Research Article A Filtering Algorithm for Maneuvering Target ...Model (IMM) Algorithm e IMM algorithm has become a standard tool for maneu-vering target tracking [ , , ]. It has a

Research ArticleA Filtering Algorithm for Maneuvering Target TrackingBased on Smoothing Spline Fitting

Yunfeng Liu1 Jidong Suo1 Hamid Reza Karimi2 and Xiaoming Liu1

1 College of Information Science and Technology Dalian Maritime University Dalian 116026 China2Department of Engineering Faculty of Technology and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Yunfeng Liu jzedulyfgmailcom

Received 15 October 2013 Revised 14 December 2013 Accepted 20 December 2013 Published 6 February 2014

Academic Editor Zexu Zhang

Copyright copy 2014 Yunfeng Liu 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

Maneuvering target tracking is a challenge Targetrsquos sudden speed or direction changing would make the common filtering trackerdivergence To improve the accuracy ofmaneuvering target tracking we propose a tracking algorithm based on spline fitting Curvefitting based on historical point trace reflects the mobility information The innovation of this paper is assuming that there is nodynamic motion model and prediction is only based on the curve fitting over the measured data Monte Carlo simulation resultsshow that when sea targets are maneuvering the proposed algorithm has better accuracy than the conventional Kalman filteralgorithm and the interactive multiple model filtering algorithm maintaining simple structure and small amount of storage

1 Introduction

In modern society radar plays an increasingly importantrole where the sea target detection and tracking is a veryimportant aspect and has great practical value Low-speedobjects such as sea ice and others bring great danger tothe ship sailing on the sea and the performance of seatarget detection directly affects the safety of navigationTracking is an important part of radar system Especiallythe maneuvering target tracking is more complex in clutterenvironment

In conventional target tracking the most commonmethod is the standard Kalman filter algorithm [1]When thetarget state and observation equations are linear equationsand the observation noise is Gaussian white noise theKalman filter algorithm is the optimal algorithm in thesense of minimum variance When the target maneuveroccurs Kalman filtering accuracy will be reduced or evendiverged due to themodelmismatch and noise characteristicsthat cannot be known exactly In view of the limitationsof the Kalman filter algorithm the multimodel filteringalgorithm [2ndash8] and the adaptive Kalman filter algorithm [9ndash12] emerged The multiple-model algorithm is the algorithmusing the weighted sum of multiple models to approximate

the actual motion model of the target Subsequently theinteracting multiple model algorithm emerged [5 6] It isthe fusion result of multimodel by updating probability ofmodels It has better performance than multimodel filteringalgorithm

During maneuvering target maneuver has still regularitywhich is reflected in the measured values and cannot beproperly represented by the Kalman filter algorithm There-fore this paper proposes a filtering algorithm based on thesmoothing spline fittingThe proposed algorithmfits throughthe measured values reflecting the movement regularityThe difference between the proposed algorithm and thestandard Kalman filter algorithm is that the latter calculatesestimations through the sum of the predicted values andthe measured values weighted and the former algorithm fitsvalues individually from the measured values to determinethe estimations

The paper is organized as follows In Section 2 Kalmanfilter algorithm is proposed In Sections 3 and 4 interactingmultiple model algorithm and spline fitting filtering algo-rithm are introduced respectively In Section 5 simulationresults are given to illustrate the effectiveness of the proposedalgorithms Finally concluding remarks are provided inSection 6

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 127643 6 pageshttpdxdoiorg1011552014127643

2 Abstract and Applied Analysis

Model

Sum of weights

Weights Covariance update

Measurements Estimations

Initial covariance

Figure 1 The Kalman filter algorithm flowchart

Interact

Model 1filter

Model 2filter

Model N filter

Model probability

update

Sum of state

estimation

p(ij)

Z(k)

ΛNk

Λ1k

Λ2k

middot middot middot




X1(k|k) X2

(k|k) XN(k|k) X(k|k)u(k)

Xo1(kminus1|kminus1) Xo2

(kminus1|kminus1)

X1(kminus1|kminus1) X2

(kminus1|kminus1) XN(kminus1|kminus1)

u(kminus1|kminus1)

XoN(kminus1|kminus1)

Figure 2 The IMM algorithm flowchart

2 Kalman Filtering (KF) Algorithm

The Kalman filtering (KF) flowchart is shown in Figure 1As can be seen from Figure 1 the weighted predicted

values and the weighted measured values determine theestimate value of theKF the predicted value is generated fromthe motion model However a single model cannot meet allthe maneuvering of the target the covariance plays a linkbetween the model and the weighting When filtering theweighting of the measured value tends to be a constant valueand contribution to the estimate becomes smaller So the KFalgorithm cannot escape from the shackles of a single model

Application of the KF algorithm to real-time problemdepends on a priori knowledge of the mean square errorof the state and the observation process [13] Moreover theoptimal estimation of covariance matrix for the state andobservation processes does not exist In order to solve thisproblem some research proposed suboptimal solutions theyestimate one or more parameters of mean variance matrix

of the state and the observe process [14ndash16] or estimate thegain adaptively [17 18] Also the authors [19 20] proposedalgorithmswith the divergence criterion and forgetting factorpreventing filter divergence but the divergence criterioncannot reflect filter divergence exactly

The KF algorithm uses a single model which cannotmeet every maneuver only suitable for the steady situationWhen the model does not suit the motion model the gaincoefficient or the covariance will be adjusted to remedythe motion model These methods do not get rid of thecomputing framework of the Kalman system nothing morethan a feedback loop [21]

3 Interacting MultipleModel (IMM) Algorithm

The IMM algorithm has become a standard tool for maneu-vering target tracking [5 6 22] It has a recursive algorithmstructure which using finite state model represents the entiresystem to try to solve the problem of a single target motionmodel mismatch The IMM approximates nonlinear systemsby finite linear model and estimates the system state by thesum of weighted estimation of models The IMM requiresmodels to represent the entire state of the system usually thisis difficult to achieve

While maneuvering occurs the accuracy of the IMM issuperior to the KF However the shortcomings of the IMMare more models more calculation the competition betweenmodels All of these affect performance of the IMMThe IMMhas multiple models to remedy the motion model when themodel does not suit the motion model The similarity of thestandard Kalman filter algorithm and the IMM algorithm isusing motion model to predict movement position filteringresults are decided by the weighted values of predicted andmeasured When maneuvering the measured values reflectbetter regularity of motion but the weight of the measuredvalues does not increase more in both algorithms

The IMM flowchart is shown in Figure 2Where

119883(119896|119896) represents the output estimate of time119896 119885(119896) represents the measurement data of time 119896 Λ119895

119896

represents the possibility vector of model 119895 119906(119896) representsthemodel probability vector 119901(119894119895) represents transition prob-ability for model 119894 to model 119895 119883119895

(119896|119896)represents state estimate

of model 119895 at time 119896 and

119883

119900119895

(119896minus1|119896minus1)is the result of the

interaction of 119883119895(119896minus1|119896minus1)

(119895 = 1 2 119873)As can be seen from Figure 2 IMM algorithm has four

steps

Step 1 (input interact) At time 119896 minus 1 set state estimation andcovariance matrix value are 119883119895

(119896minus1|119896minus1)and 119875119895(119896 minus 1119896 minus 1) 119895 =

1 2 119873After the interaction at time 119896 the input of filter is

119883

119900119895

(119896minus1|119896minus1)=

119873

sum

119894=1

119883

119894

(119896minus1|119896minus1)119906(119896minus1|119896minus1) (119894 | 119895)

Abstract and Applied Analysis 3

119901

119900119895(119896 minus 1 | 119896 minus 1)

=

119873

sum

119894=1

119901

119894(119896 minus 1 | 119896 minus 1) + [

119883

119894

(119896minus1|119896minus1) minus

119883

119900119895

(119896minus1|119896minus1)]

times [

119883

119894


119883

119900119895

(119896minus1|119896minus1)]

119879 119906(119896minus1|119896minus1) (119894 | 119895)

(1)

where

119906119896minus1|119896minus1 (119894 | 119895) =

119875(119894119895)119906119896minus1 (119894)

119862119895

119862119895 =

119873

sum

119894=1

119901(119894119895)119906119896minus1 (119894)

(2)

Step 2 (filtering predict) The first step result is the modelinput at time 119896 We get the output from KF filter as follows

119883

119895

(119896|119896) 119901

119895(119896 | 119896) (3)

Step 3 (probability update) Update probability of model 119899 asfollows

119906119896 (119895) =

Λ

119895

119896119862119895

119862

(4)

where

119862 =

119873

sum

119894=1

Λ

119894

119896119862119894 (5)

Step 4 (data fusion) The interact outputs at time 119896 will beobtained respectively after getting output and probability ofevery model

119883 (119896 | 119896) =

119873

sum

119894=1

119883

119894(119896 | 119896) 119906119896 (119894)

119875 (119896 | 119896)

=

119873

sum

119894=1

119906119896 (119894) (119875119894(119896 | 119896) + (

119883

119894(119896 | 119896) minus

119883 (119896 | 119896))

times (

119883

119894(119896 | 119896) minus

119883(119896 | 119896))

119879)

(6)

where 119875(119896 | 119896) is the state covarianceThen we perform the first interactive multiple model

algorithm recursive process

4 Spline Fitting Filtering (SFF) Algorithm

The KF algorithm requires the system model to be linearwhich greatly limits the scope of application of the KFalgorithm The IMM increases the number of model to closethe motion model And predicted and measured weightedvalues decide the estimate values in both algorithms When

Smoothing spline

Measurements

Estimations

Interpolation

Average

filter

Figure 3 The SFF algorithm flowchart

the target maneuver occurs the predicted value cannotimmediately react because both algorithms did not accuratelyand rapidly adjust the weight of the measured values How-ever the measured values can give faster and more sensitiveresponse To improve on this the paper proposes in thefiltering process the measured value that contains the trueinformation of the motion The measured value reflects theregularity of target maneuvering more promptly than thepredicted value and the regularity reflects from a certainsmoothing of the curve Therefore the estimated value of thefilter should be closely linked to the measured value Whensea targets maneuver they still have regularity of motiondespite the measurement errors and systematic errors Thecubic spline is used to approximate the mathematical modelof targets maneuvering and the smoothing spline functionreflects the regularity of maneuvering Therefore the novelpoint of SFF is the using of spline function to reflect seatargetsmaneuvering abandonment of themotionmodelTheadvantage of SFF can be seen in Figures 4 and 5 The SFFis closer to the true value than the KF and IMM When thetarget maneuver occurs the error of the SFF is the smallestone in three algorithms

Assumed measured point (119909(119895) 119910(119895)) 119895 = 1 2 here119891 is a cubic spline functionThe problem is to find a function119891 that minimizes

120588

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119910 (119895) minus 119891 (119909119895)1003816

1003816

1003816

1003816

1003816

2+ (1 minus 120588)int

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119889

2119891 (119909)

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

119889119909 (7)

where 119909(119895) 119910(119895) represent the horizontal and vertical coor-dinates of the measured value respectively 119899 representsthe number of nodes 1198892119891(119909)1198891199092 represents the secondderivative of119891(119909) 120588 is a smoothing parameter and 120588 isin [0 1]When 120588 = 0119891(sdot) is a spline function to fit the data in the leastsquares sense when 120588 = 1119891 is the cubic spline interpolationFormula (7) is a least-squares problem where the first termquantifies the error between the measured data points 119910(119895)and themodel119891(119909119895)The second term imposes a smoothness


6000

6500

7000

7500

8000

times104

5500

TrueMeasuringKalman

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

Figure 4 Point track

constraint on the solution In practice we set 120588 = 08 forbetter reflecting the regularity of maneuvering by smoothingspline curve

The SFF algorithm is as follows suppose the measuredvalue is 119911(119899) = (119909(119899) 119910(119899))

First mean the measured values 119911(119899 minus 1) = (119911(119899 minus 2) +

119911(119899))2 (119899 ge 3) Second the filtering estimate results arefitted by the spline function from 119899 filtered measured valuesSo the cycle continues Taking into account the experimentalresults and the complexity of the algorithm three points areused for the spline fitting

The SFF flowchart is shown in Figure 3

5 Simulation Results

The simulation environment is Intel CPU 253GHz 1 GBMemory and MATLAB R2009b Assume sea target depar-tures at coordinates (8000m and 8000m)

(1) 0ndash49 s the target uniform linear motion (30ms at 119909-axis direction 40ms at 119910 axis direction)

(2) 50ndash90 s the target turning movement at the turn rate3∘s 50ms speed

(3) 91ndash119 s the target uniform linear motion at 50msspeed

(4) 120ndash160 s the target turning movement at the turnrate 3∘s 50ms speed


The results of three filtering algorithms are shown inFigure 4 And the results of Monte Carlo of 100 times aredepicted in Figures 5(a) 5(b) and 5(c)

As can be seen fromFigure 4 theKFmotionmodelmeetsthe target motion model during the uniform linear motionand filtering results converge as other periods the filteringresults of theKFwill increase errors or divergenceThe reasonis the weight of the measured values is much smaller that

cannot influence the filtering results The filtering resultsdetermined by the weighted sum of the prediction values andthe measured values As the target maneuvers the algorithmmodel mismatches the uniform linear motionmodel and thepredictive value is still the main content of the filtering valueIt cannot keep up with the change of motion which leads toincreasing error and even divergence For the same set of datathe IMM has three corresponding models

The filtering result of the IMM is better than the KFduring the turningmovement but the filter effect of the IMMis reduced or even diverged as the IMM model mismatchThe SFF filtering effects is the same with the IMM Whenthe target maneuvers the SFF is superior to the formeralgorithms

As can be seen from Figure 5(a) the filtering trajectoryof the SFF is closer to the true value than the KF and IMMwhen the target maneuver occurs the average error of theSFF is smaller than that of the KF and IMM as can be seenfrom Figure 5(b) from Figure 5(c) there is a large fluctuationof standard deviation for the KF and IMM as target motionwhile the standard deviation of the SFF has small change

When selecting the type of steering ratio the conclusionis consistent and uniform

In the three algorithms the KF is optimal in uniformlinear motion when its model meets the motion model Onthe other movement the accuracy of the SFF is better thanthe accuracy of the KF and IMM because the impact of themeasured value of the SFF is more prominent in filtering andmeasurements reflect the movement regularity in time

6 Conclusion

The paper proposed a target tracking filtering algorithmbased on spline function fitting the measured value Forthe same set of data the estimation error of the proposedalgorithm compares with the estimation error of KF algo-rithm and IMM algorithm Monte Carlo results clearly showthat the SFF has higher estimation accuracy Particularlyit has obvious advantages to the standard Kalman filter


6000

6500

7000

7500

8000

8500

times104

5500

TrueMeasuringKF

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

(a)

0 20 40 60 80 100 120 140 160 180 2005

10152025303540455055

Time (s)

Mea

n er

ror (

m)

KFIMMSFF

(b)

Time (s)

4

6

8

10

12

14

16

18

20

RMS

erro

r (m

)

KFIMMSFF

0 20 40 60 80 100 120 140 160 180 200

(c)

Figure 5 (a) Average point track (b) Mean error (c) Root-mean-square error

algorithmandmaintains a simple structure and small amountof storage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author likes to thank Professor Jidong Suo and ProfessorXiaoming Liu for their useful support throughout this workThe authors are grateful to their collaborators for theirconstructive comments on the paper

References

[1] R Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 pp 35ndash45 1960

[2] D T Magill ldquoOptimal adaptive estimation of sampled stochas-tic processesrdquo IEEE Transactions on Automatic Control vol 10no 4 pp 434ndash439 1965

[3] D G Lainiotis ldquoOptimal adaptive estimation structure andparameter adaptationrdquo IEEE Transactions on Automatic Con-trol vol 16 no 2 pp 160ndash170 1971

[4] H Blom ldquoAn efficient filter for abruptly changing systemsrdquoin Proceedings of the 23rd IEEE Conference on Decision andControl pp 656ndash658 1984

[5] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[6] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[7] X R Li ldquoMultiple-model estimation with variable structuresome theoretical considerationsrdquo inProceedings of the 33rd IEEEConference on Decision and Control pp 1199ndash1204 December1994

[8] Y Wan S Y Wang and X Qin ldquoIMM iterated extended 119867infin

particle filter algorithmrdquoMathematical Problems in Engineeringvol 2013 Article ID 970158 8 pages 2013

[9] R E Kalman and R S Bucy ldquoNew results in linear filtering andprediction theoryrdquo Journal of Basic Engineering vol 83 pp 95ndash108 1961

[10] S Sage and G Husa ldquoAdaptive filtering with unkown priorstatisticsrdquo in Proceedings of the Joint Automatic Control Confer-ence pp 760ndash769 1969

[11] R K Mehra ldquoApproaches to adaptive filteringrdquo IEEE Transac-tions on Automatic Control vol 17 no 5 pp 693ndash698 1972

[12] XKanH S Shu andYChe ldquoAsymptotic parameter estimationfor a class of linear stochastic systems using Kelman-Bucyfilteringrdquo Mathematical Problems in Engineering vol 2012Article ID 342705 15 pages 2012

[13] Z Ma ldquoSpacecraft attitude determination by adaptive Kalmanfilteringrdquo in Proceedings of the 23rd Chinese Control Conferencepp 255ndash259 2004

[14] L Weiss ldquoA survey of discrete Kalman-Bucy filtering withunknown noise covariancerdquo in Proceedings of the Control andFlight Mechanics Conference pp 17ndash19 1970


[15] E Lichtfuss Non-GPS Navigation Using Vision-Aiding andActive Radio Range Measurements Department of the AirForce Air University Air Force Institute of TechnologyWright-Patterson Air Force Base Wright-Patterson Ohio USA 2011

[16] W Su C Huang P Liu and M Ma ldquoApplication of adaptiveKalman filter technique in initial alignment of inertial naviga-tion systemrdquo Journal of Chinese Inertial Technology vol 18 no1 pp 44ndash47 2010

[17] H Fu YWu and T Lou ldquoAdaptive unscented incremental filtermethodrdquo Journal of Aerospace Power vol 28 no 2 pp 259ndash2632013

[18] KW Chiang C Lin andK Y Peng ldquoThe performance analysisof an AKF based tightly coupled INSGNSS sensor fusionscheme with non-holonomic constraints for land vehicularapplicationsrdquo Innovation for Applied Science and Technologyvol 284ndash287 pp 1956ndash1960 2012

[19] Q Liu and L Chen ldquoA new compensation method for filteringdivergence caused by model errorsrdquo Command Control andSimulation vol 34 no 3 pp 95ndash101 2012

[20] S Xu X Lin and D Zhao ldquoStrong tracking SRCKF and itsapplication in vessel dynamic positioningrdquo Chinese Journal ofScientific Instrument vol 34 no 6 pp 1266ndash1272 2013

[21] J Wang H Shui and H Ma ldquoImplementation framework offilters in Kalman structurerdquo Journal of Data Acquisition andProcessing vol 24 no 1 pp 61ndash66 2009

[22] X R Li and V P Jilkov ldquoSurvey of maneuvering targettrackingmdashpart V multiple-model methodsrdquo IEEE Transactionson Aerospace and Electronic Systems vol 41 no 4 pp 1255ndash13212005

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Model

Sum of weights

Weights Covariance update

Measurements Estimations

Initial covariance

Figure 1 The Kalman filter algorithm flowchart

Interact

Model 1filter

Model 2filter

Model N filter

Model probability

update

Sum of state

estimation

p(ij)

Z(k)

ΛNk

Λ1k

Λ2k





X1(k|k) X2

(k|k) XN(k|k) X(k|k)u(k)

Xo1(kminus1|kminus1) Xo2

(kminus1|kminus1)

X1(kminus1|kminus1) X2

(kminus1|kminus1) XN(kminus1|kminus1)

u(kminus1|kminus1)

XoN(kminus1|kminus1)

Figure 2 The IMM algorithm flowchart

2 Kalman Filtering (KF) Algorithm

The Kalman filtering (KF) flowchart is shown in Figure 1As can be seen from Figure 1 the weighted predicted

values and the weighted measured values determine theestimate value of theKF the predicted value is generated fromthe motion model However a single model cannot meet allthe maneuvering of the target the covariance plays a linkbetween the model and the weighting When filtering theweighting of the measured value tends to be a constant valueand contribution to the estimate becomes smaller So the KFalgorithm cannot escape from the shackles of a single model

Application of the KF algorithm to real-time problemdepends on a priori knowledge of the mean square errorof the state and the observation process [13] Moreover theoptimal estimation of covariance matrix for the state andobservation processes does not exist In order to solve thisproblem some research proposed suboptimal solutions theyestimate one or more parameters of mean variance matrix

of the state and the observe process [14ndash16] or estimate thegain adaptively [17 18] Also the authors [19 20] proposedalgorithmswith the divergence criterion and forgetting factorpreventing filter divergence but the divergence criterioncannot reflect filter divergence exactly

The KF algorithm uses a single model which cannotmeet every maneuver only suitable for the steady situationWhen the model does not suit the motion model the gaincoefficient or the covariance will be adjusted to remedythe motion model These methods do not get rid of thecomputing framework of the Kalman system nothing morethan a feedback loop [21]

3 Interacting MultipleModel (IMM) Algorithm

The IMM algorithm has become a standard tool for maneu-vering target tracking [5 6 22] It has a recursive algorithmstructure which using finite state model represents the entiresystem to try to solve the problem of a single target motionmodel mismatch The IMM approximates nonlinear systemsby finite linear model and estimates the system state by thesum of weighted estimation of models The IMM requiresmodels to represent the entire state of the system usually thisis difficult to achieve

While maneuvering occurs the accuracy of the IMM issuperior to the KF However the shortcomings of the IMMare more models more calculation the competition betweenmodels All of these affect performance of the IMMThe IMMhas multiple models to remedy the motion model when themodel does not suit the motion model The similarity of thestandard Kalman filter algorithm and the IMM algorithm isusing motion model to predict movement position filteringresults are decided by the weighted values of predicted andmeasured When maneuvering the measured values reflectbetter regularity of motion but the weight of the measuredvalues does not increase more in both algorithms

The IMM flowchart is shown in Figure 2Where

119883(119896|119896) represents the output estimate of time119896 119885(119896) represents the measurement data of time 119896 Λ119895

119896

represents the possibility vector of model 119895 119906(119896) representsthemodel probability vector 119901(119894119895) represents transition prob-ability for model 119894 to model 119895 119883119895

(119896|119896)represents state estimate

of model 119895 at time 119896 and

119883

119900119895

(119896minus1|119896minus1)is the result of the

interaction of 119883119895(119896minus1|119896minus1)

(119895 = 1 2 119873)As can be seen from Figure 2 IMM algorithm has four

steps

Step 1 (input interact) At time 119896 minus 1 set state estimation andcovariance matrix value are 119883119895

(119896minus1|119896minus1)and 119875119895(119896 minus 1119896 minus 1) 119895 =

1 2 119873After the interaction at time 119896 the input of filter is

119883

119900119895

(119896minus1|119896minus1)=

119873

sum

119894=1

119883

119894

(119896minus1|119896minus1)119906(119896minus1|119896minus1) (119894 | 119895)


119901

119900119895(119896 minus 1 | 119896 minus 1)

=

119873

sum

119894=1

119901

119894(119896 minus 1 | 119896 minus 1) + [

119883

119894


119883

119900119895

(119896minus1|119896minus1)]

times [

119883

119894


119883

119900119895

(119896minus1|119896minus1)]

119879 119906(119896minus1|119896minus1) (119894 | 119895)

(1)

where

119906119896minus1|119896minus1 (119894 | 119895) =

119875(119894119895)119906119896minus1 (119894)

119862119895

119862119895 =

119873

sum

119894=1

119901(119894119895)119906119896minus1 (119894)

(2)


119883

119895

(119896|119896) 119901

119895(119896 | 119896) (3)


119906119896 (119895) =

Λ

119895

119896119862119895

119862

(4)

where

119862 =

119873

sum

119894=1

Λ

119894

119896119862119894 (5)


119883 (119896 | 119896) =

119873

sum

119894=1

119883

119894(119896 | 119896) 119906119896 (119894)

119875 (119896 | 119896)

=

119873

sum

119894=1

119906119896 (119894) (119875119894(119896 | 119896) + (

119883

119894(119896 | 119896) minus

119883 (119896 | 119896))

times (

119883

119894(119896 | 119896) minus

119883(119896 | 119896))

119879)

(6)





Smoothing spline

Measurements

Estimations

Interpolation

Average

filter




120588

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119910 (119895) minus 119891 (119909119895)1003816

1003816

1003816

1003816

1003816

2+ (1 minus 120588)int

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119889

2119891 (119909)

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

119889119909 (7)



6000

6500

7000

7500

8000

times104

5500

TrueMeasuringKalman

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)





















6 Conclusion



6000

6500

7000

7500

8000

8500

times104

5500

TrueMeasuringKF

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

(a)

0 20 40 60 80 100 120 140 160 180 2005

10152025303540455055

Time (s)

Mea

n er

ror (

m)

KFIMMSFF

(b)

Time (s)

4

6

8

10

12

14

16

18

20

RMS

erro

r (m

)

KFIMMSFF

0 20 40 60 80 100 120 140 160 180 200

(c)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119901

119900119895(119896 minus 1 | 119896 minus 1)

=

119873

sum

119894=1

119901

119894(119896 minus 1 | 119896 minus 1) + [

119883

119894


119883

119900119895

(119896minus1|119896minus1)]

times [

119883

119894


119883

119900119895

(119896minus1|119896minus1)]

119879 119906(119896minus1|119896minus1) (119894 | 119895)

(1)

where

119906119896minus1|119896minus1 (119894 | 119895) =

119875(119894119895)119906119896minus1 (119894)

119862119895

119862119895 =

119873

sum

119894=1

119901(119894119895)119906119896minus1 (119894)

(2)


119883

119895

(119896|119896) 119901

119895(119896 | 119896) (3)


119906119896 (119895) =

Λ

119895

119896119862119895

119862

(4)

where

119862 =

119873

sum

119894=1

Λ

119894

119896119862119894 (5)


119883 (119896 | 119896) =

119873

sum

119894=1

119883

119894(119896 | 119896) 119906119896 (119894)

119875 (119896 | 119896)

=

119873

sum

119894=1

119906119896 (119894) (119875119894(119896 | 119896) + (

119883

119894(119896 | 119896) minus

119883 (119896 | 119896))

times (

119883

119894(119896 | 119896) minus

119883(119896 | 119896))

119879)

(6)





Smoothing spline

Measurements

Estimations

Interpolation

Average

filter




120588

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119910 (119895) minus 119891 (119909119895)1003816

1003816

1003816

1003816

1003816

2+ (1 minus 120588)int

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119889

2119891 (119909)

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

119889119909 (7)



6000

6500

7000

7500

8000

times104

5500

TrueMeasuringKalman

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)





















6 Conclusion



6000

6500

7000

7500

8000

8500

times104

5500

TrueMeasuringKF

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

(a)

0 20 40 60 80 100 120 140 160 180 2005

10152025303540455055

Time (s)

Mea

n er

ror (

m)

KFIMMSFF

(b)

Time (s)

4

6

8

10

12

14

16

18

20

RMS

erro

r (m

)

KFIMMSFF

0 20 40 60 80 100 120 140 160 180 200

(c)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6000

6500

7000

7500

8000

times104

5500

TrueMeasuringKalman

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)





















6 Conclusion



6000

6500

7000

7500

8000

8500

times104

5500

TrueMeasuringKF

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

(a)

0 20 40 60 80 100 120 140 160 180 2005

10152025303540455055

Time (s)

Mea

n er

ror (

m)

KFIMMSFF

(b)

Time (s)

4

6

8

10

12

14

16

18

20

RMS

erro

r (m

)

KFIMMSFF

0 20 40 60 80 100 120 140 160 180 200

(c)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6000

6500

7000

7500

8000

8500

times104

5500

TrueMeasuringKF

IMMSFF

08 09 1 11 12 13 14 15Xcoordinate (m)

Yco

ordi

nate

(m)

(a)

0 20 40 60 80 100 120 140 160 180 2005

10152025303540455055

Time (s)

Mea

n er

ror (

m)

KFIMMSFF

(b)

Time (s)

4

6

8

10

12

14

16

18

20

RMS

erro

r (m

)

KFIMMSFF

0 20 40 60 80 100 120 140 160 180 200

(c)





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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