research article a novel robust interval kalman filter...

Research ArticleA Novel Robust Interval Kalman Filter Algorithm for GPSINSIntegrated Navigation

Chen Jiang1 Shu-bi Zhang12 and Qiu-zhao Zhang12

1School of Environment Science and Spatial Informatics China University of Mining and Technology Xuzhou 221116 China2Collaborative Innovation Center for Resource Utilization and Ecological Restoration of Old Industrial BaseChina University of Mining and Technology Xuzhou 221116 China

Correspondence should be addressed to Qiu-zhao Zhang qiuzhaozhangcumteducn

Received 14 July 2016 Revised 28 August 2016 Accepted 18 September 2016

Academic Editor Jesus Corres

Copyright copy 2016 Chen Jiang 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

Kalman filter is widely applied in data fusion of dynamic systems under the assumption that the system and measurement noisesare Gaussian distributed In literature the interval Kalman filter was proposed aiming at controlling the influences of the systemmodel uncertainties The robust Kalman filter has also been proposed to control the effects of outliers In this paper a new intervalKalmanfilter algorithm is proposed by integrating the robust estimation and the interval Kalmanfilter inwhich the systemnoise andthe observation noise terms are considered simultaneously The noise data reduction and the robust estimation methods are bothintroduced into the proposed interval Kalman filter algorithm The new algorithm is equal to the standard Kalman filter in termsof computation but superior for managing with outliersThe advantage of the proposed algorithm is demonstrated experimentallyusing the integrated navigation of Global Positioning System (GPS) and the Inertial Navigation System (INS)

1 Introduction

The integration of GPS and INS has been widely adoptedfor dynamic navigation and positioning The Kalman filter(KF) is the most notable real-time optimal estimator whichhas found an extremely broad field of application [1] KFhas tacitly become an indispensable data fusion approach forGPSINS integrated navigation However the standard KFalgorithm can hardly deal with nonlinear and robust prob-lems Numerous nonlinear filtering and robust algorithmswere developed such as the Extended Kalman Filter (EKF)[2] the Unscented Kalman Filter (UKF) [3 4] the ParticleFilter (PF) [5 6] the 119867infin filter [7 8] the Adaptive Kalmanfilter (AKF) [9ndash12] and the robust Kalman filter (RKF) [1314] A nonlinear system can be handled by EKF but theimpacts of higher order terms are neglected The advantagesmanifested by UKF lie in the concise computation and real-time processing Nevertheless it becomes unstable in a high-dimensional system Both the PF and the 119867infin filter requirerelatively more calculations and the latter breaks down inthe presence of outliers [14] The AKF algorithm which is

suitable to balance the dynamic model information and themeasurements [15] has been widely investigated howeversome of the AKF algorithms cannot control the effects ofoutliers Closely related to the model uncertainty the RKFexerts a tremendous fascination on researchers whereas itignores the system model noise Yang et al proposed theadaptively robust filter algorithm [15ndash17] which manifestedstrong stability and flexibility and the adaptively robust filtershows better ability to control the influences of both thedynamicmodel disturbances and the outlyingmeasurements

In the standard KF algorithm the system model andstatistical properties of the Gaussian white noise must beknown in order to achieve optimal results However theactual system and noise models differ invariably from theassumptions A novel algorithm named the interval Kalmanfilter (IKF) was proposed [18] to cope with model parameteruncertainties The effects of the model errors are weakenedusing the IKF algorithm and the optimality of the KFapproach is preserved In addition the IKF algorithm is per-formedwith reasonable amount of computation On the basisof interval addition multiplication and inversion arithmetic

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 3727241 7 pageshttpdxdoiorg10115520163727241

2 Journal of Sensors

[19] the interval intersection arithmetic was introduced inan improved IKF algorithm [20] To simplify the calculationprocessmodel errorswere assumed to be included only in thea priori and a posteriori estimates Nevertheless covariancematrices of the system noise and observation noise wereconsidered to be known and fixed The KF approach is notthe optimal one once the observations become contaminatedConsequently the influences of outliers must be consideredThe robust estimation method provides a way to control theinfluences of outliers [21 22] In the field of robust estimationa series of equivalent weight functions is constructed [23ndash26] to resist the impacts of outliers Among these equivalentweight functions the concept of the IGG procedure ismore suitable for the geodetic surveying errors Rather thandisproportionately pursuing validity and unbiasedness therobust estimation method focuses on the robustness and thereliability of the estimator [27] Accordingly it is desirableto develop a new filtering scheme by combining the robustestimation method with the IKF

This paper focuses on a comprehensive filtering algorithmusing the reduced IKF and the classic equivalent weightrobust estimation method for the GPSINS integrated navi-gation With the impacts of the abnormal observations andthe model errors weakened a novel filtering algorithm isdeveloped for the GPSINS integrated navigation Practicalexperiments of GPSINS integrated dynamic navigation areimplemented Finally the performance is evaluated and thesuitability of the proposed filtering algorithm is discussed

The rest of the paper is organized as follows In Section 2the theory of the IKF is introduced and the model deviationsof the conventionKF algorithm are analyzed In Section 3 therobust interval Kalman filter (RIKF) algorithm is proposedbased on the robust estimation and the IKF Section 4presentsthe advantages of the new algorithm which are verifiedby experiments and a comparison with other differentalgorithms is performed and discussed The conclusions arepresented in Section 5

2 The Interval Kalman Filter (IKF)

It is extremely difficult to establish a physical model and noisemodel of the irregularly motional carrier Therefore fixingdynamic system parameters and noise statistical parametersbecomes a crucial assumption when the standard KF algo-rithm is applied However in most cases the system andobservation noise terms are only approximately knownThusthe IKF algorithm was proposed to process the dynamicsystem with uncertainties in the model parameters

The system equation and observation equation areexpressed as

119909119896+1 = Φ119896119909119896 + Γ119896119908119896119911119896 = 119867119896119909119896 + V119896 (1)

where 119909119896 is the system state vector 119911 is the observationvector Φ119896 is the state-transition matrix and 119908119896 and V119896 arethe system and observation noises respectively Φ119896 Γ119896 and119867119896 are constant matrices and 119896 denotes the epoch Theoptimal estimate is obtained with the standard KF algorithm

when Φ119896 Γ119896 and119867119896 are fixed Influences of these uncertainmatrices can be controlled by interval matrices

Φ119868119896 = Φ119896 + ΔΦ119896 = [Φ119896 minus 1003816100381610038161003816ΔΦ1198961003816100381610038161003816 Φ119896 + 1003816100381610038161003816ΔΦ1198961003816100381610038161003816] Γ119868119896 = Γ119896 + ΔΓ119896 = [Γ119896 minus 1003816100381610038161003816ΔΓ1198961003816100381610038161003816 Γ119896 + 1003816100381610038161003816ΔΓ1198961003816100381610038161003816] 119867119868119896 = 119867119896 + Δ119867119896 = [119867119896 minus 1003816100381610038161003816Δ1198671198961003816100381610038161003816 119867119896 + 1003816100381610038161003816Δ1198671198961003816100381610038161003816]

(2)

whereΔΦ119896 ΔΓ119896 andΔ119867119896 are constant perturbationmatriceswith respective boundaries and the process of the IKFalgorithm is stated below [18 28]

Main Process

119868119896 = Φ119868119896minus1119868119896minus1 + 119870119868119896 [119911119868119896 minus 119867119868119896Φ119868119896minus1119868119896minus1] (3)

Coprocess

119875119868119896minus1 = Φ119868119896minus1119875119868119896minus1 (Φ119868119896minus1)119879 + Γ119868119896minus1119876119896minus1 (Γ119868119896minus1)119879 (4)

119870119868119896 = 119875119868119896minus1 (119867119868119896)119879 [119867119868119896119875119868119896minus1 (119867119868119896)119879 + 119877119896]minus1 (5)

119875119868119896 = [119868 minus 119870119868119896119867119868119896] 119875119868119896minus1 (6)

where 119875119868119896minus1 is the a priori covariance matrix 119870119868119896 is the gainmatrix 119868119896 is the estimated state vector of the filter 119876119896 and119877119896 are covariance matrices and 119875119868119896 is the updated covariancematrix

Three types of arithmetic for the perturbation matricesare adopted in the IKF algorithm addition multiplicationand inversion Assume that 119860 is an interval matrix 119872(119860) isthe matrix composed of the middle points of each elementin 119860 119882(119860) is the width of the interval matrix and theaddition arithmetic andmultiplication arithmetic are definedas follows

119872(1198601 + 1198602) = 119872(1198601) + 119872(1198602) 119882 (1198601 + 1198602) = 119882(1198601) + 119882(1198602) 119872 (11986011198602) = 119872(1198601)119872 (1198602) 119882 (11986011198602) = 1003816100381610038161003816119872 (1198601)1003816100381610038161003816119882 (1198602)

+ 119882(1198601) 1003816100381610038161003816119872 (1198602)1003816100381610038161003816

(7)

where 1198601 and 1198602 are interval matrices Meanwhile theinversion arithmetic should be implemented namely[119867119868119896119875119868119896minus1(119867119868119896)119879 + 119877119896]minus1 in (5)

[119867119868119896119875119868119896minus1 (119867119868119896)119879 + 119877119896]minus1 = (119867119896119875119896minus1119867119879119896 + Δ119877119896)minus1 Δ119877119896

= 119867119896119875119896minus1 (Δ119867119896)119879 + 119867119896Δ119875119896minus1119867119879119896+ 119867119896Δ119875119896minus1 (Δ119867119896)119879 + Δ119867119896119875119896minus1119867119879119896+ Δ119867119896119875119896minus1 (Δ119867119896)119879 + Δ119867119896Δ119875119896minus1119867119879119896+ Δ119867119896Δ119875119896minus1 (Δ119867119896)119879 + 119877119896

(8)

Journal of Sensors 3

In this IKF algorithm Δ119877119896 was replaced by the matrix 1198771015840119896which was calculated by all the upper bounds of the intervalelements of Δ119877119896 [18] thus the matrix inversion arithmeticbecomes much easier Similar to the standard KF the IKFalgorithm is composed of a homogeneous recursive structure

3 The GPSINS Integrated Navigation and theRobust Interval Kalman Filter (RIKF)

31 GPSINS Integrated Navigation In loosely coupledGPSINS integrated navigation the differences between theoutputs (position and velocity) of GPS and INS are regardedas the observation information A 15-dimension state vectoris selected which includes the deviations of the position thedeviations of velocities the attitudes and the noises of thegyroscope and the accelerometer The state vector is listedbelow

= [120575119909 120575119910 120575119911 120575V119909 120575V119910 120575V119911 120575120601119890 120575120601119899 120575120601119906 120575119892119909 120575119892119910 120575119892119911120575119886119909 120575119886119910 120575119886119911]

(9)

The state equation of the continuous system is

119883 (119905) = 119865 (119905)119883 (119905) + 119866 (119905)119882 (119905) (10)

where 119882(119905) is the system noise and 119865(119905) is the dynamicmatrix

The discretization of (10) is achieved by Taylor seriesexpansion

119883119896 = Φ119896119896minus1119883119896minus1 +119882119896 (11)

where 119883119896minus1 is the state vector in epoch 119896 minus 1 119882119896 is themodel deviation andΦ119896119896minus1 is the discretized state-transitionmatrix

There are two kinds of integration for integrated navi-gation namely the tightly coupled and the loosely coupledIn the tightly coupled navigation the pseudorange differencebetween GPS and INS is considered as the measurementequation

119885120588 (119905) = 120588GPS minus 120588INS (12)

The position and velocity differences between GPS and INSare applied to construct the measurement equation 119909GPS119910GPS 119911GPS V119883GPS V119884GPS and V119885GPS are the output infor-mation of the GPS in the WGS-84 coordinate system andthe output information of the INS is denoted by 119909INS 119910INS119911INS V119883INS V119884INS and V119885INSThemeasurement equation is asfollows

119871119896 = [119903GPS minus 119903INSVGPS minus VINS

] (13)

where119871119896 is the integratedmeasurement vector and 119903GPS 119903INSVGPS and VINS are the position and the velocity informationoutput of GPS and INS respectively Compared to the looselycoupled integration the tightly coupled integration performsbetter in terms of precision On the other hand the looselycoupled integration is easier to be implemented and thecomputation process is more concise [29]

32 The Robust Interval Kalman Filter After the additionmultiplication and inversion calculations the estimationresults of the intervalmatrices are similar to that of the centralpoint of the interval matrix and the difference lies in theinterval width Furthermore the interval width may becomeinfinite or zero in some casesThus the IKF and the standardKF differ little in process

Intersection arithmetic of the interval matrix is intro-duced and model errors are reduced into the a priori anda posteriori estimates Hence required calculations can bereduced and the computational speed of the reduced IKFalgorithm becomes almost identical to the standard KFalgorithm [20] Equations of the reduced IKF algorithm arelisted below

[minus119896 minus119896 ] = Φ119896119896minus1 [minus119896minus1 minus119896minus1] (14)

119875minus119896 = Φ119896119896minus1119875119896minus1Φ119879119896119896minus1 + Γ119896minus1119876119896minus1Γ119879119896minus1 (15)

119870119896 = 119875minus119896119867119879119896 (119867119896119875minus119896119867119879119896 + 119877119896)minus1 (16)

[119896 119896] = (119868 minus 119870119896119867119896) [minus119896 minus119896 ] + 119870119896119911119896 (17)

119875119896 = (119868 minus 119870119896119867119896) 119875minus1119896 (18)

where [minus119896 minus119896 ] and [119896 119896] denote the lower and the upperbounds of the a priori and the a posteriori estimatesrespectively 119870119896 is the gain matrix and 119911119896 is the innovationvector Equation (16) shows that the covariance matrix of themeasurement noise namely 119877 is regarded as a constant andfixed matrix This may result in divergence of the filteringalgorithm due to outliers in the observations

The robust estimation method is introduced into thereduced IKF algorithm and the equations of the RIKFalgorithm are as follows

Time Update

[minus119896 minus119896 ] = Φ119896119896minus1 [minus119896minus1 minus119896minus1] 119875minus119896 = Φ119896119896minus1119875119896minus1Φ119879119896119896minus1 + Γ119896minus1119876119896minus1Γ119879119896minus1

(19)

Measurement Update

119870119896 = 119875minus119896119867119879119896 (119867119896119875minus119896119867119879119896 + 119875minus1119896 )minus1 (20)

[119896 119896] = [minus119896 minus119896 ] + 119870119896 119871119896 minus 119867119896 [minus119896 minus119896 ] (21)

119875119896 = (119868 minus 119870119896119867119896) 119875minus119896 (119868 minus 119870119896119867119896)119879 + 119870119896119875minus1119896 119870119879119896 (22)

where 119870119896 is the equivalent gain matrix 119875119896 is the equivalentweight matrix of the observation vector and 119875minus1119896 = 119877119896and 119877119896 is the equivalent covariance matrix Equation (22)named the Joseph stabilized version is chosen here becauseit is more numerically stable and robust than other formsof the covariance update equation [30] The IGGIII functionis adopted to evaluate the equivalent weight matrix 119875119896


Robust estimation

Reduced IKF

IMU data

GPS dataData reading

The equivalent covariance matrix

Time update

Mechanics arrangement

Measurement update

Interval of the a priori estimate

A priori covariance matrix

The equivalent gain matrix

Interval of the a posteriori estimate

A posterioricovariance matrix

The initial covariancematrix

Robust parameter estimation

Figure 1 A flow chart of the RIKF algorithm for a loosely coupledGPSINS integrated navigation

Considering the preceding information the IGGIII functionis defined by [25 31 32]

119875119894 =

119901119894 1003816100381610038161003816V1198941003816100381610038161003816 le 1198880119901119894 11988801003816100381610038161003816V1198941003816100381610038161003816 (

1198881 minus 1003816100381610038161003816V11989410038161003816100381610038161198881 minus 1198880 )2

1198880 lt 1003816100381610038161003816V1198941003816100381610038161003816 le 11988810 1003816100381610038161003816V1198941003816100381610038161003816 gt 1198881

(23)

where 119875119894 is the 119894th diagonal element of the equivalent weightmatrix |V119894| is the 119894th standardized residual and 1198880 and 1198881 arethe critical values determined by the significance level Ingeneral 1198880 is chosen between 15 and 20 and 1198881 is chosenbetween 30 and 85 [27]

The final filtering algorithm applied to loosely coupledGPSINS integrated navigation is shown in Figure 1

4 Practical Experiments of the GPSINSIntegrated Navigation

A vehicle mounted GPSINS integrated navigation systemwhich is composed of two GPS receivers and an inertialmeasurement unit (IMU) is implemented One of the GPSreceivers was used as a base station and another receiver aswell as the IMU is mounted on the vehicle

The loosely coupled mode was used in this experimentand the position and the speed were treated as externalobservations They were calculated by the double differencepseudorange of the GPS with variances of 025m2 and00025m2s2 respectively Errors of the initial position andthe initial speed were 10m and 01ms respectively Theprecise results given by the double difference carrier phasewere considered as references

Table 1 RMSEs of each algorithm (m)

Axis KF IKF RKF RIKF119883 0130 0092 0093 0090119884 0230 0180 0207 0171119885 0118 0102 0109 0099

minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

minus05

0

05Z

(m)

Figure 2 Position errors of the KF algorithm

Four schemes are implemented in the integrated navi-gation data processing unit These four schemes are listedbelow

Scheme 1 Kalman filter (KF)Scheme 2 interval Kalman filter (IKF)Scheme 3 robust Kalman filter (RKF)Scheme 4 robust reduced interval kalman filter(RIKF)

Case 1 (initial observations) Position errors of the fourschemes are shown in Figures 2 3 4 and 5

Thefiltering results weremainly affected by the deviationsof the system model because only few outlying observationsexist It can be seen from Figures 2 and 3 that the IKFalgorithm satisfactorily performed in resisting the deviationsof the system model and the RIKF algorithm manifested abetter performance than that of the IKF algorithm in resistingthe outliers shown in Figure 5 Comparing Figures 3 4 and5 the RKF algorithm is inferior to IKF and RIKF algorithmsin handling deviations of the system model

Statistical results of the Root Mean Squares Errors(RMSE) are listed in Table 1

Table 1 shows that there was only a small differencebetween the solutions of these algorithms in terms of RMSEHowever by integrating the advantages of the IKF and the


Table 2 RMSE of each algorithm (m)


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)

100 200 300 400 500 600 7000t (s)

Figure 3 Position errors of the IKF algorithm

RKF algorithms the RIKF algorithm achieved a slightlybetter performance than the other algorithms

Case 2 (observations with perturbation) Based on the initialobservations gross errors were added to the GPS observa-tions at the 60th 160th 260th 360th and 460th epochsrespectively Four filtering algorithms were implemented toexamine their performance Position errors of these fourschemes are shown in Figures 6 7 8 and 9

Figure 6 shows that the KF algorithm had a lessercapacity to resist outliers System stability and perturbationinfluences on themodel were improved by the IKF algorithmNevertheless the filtering results were still greatly affected bythe outliers as concluded from Figure 7 The RKF algorithmwas performed using the IGGIII scheme Figure 8 showedthat the perturbation of the model changed little with theRKF algorithm although the influences of the outliers weredecreased significantly Figure 9 indicated that the impactsof the system model and the perturbation of the noisemodel and the outliers were significantly weakened with theRIKF algorithmMoreover the filtering process becamemorestable

Statistical results of the RMSEs are listed in Table 2TheRIKF algorithmhad better fault tolerance and robust-

ness among all other algorithms The RMSEs of the RIKFalgorithm were the smallest over all the coordinate compo-nents which signified that the carrier positions calculated by

minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Z (m

)Figure 4 Position errors of the RKF algorithm

minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

Z (m

)

Figure 5 Position errors of the RIKF algorithm

the RIKF algorithmwere in better agreement with the realityIn addition filtering results of the IKF and RKF algorithmsare better than that of the KF algorithm Whereas the RKFalgorithm performed even better than the IKF algorithmwith respect to the outliers which was the main goal of theintegrated navigation filtering in this experiment

5 Conclusions

This paper presented an improved interval Kalman filteralgorithmThe recent IKF algorithms focused on controlling


minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)


the effects of the model errors However deviations of thefiltering results would appear with the inclusion of outlyingobservations This work proposed a robust interval Kalmanfilter (RIKF) algorithm based on the equivalent weight robustestimation and the interval Kalman filter to compensate forthe deviations of the models and the outlying observationssimultaneously The proposed algorithm was validated byimplementing data processing of a GPSINS integrated navi-gation A better performance was achieved with the proposedRIKF algorithm compared to previous methods Overall thefollowing can be concluded

minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)


(1) The existing IKF algorithms performed well withthe GPSINS integrated navigation system whichcontained uncertainties However the performancewould be greatly affected by the outlying observa-tions

(2) The RKF algorithm showed great robustness ifthe measurements contained outliers Neverthelessthe noise term of the model generally remainedunchanged with the RKF algorithm

(3) Comparison of the IKF the RKF and the RIKFalgorithms with and without outlying observations


showed that by integrating the advantages of theRKF and the IKF algorithms the proposed RIKFalgorithm could resist the deviations of the modeland the outlying observations simultaneously Thereis broad application for the proposed RIKF algorithmin the dynamic navigation and positioning

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful for the support of the FundamentalResearch Funds for the Central Universities (China Univer-sity of Mining and Technology no 2015QNB08) and thePriority Academic Program Development of Jiangsu HigherEducation Institutions (PAPD SA1102)

References

[1] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[2] C Shen Z Bai H Cao et al ldquoOptical flow sensorINSmagnetometer integrated navigation system for MAV in GPS-denied environmentrdquo Journal of Sensors vol 2016 Article ID6105803 10 pages 2016

[3] E A Wan and R van der Merwe ldquoThe unscented Kalmanfilter for nonlinear estimationrdquo in Proceedings of the IEEEAdaptive Systems for Signal Processing Communications andControl Symposium (AS-SPCC rsquo00) pp 153ndash158 2000

[4] R Kandepu B Foss and L Imsland ldquoApplying the unscentedKalman filter for nonlinear state estimationrdquo Journal of ProcessControl vol 18 no 7-8 pp 753ndash768 2008

[5] F Gustafsson ldquoParticle filter theory and practice with posi-tioning applicationsrdquo IEEE Aerospace and Electronic SystemsMagazine vol 25 no 7 pp 53ndash81 2010

[6] O Heirich ldquoBayesian train localization with particle filterloosely coupled GNSS IMU and a track maprdquo Journal ofSensors vol 2016 Article ID 2672640 15 pages 2016

[7] K M Nagpal and P P Khargonekar ldquoFiltering and smoothingin an119867infin settingrdquo IEEE Transactions on Automatic Control vol36 no 2 pp 152ndash166 1991

[8] C E de Souza R M Palhares and P L D Peres ldquoRobust119867infinfilter design for uncertain linear systems with multiple time-varying state delaysrdquo IEEE Transactions on Signal Processingvol 49 no 3 pp 569ndash576 2001

[9] A H Mohamed and K P Schwarz ldquoAdaptive Kalman filteringfor INSGPSrdquo Journal of Geodesy vol 73 no 4 pp 193ndash2031999

[10] W Ding J Wang C Rizos and D Kinlyside ldquoImprovingadaptive kalman estimation in GPSINS integrationrdquo Journal ofNavigation vol 60 no 3 pp 517ndash529 2007

[11] Y Yang and T Xu ldquoAn adaptive Kalman filter based onsage windowing weights and variance componentsrdquo Journal ofNavigation vol 56 no 2 pp 231ndash240 2003

[12] Y Yang andWGao ldquoAn optimal adaptiveKalmanfilterrdquo Journalof Geodesy vol 80 no 4 pp 177ndash183 2006

[13] K R Koch and Y Yang ldquoRobust Kalman filter for rank deficientobservation modelsrdquo Journal of Geodesy vol 72 no 7-8 pp436ndash441 1998

[14] M A Gandhi and L Mili ldquoRobust Kalman filter based on ageneralized maximum-likelihood-type estimatorrdquo IEEE Trans-actions on Signal Processing vol 58 no 5 pp 2509ndash2520 2010

[15] Y Yang H He and G Xu ldquoAdaptively robust filtering forkinematic geodetic positioningrdquo Journal of Geodesy vol 75 no2-3 pp 109ndash116 2001

[16] Y Yang and W Gao ldquoComparison of two fading filters andadaptively robust filterrdquoGeo-Spatial Information Science vol 10no 3 pp 200ndash203 2007

[17] Y Yang and X Cui ldquoAdaptively robust filter with multi adaptivefactorsrdquo Survey Review vol 40 no 309 pp 260ndash270 2008

[18] G Chen J Wang and L S Shieh ldquoInterval Kaiman filteringrdquoIEEE Transactions on Aerospace and Electronic Systems vol 33no 1 pp 250ndash259 1997

[19] G-L Du P Zhang and X Liu ldquoMarkerless humanmanipu-lator interface using leap motion with interval Kalman filterand improved particle filterrdquo IEEE Transactions on IndustrialInformatics vol 12 no 2 pp 694ndash704 2016

[20] Y Shen L Zhang Z-Q Fu and J-Y Wang ldquoInterval Kalmanfiltering algorithm for high dynamic navigation and position-ingrdquo Journal of Astronautics vol 34 no 3 pp 355ndash361 2013

[21] J Zhou ldquoClassical theory of errors and robust estimationrdquo ActaGeodaetica et Cartographica Sinica vol 18 no 2 pp 115ndash1201989

[22] Y Yang ldquoRobust estimation of geodetic datum transformationrdquoJournal of Geodesy vol 73 no 5 pp 268ndash274 1999

[23] P J Huber ldquoRobust estimation of a location parameterrdquo TheAnnals of Mathematical Statistics vol 35 no 2 pp 73ndash101 1964

[24] F R Hampel EM Ronchetti P J Rousseeuw andW A StahelRobust Statistics The Approach Based on Influence Functionsvol 114 John Wiley amp Sons New York NY USA 2011

[25] Y Yang ldquoRobust estimation for dependent observationsrdquoManuscripta Geodeatica vol 19 no 1 pp 10ndash17 1994

[26] Y-X Yang and F-M Wu ldquoModified equivalent weight func-tion with variable criterion for robust estimationrdquo Journal ofZhengzhou Institute of Surveying andMapping vol 23 no 5 pp317ndash320 2006

[27] Y-X Yang Adaptive Navigation and Dynamic PositioningSurveying and Mapping Press 2006

[28] X-F He and G Yang ldquoExtended interval Kalman filter andits application in nonlinear GPSINS integrated systemrdquo ActaGeodaetica et Cartographica Sinica vol 33 no 1 pp 47ndash522004

[29] W-G Gao Y-X Yang and S-C Zhang ldquoAdaptive robustKalman filtering based on the current statistical modelrdquo ActaGeodaetica et Cartographica Sinica vol 35 no 1 pp 15ndash18 2006

[30] G Chang ldquoRobust Kalman filtering based on Mahalanobisdistance as outlier judging criterionrdquo Journal of Geodesy vol 88no 4 pp 391ndash401 2014

[31] Y Yang L Song and T Xu ldquoRobust estimator for correlatedobservations based on bifactor equivalent weightsrdquo Journal ofGeodesy vol 76 no 6 pp 353ndash358 2002

[32] Y Yang and J Xu ldquoGNSS receiver autonomous integrity moni-toring (RAIM) algorithm based on robust estimationrdquo Geodesyand Geodynamics vol 7 no 2 pp 117ndash123 2016

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



[19] the interval intersection arithmetic was introduced inan improved IKF algorithm [20] To simplify the calculationprocessmodel errorswere assumed to be included only in thea priori and a posteriori estimates Nevertheless covariancematrices of the system noise and observation noise wereconsidered to be known and fixed The KF approach is notthe optimal one once the observations become contaminatedConsequently the influences of outliers must be consideredThe robust estimation method provides a way to control theinfluences of outliers [21 22] In the field of robust estimationa series of equivalent weight functions is constructed [23ndash26] to resist the impacts of outliers Among these equivalentweight functions the concept of the IGG procedure ismore suitable for the geodetic surveying errors Rather thandisproportionately pursuing validity and unbiasedness therobust estimation method focuses on the robustness and thereliability of the estimator [27] Accordingly it is desirableto develop a new filtering scheme by combining the robustestimation method with the IKF

This paper focuses on a comprehensive filtering algorithmusing the reduced IKF and the classic equivalent weightrobust estimation method for the GPSINS integrated navi-gation With the impacts of the abnormal observations andthe model errors weakened a novel filtering algorithm isdeveloped for the GPSINS integrated navigation Practicalexperiments of GPSINS integrated dynamic navigation areimplemented Finally the performance is evaluated and thesuitability of the proposed filtering algorithm is discussed

The rest of the paper is organized as follows In Section 2the theory of the IKF is introduced and the model deviationsof the conventionKF algorithm are analyzed In Section 3 therobust interval Kalman filter (RIKF) algorithm is proposedbased on the robust estimation and the IKF Section 4presentsthe advantages of the new algorithm which are verifiedby experiments and a comparison with other differentalgorithms is performed and discussed The conclusions arepresented in Section 5

2 The Interval Kalman Filter (IKF)

It is extremely difficult to establish a physical model and noisemodel of the irregularly motional carrier Therefore fixingdynamic system parameters and noise statistical parametersbecomes a crucial assumption when the standard KF algo-rithm is applied However in most cases the system andobservation noise terms are only approximately knownThusthe IKF algorithm was proposed to process the dynamicsystem with uncertainties in the model parameters

The system equation and observation equation areexpressed as

119909119896+1 = Φ119896119909119896 + Γ119896119908119896119911119896 = 119867119896119909119896 + V119896 (1)

where 119909119896 is the system state vector 119911 is the observationvector Φ119896 is the state-transition matrix and 119908119896 and V119896 arethe system and observation noises respectively Φ119896 Γ119896 and119867119896 are constant matrices and 119896 denotes the epoch Theoptimal estimate is obtained with the standard KF algorithm

when Φ119896 Γ119896 and119867119896 are fixed Influences of these uncertainmatrices can be controlled by interval matrices

Φ119868119896 = Φ119896 + ΔΦ119896 = [Φ119896 minus 1003816100381610038161003816ΔΦ1198961003816100381610038161003816 Φ119896 + 1003816100381610038161003816ΔΦ1198961003816100381610038161003816] Γ119868119896 = Γ119896 + ΔΓ119896 = [Γ119896 minus 1003816100381610038161003816ΔΓ1198961003816100381610038161003816 Γ119896 + 1003816100381610038161003816ΔΓ1198961003816100381610038161003816] 119867119868119896 = 119867119896 + Δ119867119896 = [119867119896 minus 1003816100381610038161003816Δ1198671198961003816100381610038161003816 119867119896 + 1003816100381610038161003816Δ1198671198961003816100381610038161003816]

(2)

whereΔΦ119896 ΔΓ119896 andΔ119867119896 are constant perturbationmatriceswith respective boundaries and the process of the IKFalgorithm is stated below [18 28]

Main Process

119868119896 = Φ119868119896minus1119868119896minus1 + 119870119868119896 [119911119868119896 minus 119867119868119896Φ119868119896minus1119868119896minus1] (3)

Coprocess

119875119868119896minus1 = Φ119868119896minus1119875119868119896minus1 (Φ119868119896minus1)119879 + Γ119868119896minus1119876119896minus1 (Γ119868119896minus1)119879 (4)

119870119868119896 = 119875119868119896minus1 (119867119868119896)119879 [119867119868119896119875119868119896minus1 (119867119868119896)119879 + 119877119896]minus1 (5)

119875119868119896 = [119868 minus 119870119868119896119867119868119896] 119875119868119896minus1 (6)

where 119875119868119896minus1 is the a priori covariance matrix 119870119868119896 is the gainmatrix 119868119896 is the estimated state vector of the filter 119876119896 and119877119896 are covariance matrices and 119875119868119896 is the updated covariancematrix

Three types of arithmetic for the perturbation matricesare adopted in the IKF algorithm addition multiplicationand inversion Assume that 119860 is an interval matrix 119872(119860) isthe matrix composed of the middle points of each elementin 119860 119882(119860) is the width of the interval matrix and theaddition arithmetic andmultiplication arithmetic are definedas follows

119872(1198601 + 1198602) = 119872(1198601) + 119872(1198602) 119882 (1198601 + 1198602) = 119882(1198601) + 119882(1198602) 119872 (11986011198602) = 119872(1198601)119872 (1198602) 119882 (11986011198602) = 1003816100381610038161003816119872 (1198601)1003816100381610038161003816119882 (1198602)

+ 119882(1198601) 1003816100381610038161003816119872 (1198602)1003816100381610038161003816

(7)

where 1198601 and 1198602 are interval matrices Meanwhile theinversion arithmetic should be implemented namely[119867119868119896119875119868119896minus1(119867119868119896)119879 + 119877119896]minus1 in (5)

[119867119868119896119875119868119896minus1 (119867119868119896)119879 + 119877119896]minus1 = (119867119896119875119896minus1119867119879119896 + Δ119877119896)minus1 Δ119877119896

= 119867119896119875119896minus1 (Δ119867119896)119879 + 119867119896Δ119875119896minus1119867119879119896+ 119867119896Δ119875119896minus1 (Δ119867119896)119879 + Δ119867119896119875119896minus1119867119879119896+ Δ119867119896119875119896minus1 (Δ119867119896)119879 + Δ119867119896Δ119875119896minus1119867119879119896+ Δ119867119896Δ119875119896minus1 (Δ119867119896)119879 + 119877119896

(8)





= [120575119909 120575119910 120575119911 120575V119909 120575V119910 120575V119911 120575120601119890 120575120601119899 120575120601119906 120575119892119909 120575119892119910 120575119892119911120575119886119909 120575119886119910 120575119886119911]

(9)


119883 (119905) = 119865 (119905)119883 (119905) + 119866 (119905)119882 (119905) (10)



119883119896 = Φ119896119896minus1119883119896minus1 +119882119896 (11)



119885120588 (119905) = 120588GPS minus 120588INS (12)



] (13)








119875119896 = (119868 minus 119870119896119867119896) 119875minus1119896 (18)



Time Update


(19)

Measurement Update






Robust estimation

Reduced IKF

IMU data



Time update


Measurement update










119875119894 =

119901119894 1003816100381610038161003816V1198941003816100381610038161003816 le 1198880119901119894 11988801003816100381610038161003816V1198941003816100381610038161003816 (

1198881 minus 1003816100381610038161003816V11989410038161003816100381610038161198881 minus 1198880 )2

1198880 lt 1003816100381610038161003816V1198941003816100381610038161003816 le 11988810 1003816100381610038161003816V1198941003816100381610038161003816 gt 1198881

(23)








minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

minus05

0

05Z

(m)











minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)

100 200 300 400 500 600 7000t (s)







minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Z (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

Z (m

)



5 Conclusions



minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)



minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)







Competing Interests


Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













= [120575119909 120575119910 120575119911 120575V119909 120575V119910 120575V119911 120575120601119890 120575120601119899 120575120601119906 120575119892119909 120575119892119910 120575119892119911120575119886119909 120575119886119910 120575119886119911]

(9)


119883 (119905) = 119865 (119905)119883 (119905) + 119866 (119905)119882 (119905) (10)



119883119896 = Φ119896119896minus1119883119896minus1 +119882119896 (11)



119885120588 (119905) = 120588GPS minus 120588INS (12)



] (13)








119875119896 = (119868 minus 119870119896119867119896) 119875minus1119896 (18)



Time Update


(19)

Measurement Update






Robust estimation

Reduced IKF

IMU data



Time update


Measurement update










119875119894 =

119901119894 1003816100381610038161003816V1198941003816100381610038161003816 le 1198880119901119894 11988801003816100381610038161003816V1198941003816100381610038161003816 (

1198881 minus 1003816100381610038161003816V11989410038161003816100381610038161198881 minus 1198880 )2

1198880 lt 1003816100381610038161003816V1198941003816100381610038161003816 le 11988810 1003816100381610038161003816V1198941003816100381610038161003816 gt 1198881

(23)








minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

minus05

0

05Z

(m)











minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)

100 200 300 400 500 600 7000t (s)







minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Z (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

Z (m

)



5 Conclusions



minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)



minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)







Competing Interests


Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Robust estimation

Reduced IKF

IMU data



Time update


Measurement update










119875119894 =

119901119894 1003816100381610038161003816V1198941003816100381610038161003816 le 1198880119901119894 11988801003816100381610038161003816V1198941003816100381610038161003816 (

1198881 minus 1003816100381610038161003816V11989410038161003816100381610038161198881 minus 1198880 )2

1198880 lt 1003816100381610038161003816V1198941003816100381610038161003816 le 11988810 1003816100381610038161003816V1198941003816100381610038161003816 gt 1198881

(23)








minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

minus05

0

05Z

(m)











minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)

100 200 300 400 500 600 7000t (s)







minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Z (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

Z (m

)



5 Conclusions



minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)



minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)







Competing Interests


Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)

100 200 300 400 500 600 7000t (s)







minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Z (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1

0

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

Z (m

)



5 Conclusions



minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)



minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)







Competing Interests


Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus10

minus5

0

5

X (m

)

600 700300 400 500100 2000t (s)

minus10

minus5

0

5

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus10

minus5

0

5

Z (m

)


minus4

minus2

0

2

Y (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

Z (m

)

100 200 300 400 500 600 7000t (s)

minus4

minus2

0

2

X (m

)

100 200 300 400 500 600 7000t (s)



minus1

0

1

2

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Z (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus2

0

2

4

Y (m


minus05

0

05

X (m

)

100 200 300 400 500 600 7000t (s)

minus1minus05

005

1

Y (m

)

100 200 300 400 500 600 7000t (s)

100 200 300 400 500 600 7000t (s)

minus05

0

05

1

Z (m

)







Competing Interests


Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











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 novel robust interval kalman filter...

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