Adaptive Cubature and Sigma Points Kalman Filtering Applied to MEMS IMU/GNSS Data Fusion during Measurement Outlier

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1 Joural of Sesor echology,,, 5-5 Published Olie December ( Adaptive Cubature ad Sigma Poits Kalma Filterig Applied to MEMS IMU/GNSS Data Fusio durig Measuremet Outlier Hamza Bezerrou, Hasse Salhi, Alexader Nebylov SE Laboratory (Systèmes Electriques et élecommade) of Electroic Departmet of Saad Dahlab Uiversity of Blida, Soumaa, Algeria Iteratioal Istitute for Advaced Aerospace echologies of Sait-Petersburg State Uiversity of Aerospace Istrumetatio, 67 Bolshaya Morsaya, Sait Petersburg, Russia Received October, ; revised November 5, ; accepted December, Copyright Hamza Bezerrou et al. his is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. ABSRAC I this paper, adaptive sesor fusio INS/GNSS is proposed to solve specific problem of o liear time variat state space estimatio with measuremet outliers, differet algorithms are used to solve this specific problem geerally occurs i itetioal ad o-itetioal iterfereces caused by other radio avigatio sources, or by the GNSS receiver s deterioratio. No liear approximatio techiques such as Exteded Kalma filter EKF, Sigma Poit Kalma Filters such as UKF ad are computed to estimate the avigatio states for UAV flight cotrol. Several comparisos are coduced ad aalyzed i order to compare the accuracy ad the covergece of differet approaches usually applied i avigatio data fusio purposes. he last o liear filter algorithm developed is the Cubature Kalma Filter which provides more accurate estimatio with more stability i racig data fusio applicatio. I this wor, is compared with SPKF ad EKF i ideal coditios ad durig GNSS outliers supposed to occur durig specific iterval of time, iovatio based adaptive approach is selected ad used to modify the covariace calculatio of the o liear filters performed i this paper. Iterestig results are observed, discussed with real perspectives i avigatio data fusio for real time applicatios. hree parallel modified algorithms are simulated ad compared to o-adaptive forms accordig to Root Mea Square Error (R) criteria. Keywords: IMU; MEMS; GPS; GNSS; Kalma Filterig; Cubature Rule; Sigma Poits; Usceted Kalma Filter. Itroductio Data fusio for o liear system is oe of the most importat ad challegig problems i Multisesor sigal processig ad itegrated avigatio systems today. I our wor, sesors used are iertial as the mai system with exteral aid provided by GPS ad GLONASS receivers ow recetly as Global Navigatio Satellite System GNSS solutios. Data fusio based o IMU/GNSS has bee widely explored ad experimeted i the specialized literature [,]. For iertial sesors, accelerometers ad gyroscopes, the techology of maufacturig these sesors has a great importace ad high impact o the accuracy of iertial avigatio systems. I this paper, it is assumed that accelerometers ad gyroscopes are i the category Low cost which gives more importat iterest i real time applicatios where most sesors are MEMS Micro Electrical Mechaical Systems based techology. he most icoveiet of these iertial sesors are the biases ad drifts growig durig time, which eeds to be bouded by aother techology of sesors such as used i our wor, called GNSS receivers. Satellite-based systems such as GNSS gives today s satellite trajectory ad high-precisio avigatio. Iertial sesors combied with GNSS receiver are a good alterative ad reliable itegrated system for avigatio purposes. However, GNSS ; Galileo E5a/E5b sigals ad the GPS L5 sigal lie withi the Aeroautical Radioavigatio Services (ARNS) bad. hey suffer iterferece from the services i this frequecy bad, i particular, high power pulsed sigals from Distace Measurig Equipmet (DME) ad actical Air Navigatio (ACAN) systems embeeded o most aircrafts. he pulsed iterferece degrades received JS

2 6 H. BENZERROUK E AL. Sigal to Iterferece ad Noise Ratio (SINR), lowers the acquisitio sesitivity ad eve causes the tracig loops to diverge. o maitai system accuracy ad itegrity, iterferece mitigatio is beeficial ad ecessary. Adaptive itegrated avigatio systems INS/GNSS is the proposed for differet aerospace applicatios. However i our wor, we focus o GNSS outliers caused by multipath sceario, a bad satellite visibility due to flights i caio eviromet, or due to o-itetioal iterfereces caused by multiple GSM sigals, multiple satellite commuicatio techologies such as Iridium, Globalstar etc. Nevertheless, the classic form of INS/ GNSS data fusio is ot adaptive agaist jammig, impulsive oise, missig measuremet... etc. I this paper a solutio is proposed based o adaptive exteded Kalma filter (EKF) the, compared with more advaced ad moder approaches i o liear filterig such as the adaptive SPKF ad the adaptive [].. Iertial Sesors Iertial Measuremet Uit IMU is the erel of ay iertial avigatio system. It is composed by accelerometers ad gyroscopes i additio to magetometers i most moder IMU s. he techology used durig the last 5 years has bee divided i two pricipal techologies: Gimbaled INS ad Strapdow INS. I our wor, the model used is related to strapdow techology with fixed iertial sesors MEMS based, i parallel with body axes. Most of todays iertial sesors are micro electromechaical systems (MEMS). his techology was first used for commercial purposes i the 99 s, ad eabled ew applicatios through high miiaturizatio ad cost reductio. Iertial sesors bega to be used i completely ew domais, such as Pedestria avigatio. However, this miiaturizatio ad cost reductio ifueces the performace of the accelerometers ad gyroscopes, which explais why some iertial sesors based o previous techologies are still used for high-performace purposes. Gimbaled INS are mechaical with special Horizotal stabilizatio cotrol algorithm with very expesive cost, they are usually used oboard satellites, spacecrafts, submarie etc. IMUs based o MEMS sesors are strap-dow systems, which meas the sesor s orietatio depeds of the orietatio of the object it is o. heoretically, all types of previously show MEMS iertial sesors ca be combied i a IMU... Mechaizatio of Iertial Measuremet Uit Iertial avigatio system is divided i two pricipal parts: IMU ad Digital Sigal Processig of sesors data fusio. Strapdow INS mechaizatio is described such as i Figures ad with a geeral diagram of SINS as described, based o iertial sesors output; accelerometers ad gyroscopes. his avigatio system ca t esure log term accuracy of its output, ad depeds o exteral aid such as GNSS i most of aerospace applicatios [4]... IMU Sesors Output Itegratio Iertial measuremet uits (IMUs) typically cotai three orthogoal rate-gyroscopes ad three orthogoal accelerometers, measurig agular velocity ad liear acceleratio respectively. Ideally, the output of the rate-gyroscopes is writte as t t t t () b bx by bz I practice, however, the outputs cotai errors ad are writte as the formula give below: t t t (), b b b t t t t () b bx by bz Itegratig this yields the updated attitude iformatio for the system provides the followig equatio: d Cb t Cb t t, (4) dt t t t bz by t bz t bz (5) by t bz t b C tt C b t e tt t t dt si bt Cb t I t t (6) bt cos bt t t bt Similarly, accelerometers outputs ca be writte as b bx by bz a t a t a t a t, (7) Figure. Strapdow Iertial Measuremet Uit with MEMS gyroscopes from S Microelectroic. JS

3 H. BENZERROUK E AL. 7 Figure. IMU/INS Sesors output itegratio., a t a t a t (8) b b b wo itegratios subsequetly yield velocity ad positio updates as follows Velocity itegratio: Positio itegratio: V V t a g,,,,, g, (9) Pos Pos t V () where g is the estimated gravity vector ad t is the data period. Collectively, Equatios () to () describe the system model... GNSS Global Navigatio Satellite System GNSS sigal processig is much explored based o differet algorithms tested more ad more i real time coditios ad i simulatios through the specialized literature. We focus o the effect of more satellite visibility i order improve geometry dilutio of precisio due to the high umber of satellites GPS + GLONASS (6-4). GLONASS satellites also broadcast sigals i the L ad L sub-bads of the radio frequecy spectrum as described i Figure. It is observed i some situatio several iterfereces from differet sources for GPS ad GLONASS durig static ad dyamic positioig. GNSS outages or outliers cause accuracy degradatio, ad sometimes udelivered GNSS receiver positioig. However ulie PS, GLONASS (Russia) uses frequecy divisio multiple access (FDMA) i both Lad L frequecy sub-bads. his meas that each satellite modulates the same ragig code o carrier sigals with slightly differet frequecies ad is idetified by a slot umber rather tha a Pseudo radom Noise (PRN) umber. GNSS based o GPS ad GLONASS (Europea system Galileo ad Chiese system Compass i the future), are well ow satellite avigatio ystems ad uses parallel positioig techiques; the oly differece is that GPS seds differet messages o the same frequecy (L, L, L5) ad GLONASS seds the same message o mul- tiple frequecies (L, L ). It is importat to cosider i the ear future the ew statemet of GNSS costellatio icludig Galileo future Europea system ad COMPASS the future Chiese system. Each space costellatio has slightly differet orbital plae parameters. I this paper, GPS ad GLONASS C/A codes are cosidered i INS/GNSS data fusio. his study cocers o-itetioal iterfereces ad outliers/outages i GNSS sigal for civilia GNSS receivers. All adjacet commuicatio systems to GNSS bad which is a potetial source of iterfereces ad have bee studied i the literature. I this wor direct data fusio techique is applied to a importat case whe measuremet outliers occur affectig GPS ad GLONASS receivers durig UAV avigatio..4. UAV GNC System (Guidace Navigatio & Cotrol) We ow describe the applicatio of the SPKF/ to the problem of loosely coupled GPS/INS itegratio for guidace, avigatio ad cotrol (GNC) of a umaed aerial vehicle (UAV). he mai subcompoets of such a GNC system is a vehicle cotrol system ad a guidace & avigatio system (GNS) as show i Figure 4. Fixed Wig ad Rotor Wig UAVs have bee simulated through differet dyamical parameters. he embedded system icludes a iertial MEMS based IMU, a Hz GPS/ Figure. Power spectral desities of GNSS sigals. Figure 4. UAV guidace, avigatio ad cotrol. JS

4 8 H. BENZERROUK E AL. GLONASS sychroized receiver, ad a DSP Desig Autopilot REVO board for real time implemetatio. UAV oliear cotrol system is based o o liear adaptive state estimators selected as EKF, UKF, ad. We implemeted EKF, SPKF (UKF, ) ad based sesor fusio algorithms i ormal coditios ad durig outliers i order to observe the effect o the accuracy ad the covergece of filters. We will ext discuss the UAV specific system process ad observatio (measuremet) models used iside EKF, SPKF ad based avigatio system, which is used a secod time durig GNSS outliers with modified covariace estimatio. We the focus o the applicatio of the Iovatio based adaptive /SPKF to the itegrated avigatio problem. We specifically detail the developmet of a adaptive based Cubature Kalma Filter A for loosely coupled implemetatio for itegratig GPS measuremets with a IMU withi the cotext of autoomous UAV. he ext sectio describe i detail o liear filtrig algorithms used ad implemeted i this wor.. Kalma Filter ad Its Exteded Versio I estimatio theory, it is well ow that for liear state space estimatio, affected by white Gaussia oises, the optimal filter is called Kalma filter which is also equivalet to Maximum Lielihood estimator. However i most practical avigatio applicatios, a omial trajectory does ot exist beforehad. he solutio is to use the curret estimated state from the filter at each tep as the liearizatio referece from which the estimatio procedure ca proceed. Such algorithm is ow as exteded Kalma filter (EKF). Exteded Kalma Filter EKF Let us describe below the algorithm of EKF: based o state space model as give by: x f x, u, v () y g x, w () Liearizatio usig aylor approximatio at the first order gives the state space model give i most refereced literature. F. is the Jacobia matrix of f. ad H. is the Jacobia matrix of h.. hus, followig algorithm is obtaied: Iitializatio: ˆx et P. () Predictio: ˆ ˆ Q P F x P F x (4) Update: xˆ f xˆ (5) xˆ xˆ K Z h xˆ P P KH x P (7) ˆ (8) he meaig of the exteded Kalma filter ca be uderstudied by appreciatig the same equatio of gai calculatio as i the Kalma filter at the differece that i the o liear filterig, EKF is sub-optimal filter. It requires the, more adaptive approaches i solvig both filterig ad cotrol problems i Aerospace [4,5]. here is aother versio of exteded Kalma filter which could be developed at secod order of aylor approximatio, this filter offers better results uder high o liearity of the system s dyamic ad measuremet model. I the ext sectio, aother approach based o cubature rule techique is developed. 4. Sigma Poit Kalma Filterig (Cubature Rule) Differet estimators were itroduced to solve o liear estimatio problems; Sigma poits Kalma filters (SPKF) itroduced i the recet advaces i oliear filterig. Both Usceted filters (UKF) ad () mea SPKF, i this case, the desity of probability usig a determiistic sigma poits is estimated at the first ad the secod order momets of the RGV. For Cetral Differece Filter, it adopts a alterative method called cetral differece approximatio. Lie UKF, geerates several poits about the mea based o varyig the covariace matrix alog each dimesio. It evaluates a o liear fuctio at two differet poits for each dimesio of the state vector that are divided by a appropriate chose iterval, SPKF are strog estimators ad superior alterative to the EKF i several applicatios with high o liearity. 4.. Cubature Kalma Filter is ow as the closest ow approximatio to the Bayesia filter for o liear estimatio with Gaussia assumptios. Such as for UKF ad, does t require ay Jacobia matrix computatio of liearizatio. he basic steps are give i the ext paragraphs. he ey assumptio of is that the predictive desity p x Z where Z deotes the history of iput measuremet pairs up to, ad the filter lielihood p z Z are both Gaussia, which leads to a posterior Gaussia desity p x Z. Uder these coditios, reduces the computatio of mea ad covariace ˆ ˆ ˆ K P H x H x P H x R (6) JS

5 H. BENZERROUK E AL. 9 with more accuracy. he cubature based Gaussia filter algorithms use cubature rules of the form: m I f f (9) i to approximate the itegral of the form: x x dx () g x x dx g x e π Equatio (6) is a itegral of a o liear fuctio multiplied by Gaussia weight. he usceted trasformatio ca also be iterpreted as a approximatio of the itegral of the form (7). he techique itroduced is based o Gaussia sum filters explored ad give i detail by [6]. However it is proposed to model jammig GNSS sigal by particular id of Gaussia sum oise which is twi Gaussia sum affectig oly measuremet equatio. Bellow the algorithm of Adaptive proposed ad applied i this wor: 4... Predictio Step ) Draw cubature poits i, i,,, from the itersectios of the -dimesioal uit sphere ad the Cartesia axes. Scaled by. We ca write the: ei i for i,,, i,, () ei ) Propagate cubature poits. he matrix square root is the lower triagular Cholesy factor. i, i i X P m () ) Evaluate the cubature poits with dyamic model fuctio: * i, i, 4) Estimate the predicted state mea: i f X () (4) m * i, 5) Estimate the predicted error covariace: P m * x i, i, m Q (5) 4... Update Step ) Draw cubature poits i, i,,, from the itersectios of the -dimesioal uit sphere ad the Cartesia axes. Scaled by (as i step ). ) Propagate the cubature poits. Xi, P im (6) ) Evaluate the cubature poits with the measuremet model. Y i, i, 4) Estimate the predicted measuremet: yˆ h X (7) Yi, i (8) 5) Estimate the iovatio covariace matrix. S Y ˆ ˆ i, Yi, / y y R i 6) Estimate the cross covariace matrix. (9) ˆ i, i, i P xy X Y m y () 7) Estimate the Kalma gai. K 8) Estimate the update state. Pxy, S () m m K y yˆ () 9) Estimate the error covariace P P K S K () NOE: Comparig with SPKF, there are o parameters to tue i approximatig o liear fuctios of the system ad measuremet. Aother alterative to approximate the lower boud for o liear state estimatio with additive Gaussia oises is give ad described i [7,8]. NOE: Comparig with SPKF, there are o parameters to tue i approximatig o liear fuctios of the system ad measuremet. Based o similar idea such as for sub optimal fadig factor, it is possible to combie Sigma Poit Kalma Filters (UKF,, ) ad adaptive fadig approach. 4.. Adaptive Cubature Kalma Filter A It is possible to defie the the fadig factor such as ve bellow: V R i,, tr, i i, tr P, i, yy With parameters defied below: gi- (4) vv V V vv (5), hus, the covariace matrices eed to be updated bas- JS

6 H. BENZERROUK E AL. ed o the adaptive fadig factor such as give i the followig equatios: * Px i, i, mm Q ( 6) S Y Y y y ˆ ˆ i, i, R i P X Y m ˆ x, y,, i i i (7) y (8) he ey parameter i this adaptive algorithm is the fadig factor. It requires the defied parameters, some other techiques i literature use multiple fadig factors which is ot always superior to the sigle fadig factor ad are commoly selected. Coceptually, the implemetatio priciple of SPKF/ resembles that of the EKF, the implemetatio, however, is sigificatly simpler because it is ot ecessary to formulate the Jacobia ad/or Hessia matrices of partial derivatives of the oliear dyamic ad measuremet equatios, which is very importat for real time implemetatio. Before, implemetig adaptive SPKF ad, we propose to observe the effect of Iovatio based adaptive EKF o the avigatio state durig outliers as a first experiece, thus, go i more advaced sigal processig for UAV data fusio durig GNSS outliers. st he Simulatio of EKF-Adaptive EKF he coditio of simulatio is described i able : = 5 s, N = 5; dt =.; g = 9.8 m/s/s; Outlier duratio of time OD = 9 s. North Velocity able. Iitial values of covariace matrices P, Q, R. Estimatio Covariace diqg p System Covariace diqg Q Measuremet Covariace diqg R 5 /8.5/8.5/8 /8.5/8.5/8 /8.5/8.5/ INS rue Velocity GPS EKF AEFK NORH Velocity vs Figure 5. North velocity estimatio. Iitial Motio Distace N, E, D Velocity N, E, D Attitude agles pitch, (meter) (m/s) roll, yaw (rad).5 x 4 North Velocity EKF 6 /8 7 45/8 5 /8.5 From Figure 5 to Figure, oe ca observe the state estimatio results of the o liear part described i the previous sectio, so the orth velocity with correspodig, East velocity with correspodig, vertical velocity with its correspodig ad pitch attitude observatio based o multiple GNSS atea with the correspodig. he figures cotet show how accuracy degradatio of EKF ca be caused by GNSS measuremet outliers. I the first part of avigatio durig reliable GNSS outputs, EKF ad Adaptive EKF seem to be superposed. IMU outputs without observatio diverge from the real Figure 6. orth velocity. trajectories ad EKF durig outliers is less accurate tha IMU itegratio outputs. Attitude estimatio based o multiple GNSS ateas. JS

7 H. BENZERROUK E AL. 6 4 East Velocity vs INS rue Velocity GPS EKF AEFK.5 x 4 Dow Velocity.5 EKF East Velocity Figure 7. East velocity estimatio Figure. vertical velocity estimatio..5 x 4.5 East Velocity EKF 4 Pitch vs INS rue Pitch GPS EKF AEFK.5 Pitch Figure 8. orth velocity. Dow Velocity vs Figure. Pitch agle estimatio. Vertical Velocity INS rue Velocity GPS EKF AEFK Pitch EKF Figure 9. Vertical velocity estimatio. After multiple simulatios, it is clear through state ad related estimatio that Iovatio based Adaptive EKF AEKF provide a ameliorated filterig accuracy comparig with EKF. his is cofirmed by differet state Figure. pitch agle estimatio. estimatio with differet o liearity such as velocity ad attitude states. JS

8 H. BENZERROUK E AL. 5. Simulatio of Adaptive Cubature Kalma Filter Le t us go bac to the system give by () ad by the measuremet oise affected by outliers. Let the iitial state be described as a percetage of the true iitial state. hese ext simulatios should provide two importat observatios: ) Compare EKF,UKF, ad durig outliers without modificatio of Covariaces estimatio. ) Compare Iovatio based Adaptive EKF, UKF, ad durig GNSS outliers after modificatio of covariace estimatio algorithm Yaw UKF EKF UKF 5.. Results ad Discussio he same simulatio coditios of the previous sectio are cosidered i this secod experiece: first, attitude estimatio is preseted with yaw, pitch ad roll agles estimatio. he, velocity through axes of avigatio (orth, east, dow) are observed ad discussed, with fially, the liear part i state estimatio preseted i this paper with positio state vector estimatio. Attitude estimatio uder outliers from Figure to Figure 8. Velocity estimatio uder outliers from Figure 9 to Figure 5. Liear estimatio part: Positio. he oly liear part i this wor was the locatio estimate o with orth, east ad vertical distace states are observed o Figures 6 ad Observatio of Adaptive No Liear State Estimatio for IMU/GNSS Data Fusio durig Outliers Durig simulatios, the followig algorithms are compared ad applied to MEMS IMU/GNSS data fusio durig 5 secods. I this iterval, GNSS outliers are Yaw UKF EKF UKF Figure. yaw agle estimatio i degree Figure 4. yaw agle estimatio i degree zoom UKF EKF UKF Figure 5. pitch agle estimatio i degree UKF EKF UKF Figure 6. pitch agle estimatio i degree zoom. simulated ad activated betwee s ad s. For the first part of simulatio i ormal flight coditios, JS

9 H. BENZERROUK E AL UKF EKF UKF Dow Velocity UKF EKF UKF Figure 7. roll agle estimatio i degree Figure. dow velocity estimatio (m/s) zoom UKF EKF UKF x East Velocity UKF EKF UKF Figure 8. roll agle estimatio i degree zoom. Dow Velocity Figure. east velocity estimatio (m/s) UKF EKF UKF UKF EKF UKF East Velocity Figure 9. dow velocity estimatio (m/s). betwee ad s, it is possible to observe that all filters, classical ad adaptive are superposed ad preset the same accuracy ad covergece curve. his is logic re Figure. east velocity estimatio (m/s). sults due to uuseful iovatio based adaptive fadig factors used to improve the accuracy of filterig algo- JS

10 4 H. BENZERROUK E AL East Velocity UKF EKF UKF x 5 5 Dow positio UKF EKF UKF Figure. east velocity estimatio (m/s) zoom. x North Velocity UKF EKF UKF Figure 6. vertical positio estimatio (m/s). 5 5 Dow positio UKF EKF UKF Figure 4. orth velocity estimatio (m/s) North Velocity UKF EKF UKF Figure 5. orth velocity estimatio (m/s) zoom. rithms durig outlier period. It is very iterestig to observe the clear part of all figures which describes clas Figure 7. vertical positio estimatio (m/s) zoom. sical o liear filters EKF,, ad UKF without ay adaptatio, except for velocity estimatio which i that case, UKF performs i its o adaptive forms durig outliers. his is a uexpected result betwee UKF ad. I this precise situatio, UKF has better accuracy tha which is ot usual i previous study ad experimets i the field of data fusio. his is cosidered as a iterestig ovel result. As for EKF ad, it is possible to observe too a additioal iterestig result, R is less accurate tha EKF R durig outlier period for velocity estimatio which meas that GNSS outliers affect more the covariace estimatio tha the EKF computatio. It is the possible to distiguish a classificatio of filters which is for most states estimatio accordig the accuracy ad covergece of filters: EKF,, UKF, durig outliers. It is very iterestig to observe i the secod curves of the adaptive filters the clear part of all figures which describes adaptive o liear filters AEKF, A, AUKF ad A, A performs all JS

11 H. BENZERROUK E AL. 5 adaptive ASPKF ad AEKF durig outliers. It is clear that A maitai R close to the miimum variace which is ot the case for AUKF, A ad AEKF, those algorithms preset some impulsive errors durig outliers eve with iovatio based adaptive fadig factor. his is the proof that after modificatio of its covariace ad mea estimatio presets a superior accuracy ad stability of covergece comparig with modified SPKF ad EKF. It is also clear that A performs AUKF, A ad fially AEKF. he last figures of positio estimatio, it is possible to observe the R of A, ASPKF ad AEKF durig outliers, ad it is clear that for this liear state, o liear filters are o adequate oly A provide a good results betwee 9 s ad s. It is the possible to geeralize the observatio that all trasformed filters AEKF, AUKF, A ad A outperform their classical algorithms EKF, UKF, ad respectively. his meas that the solutio we propose is a good alterative to GNSS outliers i IMU/GNSS data fusio. 6. Coclusio he desig of o liear adaptive filters ad the associ- ated data fusio based o EKF, SPKF ad were deeply studied for MEMS IMU/GNSS sesor fusio durig satellite positioig outliers. Based o the iovatio fadig factor, 4 o liear filterig approaches EKF, KF, ad were modified usig this fadig factor for covariace ad mea estimatio durig GPS/ LONASS outages. By the way, several solutios were proposed ad ameliorated the accuracy ad the time of covergece of each modified filterig algorithm. However accordig the R criteria, Cubature Kalma Filter durig outliers demostrated some limitatio to perform Usceted Kalma Filter UKF i its o adaptive form. Fially, adaptive fadig demostrated a large superiority i estimatio accuracy ad i its stability comparig with other filterig classes adaptive EKF, adaptive UKF ad adaptive. REFERENCES [] I. Arasaratam ad S. Hayi, Cubature Kalma Filterig, IEEE rasactios o Automatic Cotrol, Vol. 54, No. 6, 9, pp [] A R. Rualls, A Kullbac-Leibler Approach to Gaussia Mixture Reductio, IEEE rasactio o Aerospace ad Electroic System, 6. [] P. chiavsy, C. H. Muravchi ad A. Nehorai, Posterior Cramer Rao Bouds for Discrete ime Noliear Filterig, IEEE rasactios o Sigal Processig, Vol. 46, No. 5, 998, pp [4] H. Bezerrou ad A. V. Nebylov, Robust Noliear Filterig Applied to Itegrated Navigatio System uder No Gaussia Measuremet Noise Effect, Proceedig of IEEE AEROSPACE Coferece, Big Sy Motaa. [5] D. Karlgaard Christopher ad S. Hespeter, Huber Divided Differece Fiters, Vol., No., 7, pp [6] P. Closas ad C. Feradez-Prades, Bayesia Noliear Filters for Direct Positio Estimatio, Proceedigs of the IEEE Aerospace Coferece, Big Sy, M (USA), March, pp. -. [7] P. Closas ad C. Feradez-Prades ad J. Vilà-Valls, Multiple Quadrature Kalma Filterig, IEEE rasactios o Sigal Processig, Vol. 6, No.,, pp [8] H. Bezerrou, Gaussia vs. No-Gaussia Noise i Iertial/GNSS itegratio, GNSS Solutios, Iside GNSS Magazie, November/December, pp. -9. JS