Research ArticleAn Adaptive Unscented Kalman Filtering Algorithm forMEMSGPS Integrated Navigation Systems

Jianhua Cheng12 Daidai Chen1 Rene Jr Landry2 Lin Zhao1 and Dongxue Guan1

1 Marine Navigation Research Institute College of Automation Harbin Engineering University Harbin 150001 China2 LASSENA Laboratoire Ecole de Technologie Superieure Universite du Quebec Montreal Canada H3C 1K3

Correspondence should be addressed to Daidai Chen ins dai163com

Received 13 November 2013 Accepted 15 February 2014 Published 20 March 2014

Academic Editor Yuxin Zhao

Copyright copy 2014 Jianhua Cheng 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

MEMSGPS integrated navigation systemhas beenwidely used for land-vehicle navigationThis systemexhibits large errors becauseof its nonlinear model and uncertain noise statistic characteristics Based on the principles of the adaptive Kalman filtering (AKF)and unscented Kalman filtering (AUKF) algorithms an adaptive unscented Kalman filtering (AUKF) algorithm is proposed Byusing noise statistic estimator the uncertain noise characteristics could be online estimated to adaptively compensate the time-varying noise characteristics Employing the adaptive filtering principle into UKF the nonlinearity of system can be restrainedSimulations are conducted for MEMSGPS integrated navigation system The results show that the performance of estimation isimproved by the AUKF approach compared with both conventional AKF and UKF

1 Introduction

Microelectromechanical systems (MEMS) and Global Posi-tioning System (GPS) integrated navigation system have theadvantages of small size light weight and low cost butbecause of its low accuracy it can only be applied in lowaccuracy navigation fields such as unmanned aircrafts andland-vehicles [1 2]There are threemain factors impacting itsperformance (a) the inaccuracy of model parameters (b) theuncertainty of measurement and observation noise statisticproperties and (c) the nonlinearity of model [3 4]

The classical Kalman filter (KF) provides a recursive solu-tion for estimation of linear dynamic systemsThe optimalityof the KF algorithm is mainly dependent on a priori statisticof the process and measurement noise and the linear systemmodel However if the priori information is insufficient orbiased the precision of the estimated states will be degradedeven leading to divergences [5] In the case of MEMSGPSapplications the estimation system tends to be nonlinear aswell as variational noise properties [6] Meanwhile for thevehicle navigation there are many sudden motion changesTo deal with these problems the implement of AKF appearsto be one of suitable approaches [7]

By utilizing the innovation and residual information theAKF could adapt the filter stochastic properties online toaccommodate itself to changes in vehicle dynamics Thusthis technique could reduce the reliance of filter on theprior statistical information and obtain the noise statisticparameters of the dynamic system The essence of AKF is toadapt the filter weights so as to restrain model errors andimprove the accuracy of filters It is showed that applyingAKFto the INSGPS integrated navigation system could obtainbetter estimated performance than by using conventional KFespecially less than 20 root mean square error in attitudeestimation [8] However AKF is unreliable when it is appliedinto the nonlinear applications

UKF which is another extension of Kalman filter couldgive reliable estimates even if the nonlinearities of systemare quiet severe [9] The linearization procedure is avoidedby introducing the unscented transformation (UT) whichis a method to approximate the joint distribution of statesand measurements variables In UT a number of sigmapoints are chosen which could maintain the desired meanand covariance of states Theoretically the performance ofUKF could be close to that of the three-order Taylor seriesexpansion approximation or better than it [10] However

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 451939 8 pageshttpdxdoiorg1011552014451939

2 Journal of Applied Mathematics

many kinds of systems such as aircrafts ships and spacecraftswould be intensively disturbed by external environment Asa result the statistical properties for system process andmeasurement noises cannot be predicted and UKF cannotsolve these problems effectively

As a combination of AKF and UKF the adaptive UKF hasbeen developed and applied to nonlinear joint estimation ofboth time-varying states and parameters [11] The adaptiveprinciples have been employed to update the means andcovariances of the process and measurement noises onlineThe contributions of model predicted states and measure-ment information for filtering are balanced In this paper anew AUKF algorithm is proposed for MEMSGPS integratednavigation systems in vehicle applications A noise estimatorfor UKF is designed to estimate and update the means andcovariances of noises online Then the updated means andcovariances are propagating through the UKF The proposedAUKF has the adaptive ability to time-varying noises andthe noise estimates are unbiased The performance of AUKFapplied in land navigation is evaluated by simulations andthe results show that the integrated system exhibits excellentrobustness and navigation performance

2 Unscented Kalman Filtering Algorithm forNonzero Mean Noise

In the integrated navigation field almost all of the systemsare nonlinearThe general nonlinear discrete systemmodel isgiven as

119909119896 = 119891119896minus1 (119909119896minus1) + 119908119896minus1

119911119896 = ℎ119896 (119909119896) + V119896(1)

where 119909119896 isin 119877119899 is the state vector 119911119896 isin 119877

119898 is the measurementvector and 119891119896(sdot) isin 119877

119899times119899 and ℎ119896(sdot) isin 119877119898times119899 are the state and

measurement matrices of nonlinear system respectively 119908119896

and V119896 are the Gaussian white noise which are unrelated Themean and covariance of 119908119896 and V119896 are given as follows

119864 [119908119896] = 119902 cov [119908119896119908119879

119895] = 119876120575119896119895

119864 [V119896] = 119903 cov [V119896V119879

119895] = 119877120575119896119895

(2)

where 119902 and 119903 are nonzero constant variables and 120575119896119895 is aKronecker delta function

The initial state 1199090 is uncorrelated with both the processnoise and measurement noise These initial states exhibitGaussian normal distributions The prior mean and covari-ance matrices of 1199090 are defined by

1199090 = 119864 (1199090)

1198750 = cov (1199090) = 119864 (1199090 minus 1199090) (1199090 minus 1199090)119879

(3)

Assuming that120583119896 = 119908119896minus119902 and 120578119896 = V119896minus119903 and substitutingthem into (1) yield

119909119896 = 119891119896minus1 (119909119896minus1) + 119902 + 120583119896minus1

119911119896 = ℎ119896 (119909119896) + 119903 + 120578119896

(4)

where the mean and covariance of 120583119896 and 120578119896 are given asfollows

119864 [120583119896] = 0 cov [120583119896120583119879

119895] = 119876120575119896119895

119864 [120578119896] = 0 cov [120578119896120578119879

119895] = 119877120575119896119895

(5)

According to system model of (4) the recursive solutionof UKF algorithm is noted as follows

(1) Sigma Points Sampling In order to guarantee positivesemidefinite of state covariance the modified sigma pointssampling solution based on the scaling method is adopted[12]

Choose 2119899 + 1 sigma points 120585119894119896119896minus1 as follows

1205850119896119896minus1 = 119909

120585119894119896119896minus1 = 119909 + (120572radic(119899 + 120582) 119875)119894 119894 = 1 2 119899

120585119894119896119896minus1 = 119909 minus (120572radic(119899 + 120582) 119875)119894minus119871

119894 = 119899 + 1 119899 + 2 2119899

(6)

where 119875 is the covariance of the state vector 119909 120572radic(119899 + 120582)119875 isthe matrix square root of 119899119875 and (120572radic(119899 + 120582)119875)119894 denotes the119894th row items of 120572radic(119899 + 120582)119875

(2) PredictionPropagating these sigmapoints 120585119894119896119896minus1 throughnonlinear state function 119891119896(sdot) + 119902 we obtain [13]

120574119894119896119896minus1 = 119891119896minus1 (120585119894119896119896minus1) + 119902 119894 = 0 1 2119899 (7)

Then computing the predicted state 119909119896119896minus1 the predictedcovariance 119875119896119896minus1 is as follows

119909119896119896minus1 =

119871

sum

119894=0

119882119898

119894120574119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894119891119896minus1 (120585119894119896119896minus1) + 119902 (8)

119875119896119896minus1 =

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (120574119894119896119896minus1 minus 119909119896119896minus1)

119879+ 119876

(9)

where 119882119898

119894and 119882

119888

119894are associated weights

119882119898

0=

120582

(119899 + 120582)

119882119888

0=

120582

(119899 + 120582)

+ (1 minus 1205722

+ 120573)

119882119898

119894=

1

2 (119899 + 120582)

119894 = 1 2 2119899

119882119888

119894=

1

2 (119899 + 120582)

119894 = 1 2 2119899

(10)

Parameter 120582 is a scaling parameter which is defined by

120582 = 1205722

(119899 + 120581) minus 119899 (11)

Journal of Applied Mathematics 3

The parameters 120572 120573 and 120581 are the positive constants in thesampling method

Then propagating the sigma points 120585119894119896119896minus1 the measure-ment function ℎ119896(sdot) + 119903 yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) + 119903 119894 = 0 1 2119899 (12)

Computing the predicted measurement vector 119896119896minus1 thecovariance of the measurement 119875119911119896

and the cross-covarianceof the state and measurement 119875119909119896119896

are as follows

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120585119894119896119896minus1) + 119903 (13)

119875119911119896=

119871

sum

119894=0

119882119888

119894(119909119894119896119896minus1 minus 119896119896minus1) (119909119894119896119896minus1 minus 119896119896minus1)

119879+ 119877

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (119909119894119896119896minus1 minus 119896119896minus1)

119879

(14)

(3) UpdatingThen computing the filter gain 119870119896 the updatedstate 119909119896119896 and covariance 119875119896119896 are as follows

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(15)

3 Noise Statistic Estimator

Aiming at the uncertainty of process and measurement noisestatistic properties the measurement information are used toreal time estimate and update the means and covariances ofnoises Assume that 119908119896 and V119896 are uncorrelated and obey theGaussian distributions Based on the maximum a posterior(MAP) principle a noise statistic estimator is derived

The noise parameters 119902 119903 and the noise matrices 119876119877 are unknown and need to be estimated according tothe updated measurements The MAP estimates of 119902 119876119903 119877 and the state 119909119896 are denoted as 119902 119876 119903 and 119909119895119896respectivelyThe conditional distribution of interest based onthe measurements is expressed as

119869 = 119901 [119883119896 119902 119876 119903 119877 | 119885119896] (16)

Because 119901[119885119896] is disrelated to other parameters exceptthe estimate state the problem in calculating (16) changes tocalculate the joint conditional probability distribution

119869 = 119901 [119883119896 119902 119876 119903 119877 119885119896]

= 119901 [119883119896 | 119902 119876 119903 119877] 119901 [119885119896 | 119883119896 119902 119876 119903 119877] 119901 [119902 119876 119903 119877]

(17)

where 119901[119902 119876 119903 119877] can be treated as a constant which isprovided by the prior statistic information

The original problem has changed to calculate the condi-tional probability distributions 119901[119883119896 | 119902 119876 119903 119877] and 119901[119885119896 |

119883119896 119902 119876 119903 119877]According to the Gaussian distributions of 120583119896 the con-

ditional probability distribution 119901[119883119896 | 119902 119876 119903 119877] could beexpressed as

119901 [119883119896 | 119902 119876 119903 119877]

= 119901 [1199090]

119896

prod

119895=1

119901 [119909119895 | 119909119895minus1 119902 119876]

=

1

(2120587)119899210038161003816

100381610038161198750

1003816100381610038161003816

12exp minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

times

119896

prod

119895=1

1

(2120587)1198992

|119876|12

times exp minus

1

2

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

= 1198621

10038161003816100381610038161198750

1003816100381610038161003816

minus12|119876|minus1198962

times exp

minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

+

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

(18)

where 119899 is the dimension of system state 1198621 = 1(2120587)119899(119896+1)2

is a constant and |119860| is the determinant of 119860 and 1199062

119860=

119906119879119860119906Moreover assuming that the measurements 1199111 1199112 119911119896

are known and unrelated to each other the distribution of 120578119896

is Gaussian which can be expressed as

119901 [119885119896 | 119883119896 119902 119876 119903 119877]

=

119896

prod

119895=1

119901 [119911119895 | 119909119895 119903 119877]

=

119896

prod

119895=1

1

(2120587)1198982

|119877|12

exp minus

1

2

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1

= 1198622|119877|minus1198962 exp

minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1

(19)

where 119898 denotes the dimension of measurements and 1198622 =

1(2120587)1198981198962 is a constant

4 Journal of Applied Mathematics

Substituting (18) and (19) into (17) yields

119869 = 11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12|119876|minus1198962

|119877|minus1198962

119901 [119902 119876 119903 119877]

= 119862|119876|minus1198962

|119877|minus1198962 exp

minus

1

2

[

[

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

+

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1]

]

(20)

where

119862 = exp minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12119901 [119902 119876 119903 119877] (21)

Logarithm on both sides of (20) yields

ln 119869 = minus

119896

2

ln |119876| minus

119896

2

ln |119877| minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1+ ln119862

(22)

By the logarithmic nature 119869 and ln 119869 share the sameextreme points The partial derivative of 119869 can be calculatedby the following equations

120597 ln 119869

120597119902

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119902=119902119896

= 0

120597 ln 119869

120597119876

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119876=119896

= 0

120597 ln 119869

120597119903

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119903=119903119896

= 0

120597 ln 119869

120597119877

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119877=119896

= 0

(23)

Then the noise statistic estimator can be derived whichis defined by

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896)] (24)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

times [119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

119879

(25)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896)] (26)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896) minus 119903] [119911119895 minus ℎ119895 (119909119895119896) minus 119903]

119879

(27)

In (24) to (27) the smoothed estimates119909119895minus1119896 and119909119895119896 canbe replaced by the filtered estimate 119909119895119895 or the predicted state119909119895119895minus1 as approximating solutions

4 Noise Statistic Estimator for UKF

From the consideration for the nonlinear purposes the noisestatistic estimator derived above should be modified In thelinear applications the term of 119891119895minus1(119909119895minus1) can be obtained bypropagating each estimate 119909119895minus1 through system model andℎ119895(119909119895) by propagating 119909119895 through measurement equationHowever for the nonlinear field the scaled sigma points areinserted instead of the estimate The predicted term for UKFcould be expressed as a combination of all sigma points

119891119895minus1 (119909119895minus1)

10038161003816100381610038161003816119909119895minus1larr119909119895minus1119895minus1

=

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) (28)

where 119891119895minus1(119909119895minus1) is approximated by UT with a precisionclose to that using three-order Taylor series expansionmethod [14]

Similarly ℎ119895(119909119895) can be calculated by

ℎ119895 (119909119895)

10038161003816100381610038161003816119909119895larr119909119895119895minus1

=

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1) (29)

Submitting (28) into (24) to (27) yields the noise statisticestimator for UKF

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)] (30)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1] [119909119895119895 minus 119909119895119895minus1]

119879

(31)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1)] (32)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus 119895119895minus1] [119911119895 minus 119895119895minus1]

119879

(33)

The unbiased properties of the noise estimates for UKFare proved in the appendix

Journal of Applied Mathematics 5

118

605

118

61

118

615

118

62

118

625

118

63

118

635

118

64

40107401075

40108401085

40109401095

4011401105

40111401115

Latit

ude (

∘)

Longitude (∘)

Figure 1 The track of land vehicle

Gyroscopes

Accelerometers

GPS

MEMS

MEMSGPS

MEMSGPS

measurement

Sigma pointssampling

modelNoise

estimator

AUKF Solution

Figure 2 The architecture of MEMSGPS integrated system with AUKF

5 Recursive Equations of Adaptive UnscentedKalman Filtering Algorithm

Based on the UKF and its noise statistic estimator theprediction and update steps of AUKF algorithm are asfollows

(1) Prediction Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function 119891119896(sdot) yields

120574119894119896119896minus1 = 119891119896minus1 (120585119894119896119896minus1) 119894 = 0 1 2119899 (34)

Then according to (30) and (31) estimate the processnoise 119902119896 and covariance 119876119896 respectively

With updated process noise parameters compute thepredicted state 119909119896119896minus1 and the predicted covariance 119875119896119896minus1 as

119909119896119896minus1 =

119871

sum

119894=0

119882119898

119894120574119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894119891119896minus1 (120576119894119896minus1119896minus1) + 119902119896minus1

119875119896119896minus1 =

119871

sum

119894=0

119882119898

119894(120574119894119896minus1119896minus1 minus 119909119896119896minus1)

times (120574119894119896minus1119896minus1 minus 119909119896119896minus1)119879

+ 119876119896minus1

(35)

(2) Updating Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function ℎ119896(sdot) yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) 119894 = 0 1 2119899 (36)

According to (32) and (33) estimate the measurementnoise 119903119896 and its covariance 119896 respectively

With real-time measurement noise parameters computethe predicted measurement 119896119896minus1 and the covariance 119875119911119896

as

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120576119894119896119896minus1) + 119903119896

(37)

119875119911119896=

119871

sum

119894=0

119882119898

119894(120594119894119896119896minus1 minus 119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879+ 119896

(38)

Estimate the cross-covariance 119875119909119896119896as

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879

(39)

6 Journal of Applied Mathematics

0 005 01 015 02 025 03 035 04 045 05

0002004006008 Latitude

Time (h)

AKF

UKFAUKF

minus004minus002Er

ror o

fpo

sitio

ns (∘

)

(a)

0002004006

0 005 01 015 02 025 03 035 04 045 05Time (h)

Longitude

AKF

UKFAUKF

minus004

minus002Erro

r of

posit

ions

(∘)

(b)

Figure 3 Comparison of position errors

Erro

rs o

f

010305

minus03minus05

minus010

velo

citie

s (m

s)

0 005 01 015 02 025 03 035 04 045 05Time (h)

East

AKF

UKFAUKF

(a)

Erro

rs o

fve

loci

ties (

ms

)

0020406

minus020 005 01 015 02 025 03 035 04 045 05

Time (h)

North

AKF

UKFAUKF

(b)

Figure 4 Comparison of velocity errors

Then compute the filter gain119870119896 the estimated state vector119909119896119896 and its covariance 119875119896119896

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(40)

6 MEMSGPS Integrated Navigation forLand-Vehicle Using AUKF

Because of the highly nonlinear characteristic ofMEMSGPSthe conventional AKF based on small angle approximations islimited Meanwhile due to the time-varying noise stochasticproperties for land-vehicle the standard UKF in Section 1cannot be directly applied to integrated navigation On theother side the modified AUKF based on a statistic estimator

in Section 4 appears appropriate for the MEMSGPS inte-grated navigation Simulations are conducted to compare theperformances of AKF UKF and AUKF

In the simulation the parameters of sensor errors areshown in Table 1 The initial position of vehicle is eastlongitude 126∘ and north latitude 45∘

Figure 1 shows the trajectory of land-vehicle motion Thesolid line in this figure illustrates the real simulated trajectory

The architecture of MEMSGPS integrated navigationwith AUKF is shown in Figure 2

As shown in Figures 3 4 and 5 with the comparisonswith AKF and UKF the AUKF with noise estimator couldobviously improve the accuracy of the velocity solutions Inaddition as shown in Figure 3 because the noise statisticestimators are designed in AKF and AUKF the positionerrors of the two filters exhibit similar characteristics at mostof the time in this simulation and the performance of AUKFis slightly better However which is also seen in Figure 3 there

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

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Discrete MathematicsJournal of

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Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

The parameters 120572 120573 and 120581 are the positive constants in thesampling method

Then propagating the sigma points 120585119894119896119896minus1 the measure-ment function ℎ119896(sdot) + 119903 yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) + 119903 119894 = 0 1 2119899 (12)

Computing the predicted measurement vector 119896119896minus1 thecovariance of the measurement 119875119911119896

and the cross-covarianceof the state and measurement 119875119909119896119896

are as follows

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120585119894119896119896minus1) + 119903 (13)

119875119911119896=

119871

sum

119894=0

119882119888

119894(119909119894119896119896minus1 minus 119896119896minus1) (119909119894119896119896minus1 minus 119896119896minus1)

119879+ 119877

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (119909119894119896119896minus1 minus 119896119896minus1)

119879

(14)

(3) UpdatingThen computing the filter gain 119870119896 the updatedstate 119909119896119896 and covariance 119875119896119896 are as follows

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(15)

3 Noise Statistic Estimator

Aiming at the uncertainty of process and measurement noisestatistic properties the measurement information are used toreal time estimate and update the means and covariances ofnoises Assume that 119908119896 and V119896 are uncorrelated and obey theGaussian distributions Based on the maximum a posterior(MAP) principle a noise statistic estimator is derived

The noise parameters 119902 119903 and the noise matrices 119876119877 are unknown and need to be estimated according tothe updated measurements The MAP estimates of 119902 119876119903 119877 and the state 119909119896 are denoted as 119902 119876 119903 and 119909119895119896respectivelyThe conditional distribution of interest based onthe measurements is expressed as

119869 = 119901 [119883119896 119902 119876 119903 119877 | 119885119896] (16)

Because 119901[119885119896] is disrelated to other parameters exceptthe estimate state the problem in calculating (16) changes tocalculate the joint conditional probability distribution

119869 = 119901 [119883119896 119902 119876 119903 119877 119885119896]

= 119901 [119883119896 | 119902 119876 119903 119877] 119901 [119885119896 | 119883119896 119902 119876 119903 119877] 119901 [119902 119876 119903 119877]

(17)

where 119901[119902 119876 119903 119877] can be treated as a constant which isprovided by the prior statistic information

The original problem has changed to calculate the condi-tional probability distributions 119901[119883119896 | 119902 119876 119903 119877] and 119901[119885119896 |

119883119896 119902 119876 119903 119877]According to the Gaussian distributions of 120583119896 the con-

ditional probability distribution 119901[119883119896 | 119902 119876 119903 119877] could beexpressed as

119901 [119883119896 | 119902 119876 119903 119877]

= 119901 [1199090]

119896

prod

119895=1

119901 [119909119895 | 119909119895minus1 119902 119876]

=

1

(2120587)119899210038161003816

100381610038161198750

1003816100381610038161003816

12exp minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

times

119896

prod

119895=1

1

(2120587)1198992

|119876|12

times exp minus

1

2

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

= 1198621

10038161003816100381610038161198750

1003816100381610038161003816

minus12|119876|minus1198962

times exp

minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

+

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

(18)

where 119899 is the dimension of system state 1198621 = 1(2120587)119899(119896+1)2

is a constant and |119860| is the determinant of 119860 and 1199062

119860=

119906119879119860119906Moreover assuming that the measurements 1199111 1199112 119911119896

are known and unrelated to each other the distribution of 120578119896

is Gaussian which can be expressed as

119901 [119885119896 | 119883119896 119902 119876 119903 119877]

=

119896

prod

119895=1

119901 [119911119895 | 119909119895 119903 119877]

=

119896

prod

119895=1

1

(2120587)1198982

|119877|12

exp minus

1

2

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1

= 1198622|119877|minus1198962 exp

minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1

(19)

where 119898 denotes the dimension of measurements and 1198622 =

1(2120587)1198981198962 is a constant

4 Journal of Applied Mathematics

Substituting (18) and (19) into (17) yields

119869 = 11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12|119876|minus1198962

|119877|minus1198962

119901 [119902 119876 119903 119877]

= 119862|119876|minus1198962

|119877|minus1198962 exp

minus

1

2

[

[

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

+

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1]

]

(20)

where

119862 = exp minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12119901 [119902 119876 119903 119877] (21)

Logarithm on both sides of (20) yields

ln 119869 = minus

119896

2

ln |119876| minus

119896

2

ln |119877| minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1+ ln119862

(22)

By the logarithmic nature 119869 and ln 119869 share the sameextreme points The partial derivative of 119869 can be calculatedby the following equations

120597 ln 119869

120597119902

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119902=119902119896

= 0

120597 ln 119869

120597119876

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119876=119896

= 0

120597 ln 119869

120597119903

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119903=119903119896

= 0

120597 ln 119869

120597119877

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119877=119896

= 0

(23)

Then the noise statistic estimator can be derived whichis defined by

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896)] (24)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

times [119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

119879

(25)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896)] (26)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896) minus 119903] [119911119895 minus ℎ119895 (119909119895119896) minus 119903]

119879

(27)

In (24) to (27) the smoothed estimates119909119895minus1119896 and119909119895119896 canbe replaced by the filtered estimate 119909119895119895 or the predicted state119909119895119895minus1 as approximating solutions

4 Noise Statistic Estimator for UKF

From the consideration for the nonlinear purposes the noisestatistic estimator derived above should be modified In thelinear applications the term of 119891119895minus1(119909119895minus1) can be obtained bypropagating each estimate 119909119895minus1 through system model andℎ119895(119909119895) by propagating 119909119895 through measurement equationHowever for the nonlinear field the scaled sigma points areinserted instead of the estimate The predicted term for UKFcould be expressed as a combination of all sigma points

119891119895minus1 (119909119895minus1)

10038161003816100381610038161003816119909119895minus1larr119909119895minus1119895minus1

=

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) (28)

where 119891119895minus1(119909119895minus1) is approximated by UT with a precisionclose to that using three-order Taylor series expansionmethod [14]

Similarly ℎ119895(119909119895) can be calculated by

ℎ119895 (119909119895)

10038161003816100381610038161003816119909119895larr119909119895119895minus1

=

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1) (29)

Submitting (28) into (24) to (27) yields the noise statisticestimator for UKF

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)] (30)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1] [119909119895119895 minus 119909119895119895minus1]

119879

(31)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1)] (32)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus 119895119895minus1] [119911119895 minus 119895119895minus1]

119879

(33)

The unbiased properties of the noise estimates for UKFare proved in the appendix

Journal of Applied Mathematics 5

118

605

118

61

118

615

118

62

118

625

118

63

118

635

118

64

40107401075

40108401085

40109401095

4011401105

40111401115

Latit

ude (

∘)

Longitude (∘)

Figure 1 The track of land vehicle

Gyroscopes

Accelerometers

GPS

MEMS

MEMSGPS

MEMSGPS

measurement

Sigma pointssampling

modelNoise

estimator

AUKF Solution

Figure 2 The architecture of MEMSGPS integrated system with AUKF

5 Recursive Equations of Adaptive UnscentedKalman Filtering Algorithm

Based on the UKF and its noise statistic estimator theprediction and update steps of AUKF algorithm are asfollows

(1) Prediction Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function 119891119896(sdot) yields

120574119894119896119896minus1 = 119891119896minus1 (120585119894119896119896minus1) 119894 = 0 1 2119899 (34)

Then according to (30) and (31) estimate the processnoise 119902119896 and covariance 119876119896 respectively

With updated process noise parameters compute thepredicted state 119909119896119896minus1 and the predicted covariance 119875119896119896minus1 as

119909119896119896minus1 =

119871

sum

119894=0

119882119898

119894120574119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894119891119896minus1 (120576119894119896minus1119896minus1) + 119902119896minus1

119875119896119896minus1 =

119871

sum

119894=0

119882119898

119894(120574119894119896minus1119896minus1 minus 119909119896119896minus1)

times (120574119894119896minus1119896minus1 minus 119909119896119896minus1)119879

+ 119876119896minus1

(35)

(2) Updating Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function ℎ119896(sdot) yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) 119894 = 0 1 2119899 (36)

According to (32) and (33) estimate the measurementnoise 119903119896 and its covariance 119896 respectively

With real-time measurement noise parameters computethe predicted measurement 119896119896minus1 and the covariance 119875119911119896

as

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120576119894119896119896minus1) + 119903119896

(37)

119875119911119896=

119871

sum

119894=0

119882119898

119894(120594119894119896119896minus1 minus 119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879+ 119896

(38)

Estimate the cross-covariance 119875119909119896119896as

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879

(39)

6 Journal of Applied Mathematics

0 005 01 015 02 025 03 035 04 045 05

0002004006008 Latitude

Time (h)

AKF

UKFAUKF

minus004minus002Er

ror o

fpo

sitio

ns (∘

)

(a)

0002004006

0 005 01 015 02 025 03 035 04 045 05Time (h)

Longitude

AKF

UKFAUKF

minus004

minus002Erro

r of

posit

ions

(∘)

(b)

Figure 3 Comparison of position errors

Erro

rs o

f

010305

minus03minus05

minus010

velo

citie

s (m

s)

0 005 01 015 02 025 03 035 04 045 05Time (h)

East

AKF

UKFAUKF

(a)

Erro

rs o

fve

loci

ties (

ms

)

0020406

minus020 005 01 015 02 025 03 035 04 045 05

Time (h)

North

AKF

UKFAUKF

(b)

Figure 4 Comparison of velocity errors

Then compute the filter gain119870119896 the estimated state vector119909119896119896 and its covariance 119875119896119896

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(40)

6 MEMSGPS Integrated Navigation forLand-Vehicle Using AUKF

Because of the highly nonlinear characteristic ofMEMSGPSthe conventional AKF based on small angle approximations islimited Meanwhile due to the time-varying noise stochasticproperties for land-vehicle the standard UKF in Section 1cannot be directly applied to integrated navigation On theother side the modified AUKF based on a statistic estimator

in Section 4 appears appropriate for the MEMSGPS inte-grated navigation Simulations are conducted to compare theperformances of AKF UKF and AUKF

In the simulation the parameters of sensor errors areshown in Table 1 The initial position of vehicle is eastlongitude 126∘ and north latitude 45∘

Figure 1 shows the trajectory of land-vehicle motion Thesolid line in this figure illustrates the real simulated trajectory

The architecture of MEMSGPS integrated navigationwith AUKF is shown in Figure 2

As shown in Figures 3 4 and 5 with the comparisonswith AKF and UKF the AUKF with noise estimator couldobviously improve the accuracy of the velocity solutions Inaddition as shown in Figure 3 because the noise statisticestimators are designed in AKF and AUKF the positionerrors of the two filters exhibit similar characteristics at mostof the time in this simulation and the performance of AUKFis slightly better However which is also seen in Figure 3 there

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

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Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Substituting (18) and (19) into (17) yields

119869 = 11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12|119876|minus1198962

|119877|minus1198962

119901 [119902 119876 119903 119877]

= 119862|119876|minus1198962

|119877|minus1198962 exp

minus

1

2

[

[

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

+

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1]

]

(20)

where

119862 = exp minus

1

2

10038171003817100381710038171199090 minus 1199090

1003817100381710038171003817

2

119875minus1

0

11986211198622

10038161003816100381610038161198750

1003816100381610038161003816

minus12119901 [119902 119876 119903 119877] (21)

Logarithm on both sides of (20) yields

ln 119869 = minus

119896

2

ln |119876| minus

119896

2

ln |119877| minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119909119895 minus 119891119895minus1 (119909119895minus1) minus 119902

10038171003817100381710038171003817

2

119876minus1

minus

1

2

119896

sum

119895=1

10038171003817100381710038171003817119911119895 minus ℎ119895 (119909119895) minus 119903

10038171003817100381710038171003817

2

119877minus1+ ln119862

(22)

By the logarithmic nature 119869 and ln 119869 share the sameextreme points The partial derivative of 119869 can be calculatedby the following equations

120597 ln 119869

120597119902

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119902=119902119896

= 0

120597 ln 119869

120597119876

10038161003816100381610038161003816100381610038161003816

119909119895minus1=119909119895minus1119896 119909119895=119909119895119896

119876=119896

= 0

120597 ln 119869

120597119903

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119903=119903119896

= 0

120597 ln 119869

120597119877

10038161003816100381610038161003816100381610038161003816

119909119895=119909119895119896

119877=119896

= 0

(23)

Then the noise statistic estimator can be derived whichis defined by

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896)] (24)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

times [119909119895119896 minus 119891119895minus1 (119909119895minus1119896) minus 119902]

119879

(25)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896)] (26)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus ℎ119895 (119909119895119896) minus 119903] [119911119895 minus ℎ119895 (119909119895119896) minus 119903]

119879

(27)

In (24) to (27) the smoothed estimates119909119895minus1119896 and119909119895119896 canbe replaced by the filtered estimate 119909119895119895 or the predicted state119909119895119895minus1 as approximating solutions

4 Noise Statistic Estimator for UKF

From the consideration for the nonlinear purposes the noisestatistic estimator derived above should be modified In thelinear applications the term of 119891119895minus1(119909119895minus1) can be obtained bypropagating each estimate 119909119895minus1 through system model andℎ119895(119909119895) by propagating 119909119895 through measurement equationHowever for the nonlinear field the scaled sigma points areinserted instead of the estimate The predicted term for UKFcould be expressed as a combination of all sigma points

119891119895minus1 (119909119895minus1)

10038161003816100381610038161003816119909119895minus1larr119909119895minus1119895minus1

=

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) (28)

where 119891119895minus1(119909119895minus1) is approximated by UT with a precisionclose to that using three-order Taylor series expansionmethod [14]

Similarly ℎ119895(119909119895) can be calculated by

ℎ119895 (119909119895)

10038161003816100381610038161003816119909119895larr119909119895119895minus1

=

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1) (29)

Submitting (28) into (24) to (27) yields the noise statisticestimator for UKF

119902119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)] (30)

119876119896 =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1] [119909119895119895 minus 119909119895119895minus1]

119879

(31)

119903119896 =

1

119896

119896

sum

119895=1

[119911119895 minus

119871

sum

119894=0

119882119898

119894ℎ119895 (120585119894119895119895minus1)] (32)

119896 =

1

119896

119896

sum

119895=1

[119911119895 minus 119895119895minus1] [119911119895 minus 119895119895minus1]

119879

(33)

The unbiased properties of the noise estimates for UKFare proved in the appendix

Journal of Applied Mathematics 5

118

605

118

61

118

615

118

62

118

625

118

63

118

635

118

64

40107401075

40108401085

40109401095

4011401105

40111401115

Latit

ude (

∘)

Longitude (∘)

Figure 1 The track of land vehicle

Gyroscopes

Accelerometers

GPS

MEMS

MEMSGPS

MEMSGPS

measurement

Sigma pointssampling

modelNoise

estimator

AUKF Solution

Figure 2 The architecture of MEMSGPS integrated system with AUKF

5 Recursive Equations of Adaptive UnscentedKalman Filtering Algorithm

Based on the UKF and its noise statistic estimator theprediction and update steps of AUKF algorithm are asfollows

(1) Prediction Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function 119891119896(sdot) yields

120574119894119896119896minus1 = 119891119896minus1 (120585119894119896119896minus1) 119894 = 0 1 2119899 (34)

Then according to (30) and (31) estimate the processnoise 119902119896 and covariance 119876119896 respectively

With updated process noise parameters compute thepredicted state 119909119896119896minus1 and the predicted covariance 119875119896119896minus1 as

119909119896119896minus1 =

119871

sum

119894=0

119882119898

119894120574119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894119891119896minus1 (120576119894119896minus1119896minus1) + 119902119896minus1

119875119896119896minus1 =

119871

sum

119894=0

119882119898

119894(120574119894119896minus1119896minus1 minus 119909119896119896minus1)

times (120574119894119896minus1119896minus1 minus 119909119896119896minus1)119879

+ 119876119896minus1

(35)

(2) Updating Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function ℎ119896(sdot) yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) 119894 = 0 1 2119899 (36)

According to (32) and (33) estimate the measurementnoise 119903119896 and its covariance 119896 respectively

With real-time measurement noise parameters computethe predicted measurement 119896119896minus1 and the covariance 119875119911119896

as

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120576119894119896119896minus1) + 119903119896

(37)

119875119911119896=

119871

sum

119894=0

119882119898

119894(120594119894119896119896minus1 minus 119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879+ 119896

(38)

Estimate the cross-covariance 119875119909119896119896as

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879

(39)

6 Journal of Applied Mathematics

0 005 01 015 02 025 03 035 04 045 05

0002004006008 Latitude

Time (h)

AKF

UKFAUKF

minus004minus002Er

ror o

fpo

sitio

ns (∘

)

(a)

0002004006

0 005 01 015 02 025 03 035 04 045 05Time (h)

Longitude

AKF

UKFAUKF

minus004

minus002Erro

r of

posit

ions

(∘)

(b)

Figure 3 Comparison of position errors

Erro

rs o

f

010305

minus03minus05

minus010

velo

citie

s (m

s)

0 005 01 015 02 025 03 035 04 045 05Time (h)

East

AKF

UKFAUKF

(a)

Erro

rs o

fve

loci

ties (

ms

)

0020406

minus020 005 01 015 02 025 03 035 04 045 05

Time (h)

North

AKF

UKFAUKF

(b)

Figure 4 Comparison of velocity errors

Then compute the filter gain119870119896 the estimated state vector119909119896119896 and its covariance 119875119896119896

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(40)

6 MEMSGPS Integrated Navigation forLand-Vehicle Using AUKF

Because of the highly nonlinear characteristic ofMEMSGPSthe conventional AKF based on small angle approximations islimited Meanwhile due to the time-varying noise stochasticproperties for land-vehicle the standard UKF in Section 1cannot be directly applied to integrated navigation On theother side the modified AUKF based on a statistic estimator

in Section 4 appears appropriate for the MEMSGPS inte-grated navigation Simulations are conducted to compare theperformances of AKF UKF and AUKF

In the simulation the parameters of sensor errors areshown in Table 1 The initial position of vehicle is eastlongitude 126∘ and north latitude 45∘

Figure 1 shows the trajectory of land-vehicle motion Thesolid line in this figure illustrates the real simulated trajectory

The architecture of MEMSGPS integrated navigationwith AUKF is shown in Figure 2

As shown in Figures 3 4 and 5 with the comparisonswith AKF and UKF the AUKF with noise estimator couldobviously improve the accuracy of the velocity solutions Inaddition as shown in Figure 3 because the noise statisticestimators are designed in AKF and AUKF the positionerrors of the two filters exhibit similar characteristics at mostof the time in this simulation and the performance of AUKFis slightly better However which is also seen in Figure 3 there

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

118

605

118

61

118

615

118

62

118

625

118

63

118

635

118

64

40107401075

40108401085

40109401095

4011401105

40111401115

Latit

ude (

∘)

Longitude (∘)

Figure 1 The track of land vehicle

Gyroscopes

Accelerometers

GPS

MEMS

MEMSGPS

MEMSGPS

measurement

Sigma pointssampling

modelNoise

estimator

AUKF Solution

Figure 2 The architecture of MEMSGPS integrated system with AUKF

5 Recursive Equations of Adaptive UnscentedKalman Filtering Algorithm

Based on the UKF and its noise statistic estimator theprediction and update steps of AUKF algorithm are asfollows

(1) Prediction Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function 119891119896(sdot) yields

120574119894119896119896minus1 = 119891119896minus1 (120585119894119896119896minus1) 119894 = 0 1 2119899 (34)

Then according to (30) and (31) estimate the processnoise 119902119896 and covariance 119876119896 respectively

With updated process noise parameters compute thepredicted state 119909119896119896minus1 and the predicted covariance 119875119896119896minus1 as

119909119896119896minus1 =

119871

sum

119894=0

119882119898

119894120574119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894119891119896minus1 (120576119894119896minus1119896minus1) + 119902119896minus1

119875119896119896minus1 =

119871

sum

119894=0

119882119898

119894(120574119894119896minus1119896minus1 minus 119909119896119896minus1)

times (120574119894119896minus1119896minus1 minus 119909119896119896minus1)119879

+ 119876119896minus1

(35)

(2) Updating Propagating the sigma points 120585119894119896119896minus1 throughnonlinear state function ℎ119896(sdot) yields

119909119894119896119896minus1 = ℎ119896 (120585119894119896119896minus1) 119894 = 0 1 2119899 (36)

According to (32) and (33) estimate the measurementnoise 119903119896 and its covariance 119896 respectively

With real-time measurement noise parameters computethe predicted measurement 119896119896minus1 and the covariance 119875119911119896

as

119896119896minus1 =

119871

sum

119894=0

119882119898

119894119909119894119896119896minus1 =

119871

sum

119894=0

119882119898

119894ℎ119896 (120576119894119896119896minus1) + 119903119896

(37)

119875119911119896=

119871

sum

119894=0

119882119898

119894(120594119894119896119896minus1 minus 119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879+ 119896

(38)

Estimate the cross-covariance 119875119909119896119896as

119875119909119896119896=

119871

sum

119894=0

119882119888

119894(120574119894119896119896minus1 minus 119909119896119896minus1) (120594119894119896119896minus1 minus 119896119896minus1)

119879

(39)

6 Journal of Applied Mathematics

0 005 01 015 02 025 03 035 04 045 05

0002004006008 Latitude

Time (h)

AKF

UKFAUKF

minus004minus002Er

ror o

fpo

sitio

ns (∘

)

(a)

0002004006

0 005 01 015 02 025 03 035 04 045 05Time (h)

Longitude

AKF

UKFAUKF

minus004

minus002Erro

r of

posit

ions

(∘)

(b)

Figure 3 Comparison of position errors

Erro

rs o

f

010305

minus03minus05

minus010

velo

citie

s (m

s)

0 005 01 015 02 025 03 035 04 045 05Time (h)

East

AKF

UKFAUKF

(a)

Erro

rs o

fve

loci

ties (

ms

)

0020406

minus020 005 01 015 02 025 03 035 04 045 05

Time (h)

North

AKF

UKFAUKF

(b)

Figure 4 Comparison of velocity errors

Then compute the filter gain119870119896 the estimated state vector119909119896119896 and its covariance 119875119896119896

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(40)

6 MEMSGPS Integrated Navigation forLand-Vehicle Using AUKF

Because of the highly nonlinear characteristic ofMEMSGPSthe conventional AKF based on small angle approximations islimited Meanwhile due to the time-varying noise stochasticproperties for land-vehicle the standard UKF in Section 1cannot be directly applied to integrated navigation On theother side the modified AUKF based on a statistic estimator

in Section 4 appears appropriate for the MEMSGPS inte-grated navigation Simulations are conducted to compare theperformances of AKF UKF and AUKF

In the simulation the parameters of sensor errors areshown in Table 1 The initial position of vehicle is eastlongitude 126∘ and north latitude 45∘

Figure 1 shows the trajectory of land-vehicle motion Thesolid line in this figure illustrates the real simulated trajectory

The architecture of MEMSGPS integrated navigationwith AUKF is shown in Figure 2

As shown in Figures 3 4 and 5 with the comparisonswith AKF and UKF the AUKF with noise estimator couldobviously improve the accuracy of the velocity solutions Inaddition as shown in Figure 3 because the noise statisticestimators are designed in AKF and AUKF the positionerrors of the two filters exhibit similar characteristics at mostof the time in this simulation and the performance of AUKFis slightly better However which is also seen in Figure 3 there

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

0 005 01 015 02 025 03 035 04 045 05

0002004006008 Latitude

Time (h)

AKF

UKFAUKF

minus004minus002Er

ror o

fpo

sitio

ns (∘

)

(a)

0002004006

0 005 01 015 02 025 03 035 04 045 05Time (h)

Longitude

AKF

UKFAUKF

minus004

minus002Erro

r of

posit

ions

(∘)

(b)

Figure 3 Comparison of position errors

Erro

rs o

f

010305

minus03minus05

minus010

velo

citie

s (m

s)

0 005 01 015 02 025 03 035 04 045 05Time (h)

East

AKF

UKFAUKF

(a)

Erro

rs o

fve

loci

ties (

ms

)

0020406

minus020 005 01 015 02 025 03 035 04 045 05

Time (h)

North

AKF

UKFAUKF

(b)

Figure 4 Comparison of velocity errors

Then compute the filter gain119870119896 the estimated state vector119909119896119896 and its covariance 119875119896119896

119870119896 = 119875119909119896119896(119875119911119896

)

minus1

119909119896119896 = 119909119896119896minus1 + 119870119896 (119911119896 minus 119896119896minus1)

119875119896119896 = 119875119896119896minus1 minus 119870119896119875119911119896(119870119896)119879

(40)

6 MEMSGPS Integrated Navigation forLand-Vehicle Using AUKF

Because of the highly nonlinear characteristic ofMEMSGPSthe conventional AKF based on small angle approximations islimited Meanwhile due to the time-varying noise stochasticproperties for land-vehicle the standard UKF in Section 1cannot be directly applied to integrated navigation On theother side the modified AUKF based on a statistic estimator

in Section 4 appears appropriate for the MEMSGPS inte-grated navigation Simulations are conducted to compare theperformances of AKF UKF and AUKF

In the simulation the parameters of sensor errors areshown in Table 1 The initial position of vehicle is eastlongitude 126∘ and north latitude 45∘

Figure 1 shows the trajectory of land-vehicle motion Thesolid line in this figure illustrates the real simulated trajectory

The architecture of MEMSGPS integrated navigationwith AUKF is shown in Figure 2

As shown in Figures 3 4 and 5 with the comparisonswith AKF and UKF the AUKF with noise estimator couldobviously improve the accuracy of the velocity solutions Inaddition as shown in Figure 3 because the noise statisticestimators are designed in AKF and AUKF the positionerrors of the two filters exhibit similar characteristics at mostof the time in this simulation and the performance of AUKFis slightly better However which is also seen in Figure 3 there

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

02

minus2

0 005 01 015 02 025 03 035 04 045 05Time (h)

Pitch

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(a)

minus15

0

15

005 01 015 02 025 03 035 04 045 05Time (h)

Roll

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

minus05

05

(b)

Heading

01

minus1

0 005 01 015 02 025 03 035 04 045 05Time (h)

AKF

UKFAUKF

Erro

rs o

fat

titud

es (∘

)

(c)

Figure 5 Comparison of attitude errors

Table 1 Parameters of sensor errors

Sensor Characteristic ValueMEMS Gyro Drift 5∘hMEMS Gyro Measuring white noise 05∘hMEMS accelerometer Bias 2mgMEMS accelerometer Measuring white noise 50120583gGPS (position) Measuring white noise 6mGPS (velocity) Measuring white noise 001ms

are so notable vibrations for UKF in which stable estimateerrors of positions could not be provided which are mainlycaused by the quickly changed system noises

Figure 4 shows that the AUKF always has smaller velocityerrors than AKF and UKF And Figure 5 shows that at mostof the time the AUKF scheme has a better performanceon attitudes errors The AKF and UKF solutions have largevibration errors in both Figures 4 and 5 That is because thevelocity errors are related to the attitude errors especially tothe pitch and roll errors of AKF and UKF Because of thestrong nonlinear properties of system it is difficult that AKFcannot be effectively operated for navigation Meanwhilefor UKF solutions the curves of speed errors are extremelysimilar to those of horizontal attitude errors

The simulation results indicate that if the AUKF schemeis considered the filtering solution there are only smallvariations impacting on the performance of MEMSGPSintegrated navigation system and this system has an excellentrobustness However long processing timewould cause slightdivergence of the attitude errors sequentially the velocity andposition errors

7 Conclusions

This study has developed an AUKF approach to improvethe navigation performance ofMEMSGPS integrated systemfor land-vehicle applications By treating this problem withinconventional UKF framework the noise estimator is adoptedand could effectively estimate the process and measurementnoise characteristics online The results indicate that theproposed AUKF algorithm could efficiently improve thenavigation performance of land-vehicle integrated navigationsystem By comparing with AKF and UKF methods theAUKF solution has a more stable and superior performance

Appendix

The process noise andmeasurement noise statistic propertiesare computed by estimator in (30) to (33) Their unbiasedproperties are proved as follows

According to (8) the predicted state at time step 119895 is givenas

119909119895119895minus1 =

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1) + 119902 (A1)

Hence the mean of the process noise can be expressed as

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus

119871

sum

119894=0

119882119898

119894119891119895minus1 (120585119894119895minus1119895minus1)]

=

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

(A2)

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

From (15) we have

119909119895119895 = 119909119895119895minus1 + 119870119895 (119911119895 minus 119895119895minus1) (A3)

Substituting (A3) into (A2) yields

119864 [119902119896] =

1

119896

119896

sum

119895=1

[119909119895119895 minus 119909119895119895minus1 + 119902]

=

1

119896

119896

sum

119895=1

[119870119895 (119911119895 minus 119895119895minus1) + 119902]

(A4)

When the posteriori mean and covariance are known theoutput residual vector of UKF is zero-mean Gaussian whitenoise and we see

119864 [119911119895 minus 119895119895minus1] = 0 (A5)

From (A4) and (A5) the mean of the process noise is

119864 [119902119896] =

1

119896

119896

sum

119895=1

119864 [(119911119895 minus 119895119895minus1) + 119902] = 119902 (A6)

Thus the estimate of the process noise noted by (24) isunbiased

Similarly the estimate of the measurement noise is

119864 [119903119896] = 119903 (A7)

The unbiased properties for the estimates of noises areproved

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

Funding for this work was provided by the National NatureScience Foundation of China under Grant nos 61374007 and61104036 The authors would like to thank all the editors andanonymous reviewers for improving this paper

References

[1] S Mizukami M Sugiura Y Muramatsu and H KumagaildquoMEMS GPSINS for micro air vehicle applicationrdquo in Proceed-ings of the 17th International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS rsquo04) pp 819ndash824 September 2004

[2] X Niu S Nassar and N El-Sheimy ldquoAn accurate land-vehicleMEMS IMUGPS navigation system using 3D auxiliary velocityupdatesrdquo Navigation Journal of the Institute of Navigation vol54 no 3 pp 177ndash188 2007

[3] S Y Cho and B D Kim ldquoAdaptive IIRFIR fusion filter and itsapplication to the INSGPS integrated systemrdquoAutomatica vol44 no 8 pp 2040ndash2047 2008

[4] S Y Cho and W S Choi ldquoRobust positioning technique inlow-cost DRGPS for land navigationrdquo IEEE Transactions onInstrumentation and Measurement vol 55 no 4 pp 1132ndash11422006

[5] A Gelb Applied Optimal Estimation The MIT press 1974[6] C Hide T Moore and M Smith ldquoAdaptive Kalman filtering

for low-cost INSGPSrdquo Journal of Navigation vol 56 no 1 pp143ndash152 2003

[7] C W Hu Congwei W Chen Y Chen and D Liu ldquoAdaptiveKalman filtering for vehicle navigationrdquo Journal of GlobalPositioning Systems vol 2 no 1 pp 42ndash47 2003

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

[9] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches Wiley 2006

[10] S J Julier and K U Jeffrey ldquoA general method for approxi-mating nonlinear transformations of probability distributionsrdquoTech Rep Robotics Research Group University of OxfordOxford UK 1996

[11] Z Jiang Q Song Y He and J Han ldquoA novel adaptive unscentedkalman filter for nonlinear estimationrdquo in Proceedings of the46th IEEE Conference on Decision and Control (CDC rsquo07) pp4293ndash4298 New Orleans LA USA December 2007

[12] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control COnference pp 4555ndash4559 May2002

[13] S Sarkka ldquoOn unscented Kalman filtering for state estimationof continuous-time nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 52 no 9 pp 1631ndash1641 2007

[14] L Zhao X-X Wang H-X Xue and Q-X Xia ldquoDesign ofunscented Kalman filter with noise statistic estimatorrdquo Controland Decision vol 24 no 10 pp 1483ndash1488 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of