outages (Nassaret al. 2004).

The limitations of KF motivated researchers toinvestigate alternative methods, predominantly basedon artificial intelligence (Vanicek and Omerbasic 1999).Recently, INS/GPS integration algorithms based onmulti-layer perceptron (MLP) neural networks havebeen suggested and applied to different types and gradesof INS (Chiang et al. 2006). It has been shown that aposition and velocity update architecture (PVUA) uti-lizing two MLP networks can process the INS azimuthand velocity and provide the position components alongboth the East and North directions (Chiang et al. 2006).The parameters of the MLP networks are modified usingGPS position and velocity updates. However, the neuralnetwork model of PVUA deals with the INS positioncomponents instead of INS errors. Thus, the accuracy ofthe system cannot be quantified during the navigationmission. The system showed good performance whenneural network modules were integrated with waveletde-noising (Noureldin et al. 2004).

Because of its efficiency in dealing with uncertainty,imprecision and vagueness in input data in dynamicenvironments, we suggest here that fuzzy systemsmight be employed to establish an intelligent fuzzy-based module for temporal integration of INS/GPS.This research aims at: (1) introducing a modeless andplatform independent module for INS/GPS integrationutilizing adaptive neuro-fuzzy inference system (AN-FIS); (2) Evaluating the effectiveness of the proposedmodule using real measurements of inertial sensorsand GPS mounted on a land vehicle and (3) Exam-ining the merits and limitations of the ANFIS-basedmodule.

Merits and limitations of using fuzzy inference system 891

Fig. 1 Schematic diagram showing the basic procedure of INS mechanization

Inertial Navigation

System (INS)

Global Positioning System (GPS)

INS output + INS errors

Kalman Filter

Corrected INS output

Estimation of the INS errors

GPS Error

INS Error

12 3 4 5 67 8 9ec

Ashtech

INS Error Model

Fig. 2 Schematic diagram of classical INS/GPS integration using Kalman filter

2 Methods

2.1 Modeling INS position errors

Most researchers considered INS errors as non-deter-ministic errors and modeled them using random processtheory (e.g. Farrell 1998; Brown and Hwang 1992). Whileit is evident that INS errors are non-deterministic, it canbe argued that INS errors might not be characterizedas random errors. A major criterion for random-errorsis that they cannot be reduced as additional informa-tion becomes available (Ross 2004). Given the fact thatINS errors can be significantly reduced as further obser-vations from other sources become available (e.g. GPSsignal), INS errors miss an important criterion to beclassified as random errors. Unlike random errors, non-random errors cannot be reduced using theory of prob-ability. We thus suggest using fuzzy set theory (which

encompasses the theory of probability) (Ross 2004; Klirand Yaun 1995) to model INS errors. This indicates theassumption that INS/GPS errors might occur due to ran-domness type of uncertainty in addition to other typesof uncertainty such as fuzziness, vagueness and ambigu-ity. None of the three later types can be addressed bythe theory of probability but can evidently be handledusing fuzzy set theory (Ross 2004).

2.2 System architecture

Here we suggest a new INS/GPS integration modulenamed (P–δP) based on the estimation of INS positionerror (δPINS) by processing the INS position (PINS). Theproposed module utilizes adaptive neuro-fuzzy infer-ence system (ANFIS) (Jang et al. 1997) to provide anoptimal temporal estimation of the INS errors (δPINS).The proposed system architecture compromises three

892 R. Sharaf et al.

modes of operation: the initialization model, the updatemode (Fig. 3a) and the prediction mode (Fig. 3b). Theinitialization mode and the update mode are utilized,as long as the GPS signal is available, to initialize thefirst learning rule-base and to limit INS error growth,respectively. The prediction mode is used to correct theINS position when the GPS signal is lost. Thus, the P–δPmodule is trained during the availability of the GPS sig-nal to recognize patterns of the position error embeddedin the input position components. In case of satellite sig-nal blockage, the P–δP module mimics the latest vehicledynamics and delivers prediction of the vehicle positionerror.

The INS position (PINS) and time (T) are the inputs tothe module while the error in the INS position (δPINS)

is the module output. A schematic representation of theP–δP module for establishing the rule-base relating theINS position (PINS) and time (T) to the error in the INSposition (δPINS) is shown in Fig. 4. The estimated INSposition error provided by the P–δP module is then com-pared to the error between the INS original position andthe corresponding GPS position (δPINS|GPS). The num-ber and shape of membership functions shall be prede-fined (Gaussian shape membership functions were usedhere, however, other shapes proved possible). The effectof the number of membership functions on system per-formance will be discussed below. The original mean andspread of the membership functions are computed usingfuzzy clustering techniques (Jang et al. 1997; Passino andYurkovich 1998). The membership values are evaluatedat the first layer and the fuzzy t-norm operator (�) isimplemented at the second layer. A normalized firingstrength (W̄i) is computed at the third layer. The INSposition error is computed as

if PINS ∈ A1˜

and T ∈ B1˜

then

f1 = p1PINS + q1T + r1 (1)

if PINS ∈ A2˜

and T ∈ B2˜

then

f2 = p2PINS + q2T + r2 (2)...

...

if PINS ∈ An˜

and T ∈ Bn˜

then

fn = pnPINS + qnT + rn (3)

δPINS =n∑

i=1

Wifi (4)

where the linear parameter pi,qi and ri are the con-sequent parameters originally determined using least

squares approach. Ai˜

represents a fuzzy set defined over

the PINS domain and Bi˜

represents a fuzzy set defined

over the T domain and Wi represents the mean nor-malized weight of the ith rule as shown in Fig. 4. Thedifference between δPINS and δPINS|GPS is the trainingerror (�(δP)) of the ANFIS module. In order to min-imize the training error, the ANFIS module updatesthe learning rule-base (membership function parame-ters and consequent parameters) using a hybrid learningapproach combining the least square and back propa-gation techniques until reaching a certain minimal rootmean square estimation error (RMSE) (Jang et al. 1997;Passino and Yurkovich 1998).

When the satellite signal is blocked (during GPS out-ages), the system is switched to the prediction modewhere the P–δP module is used to predict the INSposition error using the latest learning rule-base ob-tained before losing the satellite signals. The error is thenremoved from the corresponding INS position com-ponent to obtain the corrected INS position (PINSC).Therefore, three P–δP modules are developed to pro-vide complete navigation solution in the three axes for amoving vehicle represented by the three position com-ponents. These modules are named the altitude ANFISmodule, the latitude ANFIS module and the longitudeANFIS module. The position errors of the latitude andlongitude ANFIS modules are provided in meters (in-stead of radians or degrees) to represent the errors indetermining the vehicle position along the North andEast directions, respectively.

It should be also highlighted there are several sourcesof errors that contribute to the overall position error ofthe vehicle in different ways. These sources include theinertial sensors bias drift and scale factor instability, theinitial misalignment error during INS initialization andthe error in the heading angle of the vehicle (the azi-muth error). The azimuth error, for example, can havesignificant contribution on the position error along boththe East and North position components especially athigh speeds since it is modulated by the vehicle velocity.Although all these error sources are not explicitly rep-resented in the architecture shown on Fig. 3, it is evidentthat the P–δP module incorporates the effects of sucherrors in two ways. First, it establishes the learning rule-base to pattern vehicular navigation performance usingexemplar navigation scenarios that have been affectedby the same set of sources of errors. Second, it considersthe input parameters as interval data with membershipvalues rather than crisp values which provides room foruncertainty in the input parameters due to these sourcesof errors.

Merits and limitations of using fuzzy inference system 893

Fig. 3 The proposed P–δPmodule architecture.a Update mode of operation(GPS signal is available).b Prediction mode ofoperation (GPS outage)

Pd S

Time

GPS

INS position

ANFIS

Training Criterion

?PINSPINSC

ANFIS Parameter Adjustment

Time

1 2 3 4 5 6 7 8 9ec

Ashtech

PINS

?PINS|GPSPGPS

P- Pd?

e

ec

Ashtech ?(a)

(b)

INS position ANFIS

P- P dIN

PINSC

Corrected INS position

Fig. 4 Architecture of the ANFIS–P–δP component (see Fig. 3) for temporal integration of INS-GPS showing the input parametersincluding the INS position (PINS) and time (T) represented using “n” membership functions. The component output is the INS positionerror (δPINS)

2.3 System implementation

In order to utilize the P–δP module in a temporalINS/GPS integration, a sliding window with certain win-dow size (W) is considered. For each of the threeANFIS modules, number of samples (equal to W) ofINS position component PINS and the correspondingGPS position PGPS are acquired from both systems. TheINS position is considered as the input to the P–δP mod-ule and the error between PINS and PGPS is considered

as the corresponding desired response (δPINS|GPS). Theupdate procedure of the P–δP learning rule-base startsafter collecting the Wth sample of both INS and GPSposition components. Before considering the next INSand GPS samples, the ANFIS module is trained untilreaching certain minimum RMSE or after completingcertain number of training epochs. In order to guaranteetimely operation of the system, the update procedure isterminated at the end of this number of training epochseven if the desired RMSE is not achieved.

894 R. Sharaf et al.

While the GPS signal is available, the data windowcontinues to slide collecting new samples from INS andGPS position components. The erroneous INS signalhas to be corrected before applied to the P–δP mod-ule. Therefore, assuming δPINS(i + 1) = δPINS(i), thecorrected INS position component can be obtained as

PINSC(i + 1) = PINS(i + 1) − δPINS(i), (5)

where δPINS(i) is the INS position error provided by theP–δP module at time instance i.

Since the GPS signal is available during the updatemode, both PINSC and the position error between PINSCand PGPS are used to train ANFIS to mimic the dynamicspresent within the last data window. This results in a newANFIS rule-base, that is used to provide an estimate forthe INS position error at time i+1 (δPINS(i + 1)). There-fore, the INS position is continuously updated and theINS position errors (estimated by the ANFIS module)are removed, thus keeping accurate INS position com-ponents available in case of any GPS outages.

It should be noted that the proper choice of the win-dow size is very essential to guarantee delivering thedesired accuracy while ensuring system robustness inreal-time. The complexity of choosing the window size isrelated to its dependence on the level of vehicle dynam-ics and the length of the outage periods and its signifi-cance on the update procedure. Therefore, there is atrade-off in choosing small or large window size. Largewindow sizes enables mimicking significant details of thelatest vehicle dynamics, thus the module becomes reli-able during GPS outages. However, large window sizesmay complicate the update procedure and result in longprocessing time. On the other hand, a fast and robustupdate procedure can be achieved using small windowsize due to the reduced level of the non-stationary natureof INS and GPS data. Moreover, using relatively smallwindow sizes prevents considering inaccurate positioninformation provided by the ANFIS module during thefirst outage for the prediction of the INS position compo-nents. However, relatively small window sizes may causethe system to lose reliability in case of relatively longGPS outages. As a trade-off problem, optimal windowsizes can be determined using means of multi-objectiveoptimization (Deb and Gupta 2005) (considering reli-ability and updating time requirements) or by extractinga set of heuristic learning rules based on system obser-vation (Bshouty and Burroughs 2005).

3 P–δP module testing

The performance of the proposed P–δP module is exam-ined with a field test conducted in the City of Laval

Fig. 5 Test trajectory showing the locations of the intentionallyintroduced GPS outages

(Quèbec, Canada) using the VISAT van mobilemapping system (El-Sheimy 1996). In this test, anAshtech Z12 GPS receiver and a navigation-grade INS(Honeywell LRF-III) were utilized. The minimum num-ber of available satellites over the entire experiment wasseven and the average van speed was 50 km/h. The tra-jectory of this test is given in Fig. 5 (El-Sheimy 1996).It is worth mentioning that LTN90-100 INS used in thisstudy is of a navigational grade performance with its er-rors can be properly modeled for KF-based INS/GPSintegration. This was intended to provide a benchmarktesting of the P–δP module such that its accuracy can bevalidated using other techniques.

Two cases were considered to evaluate the P–δP mod-ule performance. The first case examined the P–δP mod-ule while operating in the update mode making useof the continuously available GPS position updates tocorrect for the INS errors and to prevent their unlim-ited long-term growth. The second case examined P–δPmodule while operating in the prediction mode by inten-tionally introducing nine GPS outages at different loca-tions along the trajectory (see Fig. 5) and at differentvehicle dynamics. Since the GPS position was availableduring the entire experiment, the performance of theP–δP module is evaluated by comparing the P–δP posi-tion to the GPS position taken as reference in this study.

The first outage was selected to be early in the exper-iment while the vehicle is moving in a straight line. Asshown in Fig. 5, outage 2 was chosen at a time wherechange in the vehicle direction was performed. Out-ages 5 and 6 were chosen while the vehicle was sta-tionary, while outages 4 and 7 were chosen so that theyinclude a combined stationary and moving status of thevehicle. Thus, the performance of the P–δP module can

Merits and limitations of using fuzzy inference system 895

Fig. 6 Change of the RMSE of P − δ P module during training in two different windows. a RMSE goal was achieved b RMSE goal wasnot achieved

be examined during relatively long stationary periodand when accelerating again to restore motion. Out-ages 8 and 9 were chosen relatively close to each otherto study the performance of the P–δP module in caseof two consecutive outages. All the development madeon this study was implemented using MATLAB com-puter-aided design software (Mathworks, Natick, MA)platform.

4 Results and discussion

4.1 Initialization and update modes of operation

During the update procedure, the P–δP module wastrained to mimic the latest vehicle dynamics over a win-dow of 20 s. After collecting a number of samples of INSand GPS position data equal to this window size, thefirst training procedure was performed and initial P–δPmodule was created. As the window started to slide at 1-stime step, P–δP module was updated by repeating thetraining process while considering the new INS and GPSposition samples. In general, each update (new training)of the P–δP module is performed to minimize the er-ror between the module output (δPINS) and the desiredresponse (δPINS|GPS) such that a root mean square error(RMSE) of 10−4 is achieved. The P–δP module updateprocedure for each window was terminated after 100training epochs apart from the RMSE achieved for com-putational time consideration. Figure 6 demonstratesthe performance of the P–δP module update procedurefor two different windows represented by the changeof the RMSE during training. Figure 6a shows one casewhere the performance goal of 10−4 was achieved inless than 100 epochs, while the same goal could not beachieved even if the training process continues beyondthe first 100 epochs in Fig 6b. Testing of the P–δP module

showed that the ability of the module to achieve train-ing goal is strongly dependent on the size of the windowand the nature of the motion dynamics performed bythe vehicle within this window.

In fact, there is a tradeoff in choosing the windowsizes as explained above. For instance, a large windowis beneficial in mimicking considerable amount of thelatest vehicle dynamics, thus the P–δP module becomesreliable during long GPS outages. It becomes obviousthat the proper choice of the window size is very essen-tial to guarantee delivering the desired accuracy whileensuring system robustness.

Observing ANFIS training during the update modein Fig. 6a and b, it is important to point out the abilityof ANFIS to reduce the training error whether signifi-cant dynamics occurred during the update mode or not.The P–δP module can meet two distinct cases duringthe update mode: First case, if relatively low dynamicsoccurred during the update period, the error of the firstepoch of training will be considerably low and this iswhat is shown in Fig. 6a. The change in the error duringtraining has no relation to INS dynamics but to ANFIStraining algorithm which forces changing the learningrule-base to reduce the error compared with the firstepoch error. The fluctuation of the training error doesnot affect the ANFIS performance as the system is setto use the rule-base parameters that yield the lowesttraining error for the prediction mode (here epoch# 1or 100). The second case is represented in Fig. 6b whenthe data collected during the update period for trainingincludes significant dynamics that were not observed bythe P–δP module before. The first epoch error will be rel-atively high but ANFIS learning algorithm will reducethis error. The ANFIS rule-base that resulted in the low-est training error (here epoch# 100) will be used for theprediction mode. This represents a merit for the P–δPmodule as its learning will adapt to the level of dynamicscollected for update.

896 R. Sharaf et al.

Fig. 7 RMSE during the update mode of a 20 s training window for the North position component for six different step sizes

Other factors that have a major impact on the per-formance of the ANFIS training procedure is the initialstep size κint, the step increase rate η and decrease rate γ .The step size κ represents the parameter by which thegradient descent optimization approach updates themembership functions in the backward path of the train-ing process. The value of step size κ is modified duringtraining using the initial step size κint and the increaseand decrease rates η, and γ , respectively according tothe progress in training following the heuristic rulesby Jang et al. (1997) and Jang (1993). Extensive test-ing of the P–δP module was performed by using differ-

ent step sizes and step increase and decrease rates toexamine the sensitivity of the P–δP module to step sizechange. Figure 7a–f presents example results showingthe RMSE during P–δP module mode of a 20 s trainingwindow for the North position component for six differ-ent step sizes 0.01, 0.1, 1.2, 5, 10 and 50 respectively.It became obvious that the choice of a suitable step sizeis an important design criterion to achieve the desiredsystem performance. The chosen number of member-ship function Nmf, step size (κ), step increase rate (h)and step decrease rate (g) used in ANFIS training withdifferent window sizes are presented in Table 1.

Merits and limitations of using fuzzy inference system 897

Table 1 Typical step size (κ),step increase rate (η) and stepdecrease rate (γ ) and numberof membership functions(Nmf) used with three windowsizes

Window size East position North position Vertical positioncomponent component component

κ η γ Nmf κ η γ Nmf κ η γ Nmf

W = 20 1 1.1 0.8 2 1 1.1 0.8 3 1 1.1 0.8 3W = 50 0.1 1.1 0.65 2 10 1.1 0.65 3 1 1.1 0.8 3W =100 100 1.1 0.65 2 1 1.1 0.65 2 1 1.1 0.8 2

Finally, during the update procedure using 20 s win-dow size, the P–δP module for the vertical positioncomponent (altitude) was able to achieve RMSE valueof 0.084 m over the entire trajectory. This prevents theerror growth of the INS altitude that reached 20 km er-ror while operating without update. The P–δP modulewas also capable of providing similar performances forboth the East and North position components.

4.2 Prediction mode of operation

The P–δP module is also designed to predict the vehicleposition during GPS outages. As discussed earlier, dur-ing this mode of operation, the P–δP module predictsthe INS position error based on processing the corre-sponding INS position component. Accurate predictionof the vehicle position relying only on INS and the posi-tion error delivered by the P–δP module depends highlyon how efficient the module was trained and updated be-fore losing the GPS signal. Figure 8a–b and 8c show thelargest position error in the three position componentsduring all GPS outages when utilizing a window sizeof 20 s and considering 10, 20 and 40 s of GPS outagesrespectively.

It can be observed that significant deterioration inthe positioning accuracy was usually observed when theGPS outage time exceeded the window size significantly.It is thus obvious that there is a need to limit the windowsize during the update mode in order to guarantee effi-cient performance of the P–δP module during GPS out-ages. Limiting the window size during the update modeensured that the non-stationary nature of both INS andGPS data is minimal within each window. It also assuredthat appropriate updating of the P–δP module (reachingthe predefined minimum RMSE during training) couldbe granted without the need to change critical trainingparameters such as the number of membership functionsand the step size variables for each window.

As can be depicted from Fig. 8a, the vertical positioncomponent was predicted with errors less than one me-ter for the three window sizes and all the GPS outages.On the other hand, the East position component wasdelivered with errors of less than 2 m except for the twocases of GPS outages 4 and 7 as can be observed from

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 4 5 6 7 8 9 10

Outage number

1 2 4 5 6 7 8 9 10

1 2 4 5 6 7 8 9 10

Outage number

Outage number

Ver

tica

l po

siti

on

err

or

(m) 10 seconds GPS outage

20 seconds GPS outage

40 seconds GPS outage

(a)

0

1

2

3

4

5

6

7

8

Eas

t p

osi

tio

n e

rro

r (m

)

10 seconds GPS outage

20 seconds GPS outage

40 seconds GPS outage

(b)

0

1

2

3

4

5

6

No

rth

po

siti

on

err

or

(m)

10 seconds GPS outage

20 seconds GPS outage

40 seconds GPS outage

(c)

Fig. 8 Performance of the P–δP module for 20 s window sizea Maximum position error at the vertical position component dur-ing GPS outages. b Maximum position error at the East positioncomponent during GPS outages. c Maximum position error at theNorth position component during GPS outages

898 R. Sharaf et al.

Fig. 9 The vehicle dynamics along the North direction before and during the eighth GPS outage a North position component. b Northvelocity component

Fig. 10 Change of P − δ P module positioning accuracy with re-spect to the number of membership functions using small windowsize (20 s)

Fig. 8b. The increase of the East position error at GPSoutages 4 and 7 is attributed to the abrupt change inthe East velocity component just prior to both outages.Furthermore, as shown in Fig. 8c, the North positioncomponent was delivered with relatively high accuracy(about 2 m) except for the two GPS outages 8 and 9 thatlasted for 40 s, where the position error reached 5 m.This is basically because both GPS outages 8 and 9 werechosen relatively close to each other (approximately300 s in between). Moreover, prior to these outages theP–δP module was trained on a type of vehicle dynamicsdifferent from what has been experienced during bothoutages. This is illustrated in Fig. 9. Consequently, theP–δP module has not been accurately updated to mimicthese dynamics, thus relatively large position errors of3.4 m have been reached.

Finally, while testing the P–δP module using smallwindow size, it was observed that the number of mem-

Fig. 11 Comparison of the total position error obtained byKalman filtering to P − δ P module during the 10 GPS outages

bership functions utilized to describe the INS positionas an input to the ANFIS model has insignificanteffect on the position accuracy of the system. While threemembership functions were usually capable of produc-ing acceptable accuracy, the system accuracy was notsignificantly affected when the number of membershipfunctions was changed. This observation is illustrated inFig. 10.

In fact, the P–δP module can deliver relatively ade-quate to high positioning accuracy during GPS outagessimilar to those accuracies reported for the same testdata using KF-based techniques reported by (Noureldinet al. 2006). The position errors of the P–δP module werecompared with the results reported by (Noureldin et al.2006) using conventional Kalman filtering technique asshown in Fig. 11. It is evident that the P–δP module hassimilar or lower total position error than those reportedby KF method in 8 out of 9 testing cases.

The P–δP module proposed in this study has sev-eral merits over traditional data fusion techniques. Asummary of comparison between the proposed P–δPapproach and the conventional KF-based approach ispresented in Table 2. Some obvious merits of the P–δPmodule include being modeless, system and platform

Merits and limitations of using fuzzy inference system 899

Table 2 Comparison between KF and ANFIS-based modules for INS/GPS integration

Kalman filter ANFIS-based module

Model dependency Mathematical model is needed (Deterministic model + stochastic model) Empirical and adaptive modelPriori knowledge Required (the covariance of INS and GPS data) Not requiredSensor dependency Re-design of Kalman filter parameters is needed An adaptable, platform and

for different systems. system independentLinearity Linear processing Nonlinear processingDesign time Long Short

independent module that requires no a priori knowl-edge of the navigation equipment utilized. This meritis of high importance when low-end tactical grade andMEMS-based IMUs will be used for interial naviga-tion as accuracies of KF-based techniques deterioratesignificantly. Moreover, another merit in implementingANFIS is using the fuzzy set theory ability to handleuncertainty through dealing with membership values.The membership values of the INS-position µA

˜(PINS)

used in the modeling process represent our uncertaintyof the level by which PINS belongs to the fuzzy set A

˜defined over the INS-position domain. The fact thatµA

˜(PINS) ∈ [

0 , 1]

shall not be observed within the con-

text of probability but simply used for scaling handinessto address uncertainty (Laviolette et al. 1995; Singpur-walla and Booker 2004).

Nevertheless, the P–δP module has a few limitationswhich occurred when a large window size is used and/orwhen considerable vehicle dynamics that are signifi-cantly different than those observed during system up-date. There is also an inherent relation of the proposedsystem between the window size and the outage size.Further research work is underway to address theselimitations.

5 Conclusions

This article introduced a new system for temporal inte-gration of INS and GPS in vehicular navigation based onANFIS. An architecture that is based on estimating andremoval of INS position error was employed in a tem-poral fashion using special windowing technique. Theproposed system was tested with real field test exper-imental data during the availability of GPS signal andduring GPS outages.

The investigations showed that the P–δP module wascapable of reducing the INS position error and preventits long-term growth during GPS availability. The mod-ule was also capable of predicting the INS position errorsduring GPS outages with good accuracy. The accuracyof the system depends on the size of the window utilizedfor system implementation. With 20 s window size, the

proposed system was capable of predicting the vehicleposition during GPS outages with accuracies similar orbetter than KF-based modules. Merits and limitationsof the proposed system have been discussed.

Acknowledgements This research was funded by research grantsof Dr. Noureldin from the Natural Science and Engineering Re-search Council of Canada (NSERC) and Geomatics for InformedDecisions (GEOIDE)–Network Centre of Excellence (NCE). Theauthors gratefully acknowledge this support. The financial supportto Dr. Reda Taha by Defense Threat Reduction Agency (DTRA)-University Strategic Partnership is greatly appreciated.

References

Brown RG, Hwang PYC (1992) Introduction to random signals.Wiley, New York

Bshouty NH, Burroughs L (2005) Maximizing agreements withone-sided error with applications to Heuristic learning, MachLearn 59 (1–2):99–123

Chiang K-W, Noureldin A, El-Sheimy N (2006) The utilization ofartificial neural networks for multi-sensor system integration innavigation and positioning instruments: IEEE Trans InstrumMeasure 55(5).

Deb K, Gupta H (2005) Searching for robust pareto-optimalsolutions in multi-objective Optimization. In: Proceedings ofthird international conference on evolutionary multi-crite-rion optimization CAC Coello, AH, Aguirre, E, Zitzler (eds),Guanajuato, Mexico, Lecture Notes in Computer Science 3410,Springer, Berlin Heidelberg New York

El-Sheimy N (1996) The development of VISAT - a mobile surveysystem for GIS applications. Ph.D. thesis, Department of Geo-matics Engineering, The University of Calgary, UCGE ReportNo. 20101

El-Sheimy N, Schwarz KP (1994) Integrating differential GPSreceivers with an inertial navigation system (INS) and CCDcameras for a mobile GIS data collection system. In: ISPRS94,Ottawa, Canada, pp 241–248

Farrell J (1998) The global positioning system and inertial naviga-tion, McGraw-Hill, New York

Hargrave PJ (1989) A tutorial introduction to Kalman filtering.IEE colloquium on Kalman filters: introduction, applicationsand future developments, pp 1–6

Hostetler L, Andreas R (1983) Nonlinear Kalman filtering tech-niques for terrain-aided navigation. IEEE Trans AutomaticControl 28(3): 315–323

IEEE standard 647(1995) IEEE standard specification formalguide and test procedure for single-axis laser gyros

Lobo J, Lucas P, Dias J, Traca de Almeida A (1995) Inertial navi-gation system for mobile land vehicles. In: Proceedings IEEEinternational symposion on industrial electronics ISIE ’95,Vol. 2, pp 843–848

900 R. Sharaf et al.

Jang JSR (1993) ANFIS: adaptive network-based fuzzy inferencesystems. IEEE Trans Syst Man Cybern 23(3):665–685

Jang JSR, Sun CT, Mizutani E (1997) Neuro-fuzzy and soft com-puting, a computational approach to learning and machineintelligence. Prentice Hall, Englewood Cliffs

Klir G, Yaun B (1995) Fuzzy sets and fuzzy logic: theory andapplications. Prentice Hall, Upper Saddle River

Laviolette M, Seaman JW, Barrett JD, Woodall WH (1995) Aprobabilistic and statistical view of fuzzy methods. Technomet-rics 37:249–261

Mynbaev DK (1994) Errors of an inertial navigation unit causedby ring laser gyros errors In: IEEE position location and navi-gation symposium, pp 833–838

Nassar S, Noureldin A, El-Sheimy N (2004) Improving position-ing accuracy during kinematic DGPS outage periods usingSINS/DGPS integration and SINS data de-noising. Surv Rev37(292):426–438

Noureldin A, El-Shafie A, Reda Taha M (2006) Optimizing neuro-fuzzy modules for data fusion of vehicular navigation systemsusing temporal cross-validation. In: Engineering Applicationsof Artificial Intelligence, Vol.19(7). Elsevier, Amsterdam.

Noureldin A, Osman A, El-Sheimy N (2004) A neuro-waveletmethod for multi-sensor system integration for vehicular navi-gation. J Measure Sci Technol 15(2):404–412

Noureldin A, Irvine-Halliday D, Mintchev MP (2002) Accuracylimitations of FOG-based continuous measurement-while-dril-ling surveying instruments for horizontal wells. IEEE Trans InstMeasure 51(6): 1177–1191

Passino K, Yurkovich S (1998) Fuzzy control. Addison Wesley,CA

Ross TJ (2004) Fuzzy logic with engineering applications. WileyWest Sussex

Salychev O (1998) Inertial systems in navigation and geophysicsBauman MSTU Press, Moscow

Scherzinger BM, Reid DB (1994) Modified strapdown inertialnavigator error models. In: IEEE Position location and naviga-tion Symposium, pp 426–430

Singpurwalla ND, Booker JM (2004) Membership functions andprobability measures of fuzzy sets. J Ame Stat Assoc 99 (467):867–877

Vanicek P, Omerbasic M (1999) Does a navigation algorithm haveto use Kalman filter?. Can Aeronaut Space J 45 (3)