research article positioning errors predicting method of...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 737146 7 pageshttpdxdoiorg1011552013737146
Research ArticlePositioning Errors Predicting Method of Strapdown InertialNavigation Systems Based on PSO-SVM
Xunyuan Yin Yingbo Sun and Changhong Wang
Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin 150001 China
Correspondence should be addressed to Changhong Wang cwanghiteducn
Received 13 July 2013 Revised 27 July 2013 Accepted 27 July 2013
Academic Editor Hamid Reza Karimi
Copyright copy 2013 Xunyuan Yin 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
The strapdown inertial navigation systems (SINS) have been widely used for many vehicles such as commercial airplanesUnmanned Aerial Vehicles (UAVs) and other types of aircrafts In order to evaluate the navigation errors precisely and efficiently apredictionmethod based on support vectormachine (SVM) is proposed for positioning error assessment Firstly SINS errormodelsthat are used for error calculation are established considering several error resources with respect to inertial units Secondly flightpaths for simulation are designed Thirdly the 120576-SVR based prediction method is proposed to predict the positioning errors ofnavigation systems and particle swarm optimization (PSO) is used for the SVM parameters optimization Finally 600 sets of errorparameters of SINS are utilized to train the SVMmodel which is used for the performance prediction of new navigation systems Bycomparing the predicting results with the real errors the latitudinal predicting accuracy is 9273 while the longitudinal predictingaccuracy is 9164 and PSO is effective to increase the prediction accuracy compared with traditional SVMwith fixed parametersThis method is also demonstrated to be effective for error prediction for an entire flight process Moreover the prediction methodcan save 75 of calculation time compared with analyses based on error models
1 Introduction
Strapdown inertial navigation systems have been widelyutilized in a wide range of fields such as the navigation ofairplanes ships and vehicles and the guidance of missiles [1]Although the positioning accuracy of strapdown systems islower than that of the platform inertial navigation systemsthe strapdown systems have several advantages that cannotbe found in platform systems They have low cost lowweight small volume good reliability and simplemechanicalstructures [2] So far almost all the civil aviation airplanesmanufactured by Boeing and Airbus are equipped with LTN-92 or LTN-101 laser SINS [3 4] Other types of SINS thatconsist of different kinds of gyroscopes and accelerometersare used for the navigation of vehicles which require mediumor low accuracy [5]
Navigation accuracy is the main factor that is used toaccess the performance of SINS [6] To help the vehicles tocomplete the flight task or arrive at a desired destination
the strapdown systems which can meet the navigationrequirements should be selected [7] If an effective methodthat could be established to predict the velocity errors andpositioning errors by assessing error parameters a gooddeal of time for error analyses would be saved and theaircrafts would be more likely to accomplish the initial tasks[8 9] Currently some researchers have placed importanceon the error analyses and error compensation of SINS [10]A methodology based on the theory of artificial neuralnetworks has been put up to predict the positioning errorscaused by the drift error of each singe axis of gyroscope [11]But research that stresses predicting the performance of SINSby error sources can hardly be found
Nowadays several methodologies have been reported forthe classification or regression in various fields Artificialneural networks (ANNs) have been pervasively adopted andare able to achieve acceptable results in many applicationsChen developed an ANN-based model which is called Evo-lutionary Fuzzy Neural Inference Model to predict Estimate
2 Abstract and Applied Analysis
at Completion (EAC) [12] However it has disadvantages toaddress the proposed problem since it is shown by simulationthat the predicting accuracy might be low due to a localminimization problemAs a consequence it is not guaranteedfor all the models to converge to optimal solutions [13]Besides ANN is also vulnerable to the selection of networkstructure and it has a high computational expense in termsof training process Extended Kalman filter (EKF) approachwhich is also popular in industrial applications is able tomaintain relatively high accuracy in terms of estimationbut EKF estimator often requires high cost due to highcomputational complexity [14] Fuzzy logic method is alsoreported to be available for estimation For more details onstate estimation and on filtering approaches for complexsystems the reader can be referred to [15ndash19]
Support vector machine (SVM) is an effective mathe-matical method in prediction and this technique has greatlydeveloped in the past decades SVM which is based onstructural risk minimization principle [20] is adopted byresearchers to address classification and regression problems[21] So far it has been successfully used in a variety of fieldsFor instance it is used for noise estimation and the predictionof air passengers [22 23] it is also utilized for image analysesbiomedicine and bioinformatics as estimator tools
Inertial navigation systems are sophisticated nonlinearsystems [24] Therefore it is unrealistic to estimate theperformance of the navigation systems by the analyses of themodels In order to predict the navigation errors by errorcoefficients a method based on support vector machineis proposed in this paper Support vector machine unlikeother traditional methods is relatively effective in termsof combating nonlinear situations It is more robust as anestimator than least-square based method because it isinsensitive to small changes [25] Firstly the error modelof SINS is established Secondly a series of flight paths thatmeet the characteristics of real trajectories are designedThen 300 sets of random error parameters which obeyGaussian distribution are generated and results of systemerrors are obtained by simulationThese parameters are usedto train the model of SVM Finally the trained model isused to predict the system errors of SINS with different errorcoefficients This method is tested to be effective and efficientby comparing simulation results with the actual navigationerrors
The remainder of this paper is organized as follows Errormodel of SINS is given and fight path for simulation isdesigned in Section 2 In Section 3 SVM-based navigationerror estimation method is proposed Simulation verificationof proposed prediction method is given and error predictionof an entire flight path is completed in Section 4 FinallySection 5 gives the conclusions
2 Error Model of SINS
Error equations in terms of velocity errors attitude errorsand position errors should be established in order to analyzethe navigation errors caused by strapdown inertial navigationsystems [2]
21 Attitude Error Equations The attitude errors of the air-craft are represented by 120575120572 120575120573 and 120575120574 respectively asfollows
120575 = minus
120575119881119873
119877
+ (
119881119864
119877
tan120593 + 120596119894119890sin120593) 120575120573
minus (
119881119864
119877
+ 120596119894119890cos120593) + 120576
119864
120575120573 =
120575119881119864
119877
minus (
119881119864
119877
+ 120596119894119890sin120593) 120575120572
minus 120596119894119890sin120593120575120593 minus
119881119873
119877
120575120574 + 120576119873
120575 120574 =
120575119881119864
119877
tan120593 + (119881119864119877
sec2120593 + 120596119894119890cos120593) 120575120593
+ (
119881119864
119877
+ 120596119894119890cos120593) 120575120572 +
119881119873
119877
120575120573 + 120576120585
(1)
where 120576119873 120576119864 and 120576
120585represent the gyroscope drift errors of
three axes
22 Velocity Error Equations With respect to the navigationsystem which is analyzed in this passage some error sourcessuch as the ones related to acceleration can be ignored whenthe coefficients are relatively small Therefore velocity errorequations are established in the following
120575119864= (
119881119864
119877
tan120593 minus119881120585
119877
)120575119881119864+ (
119881119864
119877
tan120593 + 2120596119894119890sin120593) 120575119881
119873
minus (
119881119864
119877
+ 2120596119894119890cos120593) 120575119881
120585
+ (
119881119864119881119873
119877
sec2120593 + 2120596119894119890cos120593119881
119873+ 2120596119894119890sin120593119881
120585) 120575120593
+ 119860119873120574 minus 119860
120585120573 + Δ119860
119864
120575119873= minus 2 (
119881119864
119877
tan120593 + 120596119894119890sin120593) 120575119881
119864minus
119881120585
119877
120575119881119873minus
119881119873
119877
120575119881120585
minus (
119881119864
119877
sec2120593 + 2120596119894119890cos120593)119881
119864120575120593
+ 119860120585120572 minus 119860
119864120574 + Δ119860
119873
120575120585= 2 (
119881119864
119877
+ 120596119894119890cos120593) 120575119881
119864+ 2
119881119873
119877
120575119881119873minus 2119881119864120596119894119890sin120593120575120593
+ 119860119864120573 minus 119860
119873120572 + Δ119860
120585
(2)
where Δ119860119864 Δ119860119873 and Δ119860
120585represent the accelerometer drift
errors of three axes
23 Positioning Error Equations The positioning errors arethemain factors that are considered when evaluating the per-formance of different kinds of aircrafts such as airplanes and
Abstract and Applied Analysis 3
39 40 41 42 43 44
116118
120122
100015002000250030003500
Latitude (deg)Longitude (deg)
Hei
ght (
m)
Figure 1 Flight path for analyses
UAVs Therefore the latitudinal and longitudinal errors areimportant factors for the assessment of navigation systems
120575 =
120575119881119873
119877
minus 120575ℎ
119881119873
1198772
120575120582 = (
120575119881119864
119877
+ 120575119871
119881119873
1198772tan119871) sec 119871 minus 120575ℎ119881119864sec 119871
1198772
120575ℎ = 120575119881
119880
(3)
24 Flight Path Design According to several papers thatfocus on error analysis of inertial navigation systems theresearch is mainly based on some simple flight paths such asuniform linear motion or uniform turning motion Howeverthese flight paths cannot involve all the flight modes As aconsequence the results of these simulations or analyses maynot represent the real performance of the aircrafts that areequipped with SINS
Therefore it is necessary to design some flight paths thatnot only include the characteristics of real paths of aircraftsbut are also able to ensure for each error source to be stimu-lated As shown in Figure 1 a typical flight path for simulationis designed
3 Navigation Error Prediction
It is unrealistic to use error equations to analyze the perfor-mance of SINS especially when there is a large variety ofsystems that should be tested in short term as it will cost agood deal of time to solve the differential error equations
In order to avoid complicated calculations support vectormachine with strong generalization ability is utilized topredict system errors by assessing each single error sourceHowever a noticeable problem is that the navigation errorsare time varying and closely associated with the flight pathsTherefore characteristic vectors related to certain flight pathsand ultimate positioning errors should be established toaccomplish the prediction because positioning errors are themost important data for navigation systems
31 Support Vector Regression Support vector machine(SVM) a method closely associated with optimization algo-rithms is an effectivemethodology to address data processing
problems [12] It is demonstrated to be eligible to overcomethe traditional obstacles with respect to multidimensionalproblems and over learning So far SVM is widely used inmany fields such as biological information voice recogni-tion failure identification and prognostics
SVM consists of support vector classification and sup-port vector regression To solve problems with respect toprediction support vector regression method can be used[21] Since the problem is nonlinear a transform 119909 = 120601(119909)
should be introduced By using a nonlinear mapping thatmaps the sample data into a high dimensional space 120601
119877119899rarr 119867 linear regression method can be conducted in
the high-dimensional space 119867 to accomplish the nonlinearprediction
A training set is given as
119879 = (1199091 1199101) (1199092 1199102) (119909
119897 119910119897) isin (119877
119899times 119910)119897 (4)
Due to the varying characteristics of the error parametersof inertial navigation systems a leading problem that shouldbe overcome is that all the significant parameters with respectto positioning errors should be preprocessed as characteristicquantities for estimation which is considered to be effectiveto improve the estimation accuracy [26] Therefore 15 errorparameters that have considerable impacts on position errorsof SINS are considered for the model training In (4) 119909
119894
represent the error sources of SINS which are the zero biaserrors random walk errors of gyros zero random walk ofaccelerometers random walk errors of accelerometers andscale factor errors of gyros 119910
119894represent navigation errors
which are latitudinal and longitudinal positioning errors ofthe training models Compared with ANNs SVM has adrawback that it can only generate one output while theANNs are able to generate multiple outputs [27] Conse-quently two separate training processes should be separatelyconducted for latitudinal and longitudinal errors Specificallythe error coefficients and positioning errors of the 600 setsof navigation systems are used to train the SVM modelThen a suitable kernel function is selected as radial BasisFunction (RBF) As there are many kernel functions availablefor analyses [28 29] RBF is demonstrated to be effective forthis problem by contrasting with other kernel functions
119870(119909119894 119909119895) = exp(minus
10038171003817100381710038171003817119909119894minus 119909119895
10038171003817100381710038171003817
2
21205902
) (5)
In the next step following convex quadratic program-ming problem (5) is resolved to obtain that 120572(lowast) =
(1205721 120572lowast
1 120572
119897120572lowast
119897)119879 as follows
min 12
119897
sum
119894119895=1
(120572lowast
119894minus 120572119894) (120572lowast
119894minus 120572119894)119870 (119909
119894 119909119894)
+ 120576
119897
sum
119894=1
(120572lowast
119894+ 120572119894) minus
119897
sum
119894=1
119910119894(120572lowast
119894minus 120572119894)
(6)
If 120572119894is picked and then 119887 = 119910
119894minussum119897
119894=1(120572lowast
119894minus 120572119894)119870 + 120576 if 120572lowast
119896
is picked then 119887 = 119910119896minus sum119897
119894=1(120572lowast
119894minus 120572119894)119870 minus 120576
4 Abstract and Applied Analysis
Decision function is established as
119910 = 119892 (119909) =
119897
sum
119894=1
(120572lowast
119894minus 120572119894)119870 (119909
119894 119909) = 119887 (7)
32 PSO-Based Optimal SVR Parameters Selection Impor-tant factors of SVM are the constant 119862 the accuracy parame-ter 120576 and the kernel function As the kernel function has beenselected as RBF appropriate 119862 and 120576 should be selected inorder to increase the estimation accuracy Since itmay take anextremely long time to seek the best parameters and desiredresults may not be achieved by simply making differenttests an efficient method called particle swarm algorithm isadopted for the optimization
Particle swarm algorithm (PSO) is an algorithm thatis inspired by birdsrsquo foraging behavior and widely used toaddress optimization problems [30ndash32] The PSO algorithmis introduced to optimize the parameters of 120576 and 119862 ThePSO is initialized with random particles and then it worksto find optimal parameters by iterative methods Practicallythe initial parameters of 120576 and 119862 should be given and thenthe optimal values will be generated by calculation
4 Error Analyses for SINS
Before predicting the performance of certain inertial naviga-tion systems error analyses for SINS which are indispensablefor the process of error prediction should be conducted Asfor the strapdown navigation systems studied in this paperthe drift errors of the gyroscopes the white noise of thegyroscopes and the drift errors of accelerometers are consid-ered whereas the error coefficients related to acceleration andthe coefficients related to quadratic acceleration are ignoredsince such coefficients are comparatively small
Two criteria should be obeyed for the selection of trainingdata
(1) The training data should be not the same as the datafor test If there were no differences between thetraining data and testing data the prediction accuracywould be relatively high but biased
(2) The dimension of the inputs should be increased ifpossible If the training data had a high dimensionmore useful characteristics could be used for modeltraining
(3) The training data should represent different systemswith vastly different error coefficients With respectto this criterion suitable standard deviation for errorsources should be assigned
The error sources are assigned by the given parameterswhich are listed in Table 1 The navigation system is affectedby multiple error sources when it performs a navigationtask so it is necessary to assign all the error parametersthat are considered in this navigation system in order tomake the results adaptive for real flight situations To makepreparation for the performance prediction of SINS it issensible to carry out error analyses for 600 sets with different
Table 1The expected value and standard deviation of error sources
Error sources Expectedvalue
Standarddeviation
Zero bias errors of gyros 001∘h 0006∘hRandom walk errors of gyros 0001∘radich 00068∘radichAccelerometer zero bias errors 3120583g 05 120583gAccelerometer random walk errors 3120583gradicHz 5120583gradicHzScale factor errors of gyros 10 ppm 0
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
Real longitudinal error (m)
Real
latit
udin
al er
ror (
m)
minus1000
minus500
Figure 2 Real distributing points of positioning errors
error parameters The assignment of different error sourcesabides by the Gaussian distribution Calculating with thesystem error functions the 600 sets of navigation errors ofthe navigation systems with different error coefficients areachieved
41 Simulation Verification of Prediction Method The errorcoefficients and the positioning error parameters of the 600sets of SINS are transformed into characteristic values whichare used to train the SVM model Other 300 sets of errorcoefficients are randomly generated and the correspondingsystem errors are calculated by error equations Then theaccuracy parameter is initially given as 120576 = 05 and thepenalty parameter is given as119862 = 60 PSO is used to generateoptimal parameters of 120576 = 00104 and 119862 = 4912
Both original SVM with fixed parameters and SVMmodel with optimal parameters generated by PSO are used inorder to predict the positioning errors The real distributionpoints of positioning errors are shown in Figure 2 while thepredicting distribution points of positioning error are shownin Figure 3
The prediction errors in terms of latitude and longitudewhich are predicted by SVMmodel with optimal parametersgenerated by PSO are shown in Figure 4
By comparing the results the predicting results are rela-tively satisfying Specifically the average north error is 343mwhile the expected value of north error by prediction is4558m The standard deviation of north error by prediction
Abstract and Applied Analysis 5
Longitudinal prediction error (m)
Latit
udin
al p
redi
ctio
n er
ror (
m)
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
minus1000
minus500
Figure 3 Prediction distributing points of positioning errors
0 50 100 150 200 250 300
0
200
400
minus200Erro
r of l
ongi
tudi
nal
pred
ictio
n (m
)
Number of SINS for test n
(a)
0 50 100 150 200 250 300
0
100
minus100
Erro
r of l
atitu
dina
lpr
edic
tion
(m)
minus200
Number of SINS for test n
(b)
Figure 4 Prediction errors generated by SVM
is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m
The predicting accuracy is defined as 119875 which is accessedwith (7) 119890
119901represents the predicting error and 119890
119903represents
the real error calculated by system error equations Letter 119899represents the number of navigation systems for test
119875 = 1 minus
119899
sum
119894=1
10038161003816100381610038161003816100381610038161003816
119890119901119894minus 119890119903119894
119890119903119894
10038161003816100381610038161003816100381610038161003816
times
1
119899
(8)
After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically
Table 2 Prediction accuracy of different methods
Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
1200
Long
itudi
nal e
rror
(m)
Time (s)
Real longitudinal errorLongitudinal error by estimation
minus200
Figure 5 Longitudinal error of a strapdown system with knownerror parameters
if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS
42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes
One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value
Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
at Completion (EAC) [12] However it has disadvantages toaddress the proposed problem since it is shown by simulationthat the predicting accuracy might be low due to a localminimization problemAs a consequence it is not guaranteedfor all the models to converge to optimal solutions [13]Besides ANN is also vulnerable to the selection of networkstructure and it has a high computational expense in termsof training process Extended Kalman filter (EKF) approachwhich is also popular in industrial applications is able tomaintain relatively high accuracy in terms of estimationbut EKF estimator often requires high cost due to highcomputational complexity [14] Fuzzy logic method is alsoreported to be available for estimation For more details onstate estimation and on filtering approaches for complexsystems the reader can be referred to [15ndash19]
Support vector machine (SVM) is an effective mathe-matical method in prediction and this technique has greatlydeveloped in the past decades SVM which is based onstructural risk minimization principle [20] is adopted byresearchers to address classification and regression problems[21] So far it has been successfully used in a variety of fieldsFor instance it is used for noise estimation and the predictionof air passengers [22 23] it is also utilized for image analysesbiomedicine and bioinformatics as estimator tools
Inertial navigation systems are sophisticated nonlinearsystems [24] Therefore it is unrealistic to estimate theperformance of the navigation systems by the analyses of themodels In order to predict the navigation errors by errorcoefficients a method based on support vector machineis proposed in this paper Support vector machine unlikeother traditional methods is relatively effective in termsof combating nonlinear situations It is more robust as anestimator than least-square based method because it isinsensitive to small changes [25] Firstly the error modelof SINS is established Secondly a series of flight paths thatmeet the characteristics of real trajectories are designedThen 300 sets of random error parameters which obeyGaussian distribution are generated and results of systemerrors are obtained by simulationThese parameters are usedto train the model of SVM Finally the trained model isused to predict the system errors of SINS with different errorcoefficients This method is tested to be effective and efficientby comparing simulation results with the actual navigationerrors
The remainder of this paper is organized as follows Errormodel of SINS is given and fight path for simulation isdesigned in Section 2 In Section 3 SVM-based navigationerror estimation method is proposed Simulation verificationof proposed prediction method is given and error predictionof an entire flight path is completed in Section 4 FinallySection 5 gives the conclusions
2 Error Model of SINS
Error equations in terms of velocity errors attitude errorsand position errors should be established in order to analyzethe navigation errors caused by strapdown inertial navigationsystems [2]
21 Attitude Error Equations The attitude errors of the air-craft are represented by 120575120572 120575120573 and 120575120574 respectively asfollows
120575 = minus
120575119881119873
119877
+ (
119881119864
119877
tan120593 + 120596119894119890sin120593) 120575120573
minus (
119881119864
119877
+ 120596119894119890cos120593) + 120576
119864
120575120573 =
120575119881119864
119877
minus (
119881119864
119877
+ 120596119894119890sin120593) 120575120572
minus 120596119894119890sin120593120575120593 minus
119881119873
119877
120575120574 + 120576119873
120575 120574 =
120575119881119864
119877
tan120593 + (119881119864119877
sec2120593 + 120596119894119890cos120593) 120575120593
+ (
119881119864
119877
+ 120596119894119890cos120593) 120575120572 +
119881119873
119877
120575120573 + 120576120585
(1)
where 120576119873 120576119864 and 120576
120585represent the gyroscope drift errors of
three axes
22 Velocity Error Equations With respect to the navigationsystem which is analyzed in this passage some error sourcessuch as the ones related to acceleration can be ignored whenthe coefficients are relatively small Therefore velocity errorequations are established in the following
120575119864= (
119881119864
119877
tan120593 minus119881120585
119877
)120575119881119864+ (
119881119864
119877
tan120593 + 2120596119894119890sin120593) 120575119881
119873
minus (
119881119864
119877
+ 2120596119894119890cos120593) 120575119881
120585
+ (
119881119864119881119873
119877
sec2120593 + 2120596119894119890cos120593119881
119873+ 2120596119894119890sin120593119881
120585) 120575120593
+ 119860119873120574 minus 119860
120585120573 + Δ119860
119864
120575119873= minus 2 (
119881119864
119877
tan120593 + 120596119894119890sin120593) 120575119881
119864minus
119881120585
119877
120575119881119873minus
119881119873
119877
120575119881120585
minus (
119881119864
119877
sec2120593 + 2120596119894119890cos120593)119881
119864120575120593
+ 119860120585120572 minus 119860
119864120574 + Δ119860
119873
120575120585= 2 (
119881119864
119877
+ 120596119894119890cos120593) 120575119881
119864+ 2
119881119873
119877
120575119881119873minus 2119881119864120596119894119890sin120593120575120593
+ 119860119864120573 minus 119860
119873120572 + Δ119860
120585
(2)
where Δ119860119864 Δ119860119873 and Δ119860
120585represent the accelerometer drift
errors of three axes
23 Positioning Error Equations The positioning errors arethemain factors that are considered when evaluating the per-formance of different kinds of aircrafts such as airplanes and
Abstract and Applied Analysis 3
39 40 41 42 43 44
116118
120122
100015002000250030003500
Latitude (deg)Longitude (deg)
Hei
ght (
m)
Figure 1 Flight path for analyses
UAVs Therefore the latitudinal and longitudinal errors areimportant factors for the assessment of navigation systems
120575 =
120575119881119873
119877
minus 120575ℎ
119881119873
1198772
120575120582 = (
120575119881119864
119877
+ 120575119871
119881119873
1198772tan119871) sec 119871 minus 120575ℎ119881119864sec 119871
1198772
120575ℎ = 120575119881
119880
(3)
24 Flight Path Design According to several papers thatfocus on error analysis of inertial navigation systems theresearch is mainly based on some simple flight paths such asuniform linear motion or uniform turning motion Howeverthese flight paths cannot involve all the flight modes As aconsequence the results of these simulations or analyses maynot represent the real performance of the aircrafts that areequipped with SINS
Therefore it is necessary to design some flight paths thatnot only include the characteristics of real paths of aircraftsbut are also able to ensure for each error source to be stimu-lated As shown in Figure 1 a typical flight path for simulationis designed
3 Navigation Error Prediction
It is unrealistic to use error equations to analyze the perfor-mance of SINS especially when there is a large variety ofsystems that should be tested in short term as it will cost agood deal of time to solve the differential error equations
In order to avoid complicated calculations support vectormachine with strong generalization ability is utilized topredict system errors by assessing each single error sourceHowever a noticeable problem is that the navigation errorsare time varying and closely associated with the flight pathsTherefore characteristic vectors related to certain flight pathsand ultimate positioning errors should be established toaccomplish the prediction because positioning errors are themost important data for navigation systems
31 Support Vector Regression Support vector machine(SVM) a method closely associated with optimization algo-rithms is an effectivemethodology to address data processing
problems [12] It is demonstrated to be eligible to overcomethe traditional obstacles with respect to multidimensionalproblems and over learning So far SVM is widely used inmany fields such as biological information voice recogni-tion failure identification and prognostics
SVM consists of support vector classification and sup-port vector regression To solve problems with respect toprediction support vector regression method can be used[21] Since the problem is nonlinear a transform 119909 = 120601(119909)
should be introduced By using a nonlinear mapping thatmaps the sample data into a high dimensional space 120601
119877119899rarr 119867 linear regression method can be conducted in
the high-dimensional space 119867 to accomplish the nonlinearprediction
A training set is given as
119879 = (1199091 1199101) (1199092 1199102) (119909
119897 119910119897) isin (119877
119899times 119910)119897 (4)
Due to the varying characteristics of the error parametersof inertial navigation systems a leading problem that shouldbe overcome is that all the significant parameters with respectto positioning errors should be preprocessed as characteristicquantities for estimation which is considered to be effectiveto improve the estimation accuracy [26] Therefore 15 errorparameters that have considerable impacts on position errorsof SINS are considered for the model training In (4) 119909
119894
represent the error sources of SINS which are the zero biaserrors random walk errors of gyros zero random walk ofaccelerometers random walk errors of accelerometers andscale factor errors of gyros 119910
119894represent navigation errors
which are latitudinal and longitudinal positioning errors ofthe training models Compared with ANNs SVM has adrawback that it can only generate one output while theANNs are able to generate multiple outputs [27] Conse-quently two separate training processes should be separatelyconducted for latitudinal and longitudinal errors Specificallythe error coefficients and positioning errors of the 600 setsof navigation systems are used to train the SVM modelThen a suitable kernel function is selected as radial BasisFunction (RBF) As there are many kernel functions availablefor analyses [28 29] RBF is demonstrated to be effective forthis problem by contrasting with other kernel functions
119870(119909119894 119909119895) = exp(minus
10038171003817100381710038171003817119909119894minus 119909119895
10038171003817100381710038171003817
2
21205902
) (5)
In the next step following convex quadratic program-ming problem (5) is resolved to obtain that 120572(lowast) =
(1205721 120572lowast
1 120572
119897120572lowast
119897)119879 as follows
min 12
119897
sum
119894119895=1
(120572lowast
119894minus 120572119894) (120572lowast
119894minus 120572119894)119870 (119909
119894 119909119894)
+ 120576
119897
sum
119894=1
(120572lowast
119894+ 120572119894) minus
119897
sum
119894=1
119910119894(120572lowast
119894minus 120572119894)
(6)
If 120572119894is picked and then 119887 = 119910
119894minussum119897
119894=1(120572lowast
119894minus 120572119894)119870 + 120576 if 120572lowast
119896
is picked then 119887 = 119910119896minus sum119897
119894=1(120572lowast
119894minus 120572119894)119870 minus 120576
4 Abstract and Applied Analysis
Decision function is established as
119910 = 119892 (119909) =
119897
sum
119894=1
(120572lowast
119894minus 120572119894)119870 (119909
119894 119909) = 119887 (7)
32 PSO-Based Optimal SVR Parameters Selection Impor-tant factors of SVM are the constant 119862 the accuracy parame-ter 120576 and the kernel function As the kernel function has beenselected as RBF appropriate 119862 and 120576 should be selected inorder to increase the estimation accuracy Since itmay take anextremely long time to seek the best parameters and desiredresults may not be achieved by simply making differenttests an efficient method called particle swarm algorithm isadopted for the optimization
Particle swarm algorithm (PSO) is an algorithm thatis inspired by birdsrsquo foraging behavior and widely used toaddress optimization problems [30ndash32] The PSO algorithmis introduced to optimize the parameters of 120576 and 119862 ThePSO is initialized with random particles and then it worksto find optimal parameters by iterative methods Practicallythe initial parameters of 120576 and 119862 should be given and thenthe optimal values will be generated by calculation
4 Error Analyses for SINS
Before predicting the performance of certain inertial naviga-tion systems error analyses for SINS which are indispensablefor the process of error prediction should be conducted Asfor the strapdown navigation systems studied in this paperthe drift errors of the gyroscopes the white noise of thegyroscopes and the drift errors of accelerometers are consid-ered whereas the error coefficients related to acceleration andthe coefficients related to quadratic acceleration are ignoredsince such coefficients are comparatively small
Two criteria should be obeyed for the selection of trainingdata
(1) The training data should be not the same as the datafor test If there were no differences between thetraining data and testing data the prediction accuracywould be relatively high but biased
(2) The dimension of the inputs should be increased ifpossible If the training data had a high dimensionmore useful characteristics could be used for modeltraining
(3) The training data should represent different systemswith vastly different error coefficients With respectto this criterion suitable standard deviation for errorsources should be assigned
The error sources are assigned by the given parameterswhich are listed in Table 1 The navigation system is affectedby multiple error sources when it performs a navigationtask so it is necessary to assign all the error parametersthat are considered in this navigation system in order tomake the results adaptive for real flight situations To makepreparation for the performance prediction of SINS it issensible to carry out error analyses for 600 sets with different
Table 1The expected value and standard deviation of error sources
Error sources Expectedvalue
Standarddeviation
Zero bias errors of gyros 001∘h 0006∘hRandom walk errors of gyros 0001∘radich 00068∘radichAccelerometer zero bias errors 3120583g 05 120583gAccelerometer random walk errors 3120583gradicHz 5120583gradicHzScale factor errors of gyros 10 ppm 0
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
Real longitudinal error (m)
Real
latit
udin
al er
ror (
m)
minus1000
minus500
Figure 2 Real distributing points of positioning errors
error parameters The assignment of different error sourcesabides by the Gaussian distribution Calculating with thesystem error functions the 600 sets of navigation errors ofthe navigation systems with different error coefficients areachieved
41 Simulation Verification of Prediction Method The errorcoefficients and the positioning error parameters of the 600sets of SINS are transformed into characteristic values whichare used to train the SVM model Other 300 sets of errorcoefficients are randomly generated and the correspondingsystem errors are calculated by error equations Then theaccuracy parameter is initially given as 120576 = 05 and thepenalty parameter is given as119862 = 60 PSO is used to generateoptimal parameters of 120576 = 00104 and 119862 = 4912
Both original SVM with fixed parameters and SVMmodel with optimal parameters generated by PSO are used inorder to predict the positioning errors The real distributionpoints of positioning errors are shown in Figure 2 while thepredicting distribution points of positioning error are shownin Figure 3
The prediction errors in terms of latitude and longitudewhich are predicted by SVMmodel with optimal parametersgenerated by PSO are shown in Figure 4
By comparing the results the predicting results are rela-tively satisfying Specifically the average north error is 343mwhile the expected value of north error by prediction is4558m The standard deviation of north error by prediction
Abstract and Applied Analysis 5
Longitudinal prediction error (m)
Latit
udin
al p
redi
ctio
n er
ror (
m)
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
minus1000
minus500
Figure 3 Prediction distributing points of positioning errors
0 50 100 150 200 250 300
0
200
400
minus200Erro
r of l
ongi
tudi
nal
pred
ictio
n (m
)
Number of SINS for test n
(a)
0 50 100 150 200 250 300
0
100
minus100
Erro
r of l
atitu
dina
lpr
edic
tion
(m)
minus200
Number of SINS for test n
(b)
Figure 4 Prediction errors generated by SVM
is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m
The predicting accuracy is defined as 119875 which is accessedwith (7) 119890
119901represents the predicting error and 119890
119903represents
the real error calculated by system error equations Letter 119899represents the number of navigation systems for test
119875 = 1 minus
119899
sum
119894=1
10038161003816100381610038161003816100381610038161003816
119890119901119894minus 119890119903119894
119890119903119894
10038161003816100381610038161003816100381610038161003816
times
1
119899
(8)
After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically
Table 2 Prediction accuracy of different methods
Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
1200
Long
itudi
nal e
rror
(m)
Time (s)
Real longitudinal errorLongitudinal error by estimation
minus200
Figure 5 Longitudinal error of a strapdown system with knownerror parameters
if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS
42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes
One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value
Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
39 40 41 42 43 44
116118
120122
100015002000250030003500
Latitude (deg)Longitude (deg)
Hei
ght (
m)
Figure 1 Flight path for analyses
UAVs Therefore the latitudinal and longitudinal errors areimportant factors for the assessment of navigation systems
120575 =
120575119881119873
119877
minus 120575ℎ
119881119873
1198772
120575120582 = (
120575119881119864
119877
+ 120575119871
119881119873
1198772tan119871) sec 119871 minus 120575ℎ119881119864sec 119871
1198772
120575ℎ = 120575119881
119880
(3)
24 Flight Path Design According to several papers thatfocus on error analysis of inertial navigation systems theresearch is mainly based on some simple flight paths such asuniform linear motion or uniform turning motion Howeverthese flight paths cannot involve all the flight modes As aconsequence the results of these simulations or analyses maynot represent the real performance of the aircrafts that areequipped with SINS
Therefore it is necessary to design some flight paths thatnot only include the characteristics of real paths of aircraftsbut are also able to ensure for each error source to be stimu-lated As shown in Figure 1 a typical flight path for simulationis designed
3 Navigation Error Prediction
It is unrealistic to use error equations to analyze the perfor-mance of SINS especially when there is a large variety ofsystems that should be tested in short term as it will cost agood deal of time to solve the differential error equations
In order to avoid complicated calculations support vectormachine with strong generalization ability is utilized topredict system errors by assessing each single error sourceHowever a noticeable problem is that the navigation errorsare time varying and closely associated with the flight pathsTherefore characteristic vectors related to certain flight pathsand ultimate positioning errors should be established toaccomplish the prediction because positioning errors are themost important data for navigation systems
31 Support Vector Regression Support vector machine(SVM) a method closely associated with optimization algo-rithms is an effectivemethodology to address data processing
problems [12] It is demonstrated to be eligible to overcomethe traditional obstacles with respect to multidimensionalproblems and over learning So far SVM is widely used inmany fields such as biological information voice recogni-tion failure identification and prognostics
SVM consists of support vector classification and sup-port vector regression To solve problems with respect toprediction support vector regression method can be used[21] Since the problem is nonlinear a transform 119909 = 120601(119909)
should be introduced By using a nonlinear mapping thatmaps the sample data into a high dimensional space 120601
119877119899rarr 119867 linear regression method can be conducted in
the high-dimensional space 119867 to accomplish the nonlinearprediction
A training set is given as
119879 = (1199091 1199101) (1199092 1199102) (119909
119897 119910119897) isin (119877
119899times 119910)119897 (4)
Due to the varying characteristics of the error parametersof inertial navigation systems a leading problem that shouldbe overcome is that all the significant parameters with respectto positioning errors should be preprocessed as characteristicquantities for estimation which is considered to be effectiveto improve the estimation accuracy [26] Therefore 15 errorparameters that have considerable impacts on position errorsof SINS are considered for the model training In (4) 119909
119894
represent the error sources of SINS which are the zero biaserrors random walk errors of gyros zero random walk ofaccelerometers random walk errors of accelerometers andscale factor errors of gyros 119910
119894represent navigation errors
which are latitudinal and longitudinal positioning errors ofthe training models Compared with ANNs SVM has adrawback that it can only generate one output while theANNs are able to generate multiple outputs [27] Conse-quently two separate training processes should be separatelyconducted for latitudinal and longitudinal errors Specificallythe error coefficients and positioning errors of the 600 setsof navigation systems are used to train the SVM modelThen a suitable kernel function is selected as radial BasisFunction (RBF) As there are many kernel functions availablefor analyses [28 29] RBF is demonstrated to be effective forthis problem by contrasting with other kernel functions
119870(119909119894 119909119895) = exp(minus
10038171003817100381710038171003817119909119894minus 119909119895
10038171003817100381710038171003817
2
21205902
) (5)
In the next step following convex quadratic program-ming problem (5) is resolved to obtain that 120572(lowast) =
(1205721 120572lowast
1 120572
119897120572lowast
119897)119879 as follows
min 12
119897
sum
119894119895=1
(120572lowast
119894minus 120572119894) (120572lowast
119894minus 120572119894)119870 (119909
119894 119909119894)
+ 120576
119897
sum
119894=1
(120572lowast
119894+ 120572119894) minus
119897
sum
119894=1
119910119894(120572lowast
119894minus 120572119894)
(6)
If 120572119894is picked and then 119887 = 119910
119894minussum119897
119894=1(120572lowast
119894minus 120572119894)119870 + 120576 if 120572lowast
119896
is picked then 119887 = 119910119896minus sum119897
119894=1(120572lowast
119894minus 120572119894)119870 minus 120576
4 Abstract and Applied Analysis
Decision function is established as
119910 = 119892 (119909) =
119897
sum
119894=1
(120572lowast
119894minus 120572119894)119870 (119909
119894 119909) = 119887 (7)
32 PSO-Based Optimal SVR Parameters Selection Impor-tant factors of SVM are the constant 119862 the accuracy parame-ter 120576 and the kernel function As the kernel function has beenselected as RBF appropriate 119862 and 120576 should be selected inorder to increase the estimation accuracy Since itmay take anextremely long time to seek the best parameters and desiredresults may not be achieved by simply making differenttests an efficient method called particle swarm algorithm isadopted for the optimization
Particle swarm algorithm (PSO) is an algorithm thatis inspired by birdsrsquo foraging behavior and widely used toaddress optimization problems [30ndash32] The PSO algorithmis introduced to optimize the parameters of 120576 and 119862 ThePSO is initialized with random particles and then it worksto find optimal parameters by iterative methods Practicallythe initial parameters of 120576 and 119862 should be given and thenthe optimal values will be generated by calculation
4 Error Analyses for SINS
Before predicting the performance of certain inertial naviga-tion systems error analyses for SINS which are indispensablefor the process of error prediction should be conducted Asfor the strapdown navigation systems studied in this paperthe drift errors of the gyroscopes the white noise of thegyroscopes and the drift errors of accelerometers are consid-ered whereas the error coefficients related to acceleration andthe coefficients related to quadratic acceleration are ignoredsince such coefficients are comparatively small
Two criteria should be obeyed for the selection of trainingdata
(1) The training data should be not the same as the datafor test If there were no differences between thetraining data and testing data the prediction accuracywould be relatively high but biased
(2) The dimension of the inputs should be increased ifpossible If the training data had a high dimensionmore useful characteristics could be used for modeltraining
(3) The training data should represent different systemswith vastly different error coefficients With respectto this criterion suitable standard deviation for errorsources should be assigned
The error sources are assigned by the given parameterswhich are listed in Table 1 The navigation system is affectedby multiple error sources when it performs a navigationtask so it is necessary to assign all the error parametersthat are considered in this navigation system in order tomake the results adaptive for real flight situations To makepreparation for the performance prediction of SINS it issensible to carry out error analyses for 600 sets with different
Table 1The expected value and standard deviation of error sources
Error sources Expectedvalue
Standarddeviation
Zero bias errors of gyros 001∘h 0006∘hRandom walk errors of gyros 0001∘radich 00068∘radichAccelerometer zero bias errors 3120583g 05 120583gAccelerometer random walk errors 3120583gradicHz 5120583gradicHzScale factor errors of gyros 10 ppm 0
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
Real longitudinal error (m)
Real
latit
udin
al er
ror (
m)
minus1000
minus500
Figure 2 Real distributing points of positioning errors
error parameters The assignment of different error sourcesabides by the Gaussian distribution Calculating with thesystem error functions the 600 sets of navigation errors ofthe navigation systems with different error coefficients areachieved
41 Simulation Verification of Prediction Method The errorcoefficients and the positioning error parameters of the 600sets of SINS are transformed into characteristic values whichare used to train the SVM model Other 300 sets of errorcoefficients are randomly generated and the correspondingsystem errors are calculated by error equations Then theaccuracy parameter is initially given as 120576 = 05 and thepenalty parameter is given as119862 = 60 PSO is used to generateoptimal parameters of 120576 = 00104 and 119862 = 4912
Both original SVM with fixed parameters and SVMmodel with optimal parameters generated by PSO are used inorder to predict the positioning errors The real distributionpoints of positioning errors are shown in Figure 2 while thepredicting distribution points of positioning error are shownin Figure 3
The prediction errors in terms of latitude and longitudewhich are predicted by SVMmodel with optimal parametersgenerated by PSO are shown in Figure 4
By comparing the results the predicting results are rela-tively satisfying Specifically the average north error is 343mwhile the expected value of north error by prediction is4558m The standard deviation of north error by prediction
Abstract and Applied Analysis 5
Longitudinal prediction error (m)
Latit
udin
al p
redi
ctio
n er
ror (
m)
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
minus1000
minus500
Figure 3 Prediction distributing points of positioning errors
0 50 100 150 200 250 300
0
200
400
minus200Erro
r of l
ongi
tudi
nal
pred
ictio
n (m
)
Number of SINS for test n
(a)
0 50 100 150 200 250 300
0
100
minus100
Erro
r of l
atitu
dina
lpr
edic
tion
(m)
minus200
Number of SINS for test n
(b)
Figure 4 Prediction errors generated by SVM
is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m
The predicting accuracy is defined as 119875 which is accessedwith (7) 119890
119901represents the predicting error and 119890
119903represents
the real error calculated by system error equations Letter 119899represents the number of navigation systems for test
119875 = 1 minus
119899
sum
119894=1
10038161003816100381610038161003816100381610038161003816
119890119901119894minus 119890119903119894
119890119903119894
10038161003816100381610038161003816100381610038161003816
times
1
119899
(8)
After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically
Table 2 Prediction accuracy of different methods
Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
1200
Long
itudi
nal e
rror
(m)
Time (s)
Real longitudinal errorLongitudinal error by estimation
minus200
Figure 5 Longitudinal error of a strapdown system with knownerror parameters
if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS
42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes
One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value
Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Decision function is established as
119910 = 119892 (119909) =
119897
sum
119894=1
(120572lowast
119894minus 120572119894)119870 (119909
119894 119909) = 119887 (7)
32 PSO-Based Optimal SVR Parameters Selection Impor-tant factors of SVM are the constant 119862 the accuracy parame-ter 120576 and the kernel function As the kernel function has beenselected as RBF appropriate 119862 and 120576 should be selected inorder to increase the estimation accuracy Since itmay take anextremely long time to seek the best parameters and desiredresults may not be achieved by simply making differenttests an efficient method called particle swarm algorithm isadopted for the optimization
Particle swarm algorithm (PSO) is an algorithm thatis inspired by birdsrsquo foraging behavior and widely used toaddress optimization problems [30ndash32] The PSO algorithmis introduced to optimize the parameters of 120576 and 119862 ThePSO is initialized with random particles and then it worksto find optimal parameters by iterative methods Practicallythe initial parameters of 120576 and 119862 should be given and thenthe optimal values will be generated by calculation
4 Error Analyses for SINS
Before predicting the performance of certain inertial naviga-tion systems error analyses for SINS which are indispensablefor the process of error prediction should be conducted Asfor the strapdown navigation systems studied in this paperthe drift errors of the gyroscopes the white noise of thegyroscopes and the drift errors of accelerometers are consid-ered whereas the error coefficients related to acceleration andthe coefficients related to quadratic acceleration are ignoredsince such coefficients are comparatively small
Two criteria should be obeyed for the selection of trainingdata
(1) The training data should be not the same as the datafor test If there were no differences between thetraining data and testing data the prediction accuracywould be relatively high but biased
(2) The dimension of the inputs should be increased ifpossible If the training data had a high dimensionmore useful characteristics could be used for modeltraining
(3) The training data should represent different systemswith vastly different error coefficients With respectto this criterion suitable standard deviation for errorsources should be assigned
The error sources are assigned by the given parameterswhich are listed in Table 1 The navigation system is affectedby multiple error sources when it performs a navigationtask so it is necessary to assign all the error parametersthat are considered in this navigation system in order tomake the results adaptive for real flight situations To makepreparation for the performance prediction of SINS it issensible to carry out error analyses for 600 sets with different
Table 1The expected value and standard deviation of error sources
Error sources Expectedvalue
Standarddeviation
Zero bias errors of gyros 001∘h 0006∘hRandom walk errors of gyros 0001∘radich 00068∘radichAccelerometer zero bias errors 3120583g 05 120583gAccelerometer random walk errors 3120583gradicHz 5120583gradicHzScale factor errors of gyros 10 ppm 0
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
Real longitudinal error (m)
Real
latit
udin
al er
ror (
m)
minus1000
minus500
Figure 2 Real distributing points of positioning errors
error parameters The assignment of different error sourcesabides by the Gaussian distribution Calculating with thesystem error functions the 600 sets of navigation errors ofthe navigation systems with different error coefficients areachieved
41 Simulation Verification of Prediction Method The errorcoefficients and the positioning error parameters of the 600sets of SINS are transformed into characteristic values whichare used to train the SVM model Other 300 sets of errorcoefficients are randomly generated and the correspondingsystem errors are calculated by error equations Then theaccuracy parameter is initially given as 120576 = 05 and thepenalty parameter is given as119862 = 60 PSO is used to generateoptimal parameters of 120576 = 00104 and 119862 = 4912
Both original SVM with fixed parameters and SVMmodel with optimal parameters generated by PSO are used inorder to predict the positioning errors The real distributionpoints of positioning errors are shown in Figure 2 while thepredicting distribution points of positioning error are shownin Figure 3
The prediction errors in terms of latitude and longitudewhich are predicted by SVMmodel with optimal parametersgenerated by PSO are shown in Figure 4
By comparing the results the predicting results are rela-tively satisfying Specifically the average north error is 343mwhile the expected value of north error by prediction is4558m The standard deviation of north error by prediction
Abstract and Applied Analysis 5
Longitudinal prediction error (m)
Latit
udin
al p
redi
ctio
n er
ror (
m)
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
minus1000
minus500
Figure 3 Prediction distributing points of positioning errors
0 50 100 150 200 250 300
0
200
400
minus200Erro
r of l
ongi
tudi
nal
pred
ictio
n (m
)
Number of SINS for test n
(a)
0 50 100 150 200 250 300
0
100
minus100
Erro
r of l
atitu
dina
lpr
edic
tion
(m)
minus200
Number of SINS for test n
(b)
Figure 4 Prediction errors generated by SVM
is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m
The predicting accuracy is defined as 119875 which is accessedwith (7) 119890
119901represents the predicting error and 119890
119903represents
the real error calculated by system error equations Letter 119899represents the number of navigation systems for test
119875 = 1 minus
119899
sum
119894=1
10038161003816100381610038161003816100381610038161003816
119890119901119894minus 119890119903119894
119890119903119894
10038161003816100381610038161003816100381610038161003816
times
1
119899
(8)
After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically
Table 2 Prediction accuracy of different methods
Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
1200
Long
itudi
nal e
rror
(m)
Time (s)
Real longitudinal errorLongitudinal error by estimation
minus200
Figure 5 Longitudinal error of a strapdown system with knownerror parameters
if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS
42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes
One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value
Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Longitudinal prediction error (m)
Latit
udin
al p
redi
ctio
n er
ror (
m)
600 800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
minus1000
minus500
Figure 3 Prediction distributing points of positioning errors
0 50 100 150 200 250 300
0
200
400
minus200Erro
r of l
ongi
tudi
nal
pred
ictio
n (m
)
Number of SINS for test n
(a)
0 50 100 150 200 250 300
0
100
minus100
Erro
r of l
atitu
dina
lpr
edic
tion
(m)
minus200
Number of SINS for test n
(b)
Figure 4 Prediction errors generated by SVM
is 457mThe average east error is 964m while the expectedvalue of east error by prediction is 11946m The standarddeviation of east error by prediction is 1037m
The predicting accuracy is defined as 119875 which is accessedwith (7) 119890
119901represents the predicting error and 119890
119903represents
the real error calculated by system error equations Letter 119899represents the number of navigation systems for test
119875 = 1 minus
119899
sum
119894=1
10038161003816100381610038161003816100381610038161003816
119890119901119894minus 119890119903119894
119890119903119894
10038161003816100381610038161003816100381610038161003816
times
1
119899
(8)
After calculation the predicting accuracy is seen inTable 2 the latitudinal predicting accuracy is 9273 whilethe longitudinal predicting accuracy is 9164 The accuracyof PSO-based method is noticeably higher than that oforiginal SVMThere is also a substantial decrease in the timewhich is spent assessing the SINS performance Specifically
Table 2 Prediction accuracy of different methods
Method Latitudinal accuracy Longitudinal accuracyOriginal-SVM 8639 8420PSO-SVM 9273 9164
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
1200
Long
itudi
nal e
rror
(m)
Time (s)
Real longitudinal errorLongitudinal error by estimation
minus200
Figure 5 Longitudinal error of a strapdown system with knownerror parameters
if 100 systems are analyzed the error model based methodtakes 2510 s whereas the SVM-basedmethod only takes 623 sCompared with analysis method based on error model thepredicting method based on SVM is able to save up to 75of calculating time Therefore it is effective and efficient toevaluate the navigation errors of SINS
42 Error Prediction of an Entire Flight Process It is demon-strated that the accuracy of proposed method is satisfyingin terms of positioning error prediction However it is onlyavailable for the prediction with fixed flight time So it isnecessary to seek a solution to conditions with different flighttimes
One strapdown system with known error parameters isanalyzed during 3600 s flight time 100 models with respectto this system are trained by the given parameters onemodel is generated for every 36 s flight So it is achievableto predict the positioning error during the entire flight Thelongitudinal error and latitudinal error are seen in Figures5 and 6 respectively which show that at the beginning ofthe flight when the positioning errors are relatively smallthe prediction error is considerably low and can be ignoredThe reason is that during the initial period the trainingparameters of real positioning errors are small and similarAs a result SVR-based prediction method tends to generateestimation outputs around that value
Although several estimating results with big errors withrespect to both longitudinal and latitudinal estimation aregenerated the overall trends are coherent with real errorcurves and the overall prediction accuracy is satisfyingTherefore the proposed method is demonstrated to be
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Latit
udin
al er
ror (
m)
Time (s)
Real latitudinal errorLatitudinal error by estimation
minus200
Figure 6 Latitudinal error of a strapdown systemwith known errorparameters
effective for error prediction of an entire flight process withmedium accuracy
5 Conclusion
An SVM-based predicting method for SINS positioningerrors is proposed which can be used to assess the perfor-mance of navigation systems Error functions of strapdownnavigation systems are established to provide necessary errorparameters which are not only used to train SVM model butalso utilized to make comparisons with the predicting resultsof extra systems RBF is selected to be the kernel function ofSVM and appropriate parameters of SVM are generated byPSO method
As shown in the numerical verifications the proposedprediction method is effective in terms of predicting nav-igation errors of strapdown systems with different errorparameters The accuracy of latitudinal prediction can reach9273 while the accuracy with respect to longitudinalprediction is 9164 which is considered to be high enoughfor application In addition this method compared witherror model analysis can save up to 75 of calculation timeFinally the proposed method is demonstrated to be effectivefor error prediction for an entire flight process which makesthe method more applicable
Therefore it enables the researchers to choose appropriatesystems for different trajectories or applications by assessingnavigation errors efficiently Since this method is able toevaluate the positioning errors precisely by assessing errorparameters of inertial measurement units it will be usefulin terms of error compensation of strapdown navigationsystems which are equipped on different kinds of aircrafts
References
[1] Y Wu X Hu M Wu and D Hu ldquoStrapdown inertial nav-igation using dual quaternion algebra error analysisrdquo IEEE
Transactions on Aerospace and Electronic Systems vol 42 no1 pp 259ndash266 2006
[2] Y Qin Inertial Navigation Science Press Beijing China 2009[3] X He W Wang and J Huang ldquoCharacteristics of gyro
error propagation on FOG-SINSrdquo Journal of Chinese InertialTechnology vol 15 no 4 pp 407ndash411 2007
[4] J S Stambaugh ldquoPropagation and system accuracy impact ofmajor sensor errors on a strapdown aircraft navigatorrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 6pp 838ndash846 1973
[5] F Sun Y-Y Ben and W Gao ldquoApplication of spiral theory instrapdown inertial navigation algorithmrdquo Systems Engineeringand Electronics vol 29 no 9 pp 1532ndash1535 2007
[6] Y Hao J Gong W Gao and L Li ldquoResearch on the dynamicerror of strapdown inertial navigation systemrdquo in Proceedingsof the IEEE International Conference on Mechatronics andAutomation (ICMA rsquo08) pp 814ndash819 August 2008
[7] F Gomez-Estern and F Gordillo ldquoError analysis in strapdownINS for aircraft assembly linesrdquo in Proceedings of the 10thInternational Conference on Control Automation Robotics andVision (ICARCV rsquo08) pp 184ndash189 Hanoi Vietnam December2008
[8] W Gao B Cao Y Ben and B Xu ldquoAnalysis of gyrorsquos slope driftaffecting inertial navigation system errorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 3757ndash3762 Changchun China August 2009
[9] H Musoff and J H Murphy ldquoStudy of strapdown navigationattitude algorithmsrdquo Journal of Guidance Control and Dynam-ics vol 18 no 2 pp 287ndash290 1995
[10] J Wang and H Gu ldquoCompensation algorithm of device errorfor rate strapdown inertial navigation systemrdquo in Proceedingsof the 1st International Conference on Intelligent Networks andIntelligent Systems (ICINIS rsquo08) pp 667ndash670 Wuhan ChinaNovember 2008
[11] Y-HQiao Y Liu B-K Su andMZeng ldquoTestmethod for errormodel coefficients of pendulous integrating gyro accelerometeron centrifugerdquo Journal of Astronautics vol 28 no 4 pp 854ndash931 2007
[12] T L Chen Estimate at completion for construction projectsusing evolutionary fuzzy neural inference model [MS thesis]Department of Construction Engineering National TaiwanUniversity of Science and Technology Taipei Taiwan 2008
[13] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 4 pp85ndash105 2002
[14] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packsmdashpart 2 modelingand identificationrdquo Journal of Power Sources vol 134 no 2 pp262ndash276 2004
[15] L Zhang ldquoHinfinestimation for discrete-time piecewise homoge-
neous Markov jump linear systemsrdquo Automatica vol 45 no 11pp 2570ndash2576 2009
[16] L Zhang and E-K Boukas ldquoStability and stabilization ofMarkovian jump linear systems with partly unknown transitionprobabilitiesrdquo Automatica vol 45 no 2 pp 463ndash468 2009
[17] L Zhang and E-K Boukas ldquoMode-dependent Hinfin
filteringfor discrete-time Markovian jump linear systems with partlyunknown transition probabilitiesrdquo Automatica vol 45 no 6pp 1462ndash1467 2009
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
[18] L Zhang P Shi E-K Boukas and C Wang ldquoHinfin
modelreduction for uncertain switched linear discrete-time systemsrdquoAutomatica vol 44 no 11 pp 2944ndash2949 2008
[19] L Zhang E-K Boukas and A Haidar ldquoDelay-range-dependent control synthesis for time-delay systems withactuator saturationrdquo Automatica vol 44 no 10 pp 2691ndash26952008
[20] S R Gunn M Brown and K M Bossley ldquoNetwork perfor-mance assessment for neurofuzzy data modelingrdquo Advances inIntelligent Data Analysis Reasoning About Data vol 1280 pp313ndash323 1997
[21] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[22] J Nong ldquoParameters selection and noise estimation of SVMregressionrdquo in Proceedings of the 5th International Joint Confer-ence on Computational Sciences and Optimization pp 379ndash381Harbin China 2012
[23] K-W Yan ldquoStudy on the forecast of air passenger flow basedon SVM regression algorithmrdquo in Proceedings of the 1st Inter-national Workshop on Database Technology and Applications(DBTA rsquo09) pp 325ndash328 Wuhan China April 2009
[24] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998
[25] T Hansen and C-J Wang ldquoSupport vector based battery stateof charge estimatorrdquo Journal of Power Sources vol 141 no 2 pp351ndash358 2005
[26] J C Alvarez Anton P J Garcia Nieto C Blanco Viejo and JA Vilan Vilan ldquoSupport vector machines used to estimate thebattery state of chargerdquo IEEE Transactions on Power Electronicsvol 28 no 12 pp 5919ndash5926 2013
[27] I Steinwart D Hush and C Scovel ldquoAn explicit descriptionof the reproducing Kernel Hilbert spaces of Gaussian RBFkernelsrdquo IEEE Transactions on Information Theory vol 52 no10 pp 4635ndash4643 2006
[28] Y-J Oyang S-C Hwang Y-Y Ou C-Y Chen and Z-WChen ldquoData classification with radial basis function networksbased on a novel kernel density estimation algorithmrdquo IEEETransactions on Neural Networks vol 16 no 1 pp 225ndash2362005
[29] G F Smits and EM Jordaan ldquoImproved SVM regression usingmixtures of kernelsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo02) pp 2785ndash2790Honolulu Hawaii USA May 2002
[30] H Wang Z Hu M Hu and Z Zhang ldquoShort-term predictionof wind farm power based on PSO-SVMrdquo in Proceedings of thePower and Energy Engineering Conference pp 1ndash4 ShanghaiChina 2012
[31] Y Bazi and F Melgani ldquoSemisupervised PSO-SVM regressionfor biophysical parameter estimationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 45 no 6 pp 1887ndash18952007
[32] M Pastorino andA Randazzo ldquoThe SVM-based smart antennafor estimation of the directions of arrival of electromagneticwavesrdquo IEEE Transactions on Instrumentation and Measure-ment vol 55 no 6 pp 1918ndash1925 2006
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
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
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of