Research Article An Enhanced UWB-Based Range/GPS ...downloads.hindawi.com/journals/mpe/2015/843719.pdf · suitable for vehicle positioning. A Doppler-shi -based CP method for VANETs

Research ArticleAn Enhanced UWB-Based RangeGPS CooperativePositioning Approach Using Adaptive Variational BayesianCubature Kalman Filtering

Feng Shen and Guanghui Xu

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Feng Shen harbinsfgmailcom

Received 22 October 2014 Revised 23 December 2014 Accepted 26 December 2014

Academic Editor Hung Nguyen-Xuan

Copyright copy 2015 F Shen and G Xu 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

Precise position awareness is a fundamental requirement for advanced applications of emerging intelligent transportation systemssuch as collision warning and speed advisory system However the achievable level of positioning accuracy using global navigationsatellite systems does not meet the requirements of these applications Fortunately cooperative positioning (CP) techniques canimprove the performance of positioning in a vehicular ad hoc network (VANET) through sharing the positions between vehiclesIn this paper a novel enhanced CP technique is presented by combining additional range-ultra-wide bandwidth- (UWB-) basedmeasurements Furthermore an adaptive variational Bayesian cubature Kalman filtering (AVBCKF) algorithm is proposed andused in the enhanced CP method which can add robustness to the time-variant measurement noise Based on analytical andexperimental results the proposed AVBCKF-based CP method outperforms the cubature Kalman filtering- (CKF-) based CPmethod and extended Kalman filtering- (EKF-) based CP method

1 Introduction

Maintaining the availability of absolute position is a sig-nificant challenge for many applications such as intelligenttransportation systems (ITS) [1] and location based services(LBS) [2] The Global Navigation Satellite Systems (GNSSs)such as the Global Positioning System (GPS) do notmeet therequirements of safety-related applications such as collisionavoidance and lane-level guidance due to limited accuracyand availability Several techniques have been proposed toimprove the GPS positioning performance such as real-time kinematic (RTK) [3] differential GPS (DGPS) [4] andsatelliteground-based augmentation systems [5] Howeverthese techniques rely on the infrastructures and cannotperform well in urban environments due to limited viewof the sky multipath interference and unavailability of lineof sight Fortunately some innovative approaches calledcooperative positioning (CP) have been proposed in recentyears for positioning accuracy enhancement within vehicu-lar networks based on mutual communicating among thenodes of the network The CP process typically consists of

two phases [6] The first phase is the measurement phaseduringwhich the vehiclesmeasure internal state informationsuch as vehiclesrsquo positions by standalone GPS receivers andrange values by ranging instruments The second phase isthe position update phase during which vehicles share themeasurements with others and infer their positions based onall the measurements

In the first phase of CP radio-based-ranging methodsare the widely used ranging techniques which use signalpropagation characteristics to derive an estimation of thedistance between the transmitter and the receiver [7] Thereare four common radio-ranging techniques used for CP inVANETs time of arrival (TOA) time difference of arrival(TDOA) angle of arrival (AOA) and received signal strength(RSS) Christie et al [8] combined the RSS based intervehicledistance measurements with the road maps and vehiclekinematics for vehicle positioning However the RSS-basedrange does not always provide accurate distance measure-ment due to large variations in signal attenuation underdifferent environments particularly when multipath andshadowing effects are present [9] Benslimane [10] assumed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 843719 8 pageshttpdxdoiorg1011552015843719

2 Mathematical Problems in Engineering

that the ranges among vehicles can be estimated using RSSTOA or TDOA and proposed a CP method that uses theranges and GPS-based positions to estimate other GPS-denied positions In [11 12] two range-based CP methodsare proposed to mitigate GPS multipath error and improveaccuracy in a VANET respectively Apart from the range-based CP techniques there is another class of CP techniquesbased on radio-ranging rate which was presented in [13ndash15]In [13] a location estimation algorithm based on the Dopplereffect is presented to estimate the position of a moving targetin a wireless sensor network However the geometry of thetarget over a large distance is not consistentwith the geometryof the vehicular environment in urban streets which is notsuitable for vehicle positioning A Doppler-shift-based CPmethod for VANETs is proposed in [14] which mainly relieson the infrastructure

In the second phase measurements are aggregated andpassed to the fusing algorithm For most of the CP datafusion the system and measurement models are nonlineartherefore the nonlinear filtering techniques are used toachieve better positioning results As a widely used nonlinearfiltering technique extended Kalman filter (EKF) is usuallyused as the core of the CP algorithm [16 17] Howeverthe EKF can result in particularly poor performance if thedynamic systems are highly nonlinear As better alternativesto the EKF some other filters based on Bayesian samplingsuch as unscented Kalman filter (UKF) and particle filter[18 19] can also be used for nonlinear system However aheavy computational load and the curse of the dimension-ality in practical application are major constraints in thesefiltering algorithms The cubature Kalman filter (CKF) [20]has recently received increasing attention which employsdeterministic sampling to evaluate the intractable integralsencountered in filtering problem However the performanceof these filters mentioned above may degrade due to the factthat in practical situations the statistics ofmeasurement noisemight change

In this paper an enhanced CP method by utilizingthe ultra-wideband- (UWB-) based-range measurements isproposed It focuses on the absolute positioning in emergingintelligent transportation systems In this method the GPS-based positions are shared among the participating vehi-cles Then each vehicle fuses the GPS measurements andUWB-based range to obtain the position For data fusionan adaptive variational Bayesian cubature Kalman filtering(AVBCKF) is proposed as the core of the CP algorithmIn each update step of the AVBCKF both the system stateand time-variant measurement noise are considered as ran-dom variables to be estimated Measurement noise varianceis approximated by variational Bayesian (VB) approachthereafter system states are updated by cubature Kalmanfiltering The analytical and experimental results show thatthe AVBCKF can dynamically estimate the measurementnoise and theAVBCKF-basedCPmethodoutperformsCKF-based CP method and EKF-based CP method

The rest of the paper is organized as follows In Section 2the novel AVBCKF is described Section 3 explains theUWB-based-range technique and the estimator of the proposed CPmethod In Section 4 the experimental results are discussed

and the performance of the proposed system is evaluatedSection 5 summarizes the contributions of this work andfuture work

2 Adaptive Variational Bayesian CubatureKalman Filtering

21 Variational Bayesian Approach The VB approach alsoknown as ensemble learning takes its name from Feyn-manrsquos variational free energy method developed in statisticalphysics [21] VB is developed by the machine learningcommunity and has been applied in a variety of statisticaland signal processing domains In this paper VB is used toestimate the noise variance of measurements in the proposedCP algorithm Therefore the following will explain the VBapproach

In the parameter estimation domain after getting theobserved data setZ the posterior probability density function(PDF) of parameter set 120579 (119901(120579 | Z)) can be calculated byBayesrsquo rule as

119901 (120579 | Z) =119901 (Z | 120579) 119901 (120579)

119901 (Z) (1)

where 119901(Z) is the marginal likelihood function and is thekey parameter to obtain the value of 119901(120579 | Z) As a resultof the intractability of the integral in computing 119901(Z) itis difficult to calculate 119901(Z) to obtain 119901(120579 | Z) The keyidea of VB is to use a new distribution 119902(120579) to approximatethe true distribution 119901(120579 | Z) The form of the newdistribution 119902(120579) is selected freely owing to the conjugacyproperty Furthermore if the set 120579 is partitioned into119873 partsas 120579 = 120579

1 1205792 120579

119894 120579

119873 then each 120579

119894is assumed to be

independent of each other Naturally 119902(120579) factorizes into 119873independent 119902(120579

119894) and the joint distribution as

119902 (120579) =

119873

prod119894=1

119902 (120579119894) (2)

The technique used in VB approach is also called mean fieldvariational Bayes

In VB framework the goal is to find a 119902(120579) which isas close as to 119901(120579 | Z) Fortunately Kullback-Leibler (KL)divergence is a nonnegative dissimilarity function measuringthe discrepancy between two distributions 119902(120579) and 119901(120579 | Z)Hence 119902(120579) can be obtained byminimizing the KL divergencebetween 119902(120579) and 119901(120579 | Z) It has also been shown in [22] thatminimizing KL(119902(120579) 119901(120579 | Z)) is equivalent to maximizingthe lower bound of log119901(Z) The lower bound function canbe written as

119865 (119902 (120579)) = int 119902 (120579) log119901 (Z 120579)119902 (120579)

d120579 (3)

Maximizing 119865(119902(120579)) tightens the lower bound and gives riseto the optimal distribution The approximate distribution isoptimal in the sense of KL divergence because KL(119902(120579)

119901(120579 | Z)) equals zero if and only if 119902(120579) = 119901(120579 | Z) It is worthnoting that this optimal sense differs from the counterpart

Mathematical Problems in Engineering 3

in optimal estimation theory where the optimization ismeasured by minimum-variance rules Therefore 119902(120579

119894) can

be computed by differentiating119865(119902(120579))with respect to 119902(120579119895=119894)

General solution of 119902(120579119894)s is given as

119902 (120579119894) =

exp (119864119902(120579119895 =119894)

(log119901 (Z 120579)))

int exp (119864119902(120579119895 =119894)

(log119901 (Z 120579))) d120579119896

(4)

where 119864119902(120579119895 =119894)

(log119901(Z 120579)) is the expectation of log jointdistribution over all the parameters not in the partition

Note that (4) is an implicit solution because of the circulardependencies This forms an iterative estimation schemewhere the distribution of each parameter is estimated withthe expectation over other distributions given the appro-priate initialization of the hyperparameters Afterwards thedistribution of each parameter is updated according to (4)during the next iterations until the algorithm convergesThe convergence of VB has been analytically proved in [23]Hence in VB framework coupled equations can be solvedthrough form separable approximate distributions

22 Cubature Kalman Filtering Consider the discrete-timenonlinear filtering problems with additive process and mea-surement noise whose state space model can be expressed bythe pair difference equations given as

x119896= 119891 (x

119896minus1) + w119896 (5)

z119896= ℎ (x

119896) + v119896 (6)

Equations (5) and (6) are the process equation and themeasurement equation respectively where x

119896and z119896are the

state and measurement values at time 119896 respectively 119891(x119896minus1)

and ℎ(x119896) are some nonlinear functions w

119896and v119896are white

noise with zero mean and covariance q119896and r119896 respectively

From the perspective of probability the state spacemodel canbe reformulated according to [24] as

x119896sim 119901 (x

119896| x119896minus1) = 119873 (119891 (x

119896minus1) q119896)

z119896sim 119901 (z

119896| x119896) = 119873 (ℎ (x

119896) r119896)

(7)

Assume that x119896is Gaussian distribution with mean m

119896and

covarianceP119896Then the prior density of the state at 119896minus1 and 119896

obeys theGaussian distribution as119901(x119896| z1119896minus1

) = 119873(mminus119896Pminus119896)

and 119901(x119896| z1119896) = 119873(m

119896P119896) respectively Therefore the

prediction and update equations of the CKF can be expressedas follows

The prediction equation is as follows

120594(119894)

119896minus1= m119896minus1

+ radic119899P119896minus1

119883(119894)

119896= 119891 (120594

(119894)

119896minus1) 119894 = 1 2119899

mminus119896=

1

2119899

2119899

sum119894=1

119883(119894)

119896

Pminus119896=

1

2119899

2119899

sum119894=1

(119883(119894)

119896minusmminus119896) (119883(119894)

119896minusmminus119896)T+ q119896

(8)

The update equation is as follows

120594minus(119894)

119896= mminus119896+ radic119899Pminus

119896

z(119894)119896= ℎ (120594

minus(119894)

119896) 119894 = 1 2119899

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)

119896minusmminus119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

(9)

23 The Proposed Adaptive Variational Bayesian CubatureKalman Filtering In the traditional CKF algorithm the sta-tistical moment of measurement noise is invariant Howeverthe performance of the traditional CKF may be degradeddue to the fact that in practical situations the statistics ofmeasurement noise might change with time In order tointroduce robustness against the measurement noise in theCP an improved cubature Kalman filtering approach basedon variational Bayesian is proposed as the core component ofthe CP algorithm In each update step of the AVBCKF bothsystem state and time-variant measurement noise are con-sidered as random variables to be estimated Measurementsnoise variances are approximated by variational Bayesianapproach thereafter the system states are updated by cuba-ture Kalman filtering

In the proposed AVBCKF algorithm the system statex119896and the covariance of time-variant measurement noise

r119896are considered as random variables to be estimated

Consequently the prior distribution of the joint probabilitydensity function of x

119896and r119896at time 119896minus1 can be expressed as

119901 (x119896 r119896| z1119896minus1

)

= int119901 (x119896| x119896minus1) 119901 (r119896| r119896minus1)

sdot 119901 (x119896minus1 r119896minus1

| z1119896minus1

) dx119896minus1

dr119896minus1

(10)


At time 119896 the prior distribution of the joint probabilitydensity function of x

119896and r119896is

119901 (x119896 r119896| z1119896) =

119901 (x119896 r119896 z1119896)

119901 (z1119896)

=119901 (z119896| x119896 r119896) 119901 (x119896 r119896| z1119896minus1

)

119901 (z119896| z1119896minus1

)

(11)

From the perspective of generalized Bayesrsquo rule (10) and(11) can be seen as the prediction equation and the updateequation of Bayes filtering theory However the computationof the integral in (10) is intractability and the variationalBayesian approach is used to obtain the approximate optimalvalue Assume that the variables x

119896and r119896are independent

of each other and according to the prior knowledge [25]these can obey the Gaussian distribution and inverse-gammadistribution respectively Then at time 119896 minus 1

119901 (x119896 r119896| z1119896minus1

) = 119901 (x119896| z1119896minus1

) 119901 (r119896| z1119896minus1

)

= 119873 (x119896| mminus119896Pminus119896)

119889

prod119894=1

IG (1205902119896119894| 120572minus

119896119894 120573minus

119896119894)

(12)

After getting the measurement at time 119896 a new distribu-tion 119902(x

119896 r119896| z1119896) to replace the real posterior distribution

119901(x119896 r119896| z1119896) will be introduced to use the variational

Bayesian approach For simplicity the dependence of dis-tribution on z is omitted in the following formulations Forexample the 119902(x

119896 r119896| z1119896) will be expressed as 119902(x

119896 r119896)

Under the assumption that two variables x119896and r

119896are

independent the following relationship can be obtained119902(x119896 r119896) = 119902(x

119896)119902(r119896) Equation (4) gives the general approx-

imate solution of variational Bayesian approach Meanwhilereferring to the CKF algorithm and using the nonlinearfunctionℎ(sdot) in (6) for the state variable x

119896and the covariance

of measurement noise r119896 the following relationship can be

obtained

119902 (x119896) = 119862 exp(minus1

2xT119896((Pminus119896)minus1

+HT119896rminus1119896H119896) x119896

+ xT119896[(Pminus119896)minus1mminus119896+HT119896rminus1119896z119896])

(13)

119902 (r119896) = 119862 exp(

119889

sum119894=1

[(minus120572minus

119896119894minus1

2minus 1) ln1205902

119896119894]

minus

119889

sum119894=1

1

1205902119896119894

[120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

)])

(14)

where 119862 is the constant and is unrelated to the form ofdistribution As it can be seen from (13) and (14) the posteriordistributions 119902(x

119896) and 119902(r

119896) have the same form and dif-

ferent parameters with the prior distributions 119901(x119896| z1119896minus1

)

and 119901(r119896| z1119896minus1

) which obey the Gaussian distribution andinverse-gamma distribution respectively The reason is that

the Gaussian distribution and inverse-gamma distributionbelong to the conjugate exponential distribution [26]

The parameters m119896 P119896in the Gaussian distribution of

state variable can be derived as

m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896(1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

(15)

where

r119896= 119864r119896= diag (1198641205902

1198961 1198641205902

1198962 119864120590

2

119896119889) (16)

and the parameters in approximate inverse-gamma distribu-tion 119902(r

119896) = prod

119889

119894=1IG(1205902119896119894| 120572119896119894 120573119896119894) can also be derived as

120572119896119894= 120572minus

119896119894+1

2 (17)

120573119896119894= 120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

) (18)

Using the characteristic of inverse-gamma distribution [27]

1198641205902

119896119894=120573119896119894

120572119896119894

119894 = 1 sim 119889 (19)

According to the sample strategy of CKF the expectation in(18) can be derived as

119864x119896 [((z119896 minus ℎ119896 (x119896))119894)2

]

= ((z119896minus1

2119899

2119899

sum119894=1

ℎ (mminus119896+ radic119899Pminus

119896))

119894

)

2

(20)

The joint density of x119896and r119896can be estimated using (13)

to (20) The steps of variational Bayesian algorithm can besummarized as follows Firstly the parameters of the priordistribution in (15) (17) (18) and (19) are defined to calculatethe approximate distribution 119902(r

119896) and the parameters m

119896

P119896 Then the distribution is updated as shown in (13) and


Vehicle k

Data fusion algorithm

UWB

GPSreceiver Position

Vehicle k

Vehicle l Exchange message

Rl

(x y)

(x y)

(xl yl)

(xl yl)

Figure 1 The architecture of CP in VANET

(14) and new values are calculated using (19) and (20)Furthermore the distribution parameters are updated using(15) to (18) The estimation process is iterative and theiteration will stop when the convergence is reached It isreported in [27] that in a similar case VB will converge verysoon with only a few iterations

After using the VB to estimate the covariance ofmeasure-ment noise and combining the VB into CKF the AVBCKFproposed in this paper can be explained as follows

(i) PredictInitialize the 120572

0 1205730m0 P0

Predict the noise parameters using (17) (18) and (8)(ii) Update

Update the measurement noise using VB equations(14) and (17)-(18) and then update the state variableusing (15)

3 The Estimator of the Proposed CP

Assume that a VANET consists of a number of vehicles andall the vehicles are equipped with a GPS receiver and anUWB transceiver that can communicate the measurementdata among the vehiclesThe ultimate goal is that each vehiclecan improve the accuracy of its position using a data fusionalgorithmwhich is fed by local measurements and neighborsrsquoGPS-based positions and the UWB-based ranges Thereforethe measurements used in the proposed CP include the posi-tions and ranges of the vehicles In the method GPS-based

positions are shared between vehicles through the UWBcommunication The positions are combined with the rangemeasurements from UWB to achieve enhanced positioningThe AVBCKF algorithm proposed above is designed as thecore of the CP algorithm for data fusion For simplicity andconvenience only two vehicles 119896 and 119897 are considered whichcan be easily extended to the case of more than two vehiclesIn this paper the orthogonal axes east-north-up (ENU) areused as the coordinate frame to express the location of thevehicle For simplicity 2D positioning is considered but themethod can easily be expanded to 3D positioning Figure 1shows the typical architecture of the proposed CP with twovehicles As shown in Figure 1 assume that vehicle 119896 isthe target vehicle and it broadcasts its GPS-based position(119909 119910) to vehicle 119897 and receives the position (119909

119897 119910119897) of vehicle

119897 through UWB Meanwhile the UWB device in vehicle 119896measures the range 119877

119897between vehicles 119896 and 119897 In vehicle 119896

the positions (119909 119910) and (119909119897 119910119897) and the range 119877

119897are used as

inputs to the fusing algorithm to get the enhanced position ofvehicle 119896

The state space model of the movement of target vehicle119896 is defined as

119883 (119905 + 120591) = 119865119883 (119905) + 119866119882(119905) (21)

where 119883 = [119909 119910 V119909

V119910119886119909119886119910]

T is the state vector thatcontains the position (119909 119910) velocity (V

119909 V119910) and acceleration

(119886119909 119886119910) of the target vehicle 119909 and 119910 are along the east and

north axes respectively 119865 is the state transition model 119866 isthe process noise model119882 is the Gaussian system noise withthe STD 120590 and zero mean along each axis the covariance of


process noise is 119876 = 1205902119866119866T 120591 is the observation period and119905 is the time 119865 and 119866 are given as

119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)

where 119868 is a 2 times 2 identity matrix and119874 is a 2 times 2 zero matrixCorresponding to the system statemodel the observation

model of the CP algorithm can be defined as

119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)

where ℎ is a nonlinear observation vector in terms of119883 and 120577is the observation noiseThe observation vector119885 consists ofGPS-based positions of two vehicles and the range betweentwo vehicles measured by UWB The distance 119877

119897between

vehicles 119896 and 119897 can be expressed as

119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)

EKF can be used to implement the CP algorithm due tononlinearity of ℎ in (23) However using EKF introduces theprerequisites of definite models and known invariant noiseparameters In this paper theAVBCKF is designed as the datafusion algorithm inCPwhich is robust to the variations in themeasurement noise

4 Experimental Results

In order to evaluate the performance of CP method based onthe proposed AVBCKF the same experimental data as foundin [28] is used The experimental setup includes two vehiclesequipped with GPS receivers INSs and UWB transceiversA set of relatively expensive reference equipment and a set oflow cost sensors are fitted into each test vehicleThe referenceequipment is the Leica GS10 receiver for vehicle 1 and aNovatel INS-LCI (integrated GNSS-INS) for vehicle 2 Thecarrier-phase-based differential position estimates (RTK) ofthese receivers were used as ground truth positioning dataThe noisy GPS-based positions and the UWB- (MSSI-) basedrange were used as measurements to the AVBCKF-baseddata fusing technique in CP algorithm Figure 2 shows thephotograph of two experimental vehicles

The duration of the entire experiment was much largerthan the duration of the data selected (14 minutes) howeverthis was the largest continuous block of data with corre-sponding RTK GPS availability Meanwhile two vehiclescan communicate through UWB continuously during theexperimental data selected The GPS observations logged atthe Nottingham Geospatial Institute reference station wereused for the calculation of the DGPS corrections The testarea was the Clifton Boulevard between Derby Road andLoughborough Road in Nottingham UK which had goodopen sky to maximize the satellite visibility Figure 3 showsthe number of the common visible satellites to the vehiclesfor the selected experimental data It can be found that thenumber of common visible satellites is almost always abovethe required minimum that is four However when it falls

Figure 2 Two experimental vehicles

8 satellites325

7 satellites379

6 satellites21

5 satellites684 satellites

13

Less than 4 satellites 05

Figure 3 Common satellite visibility of the experimental dataselected

below the minimum the Kalman filter can compensate usingthe dynamic model of the system setting the innovation ofmissing observation to zero and assigning a large numberas infinity to the corresponding element in observationcovariance

In addition to the AVBCKF the EKF and CKF are alsoused as the data fusion algorithms in the CP to evaluate arelative efficiency of the proposedAVBCKF For convenienceldquoEKF-CPrdquo ldquoCKF-CPrdquo and ldquoAVBCKF-CPrdquo are used in thefollowing figures to denote the three different data fusionalgorithmsmentioned aboveThe positioning error of vehiclewas calculated as 119890(119905) = 119901

1015840(119905) minus 119901(119905) where 1199011015840(119905) and 119901(119905)are the estimated and the true positions using RTKGPS datarespectively

Figures 4 and 5 provide a comparison of positioning erroralong the east and north directions and positioning resultsagainst different data fusing algorithms

The root mean square (RMS) of 119890119889(rmse) is evaluated

according to the definition given in [28] Figure 6 showsthe rmse of positioning results using different data fusingalgorithms

It can be noticed from the figures that compared tothe EKF-based data fusion and CKF-based data fusion theproposed AVBCKF solution has improved precision of theposition The obvious reason is the consideration of bothsystem state and time-variant measurement noise as ran-dom variables that are estimated in the proposed AVBCKFTherefore the AVBCKF-based data fusion algorithm is morerobust to the variations in the measurement noise andoutperforms the other two data fusing algorithms


0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)

AVBCKF-CPCKF-CPEKF-CP

minus6

minus5

minus4

minus3

minus2

minus1

Figure 4 Comparison of positioning error in east direction withdifferent data fusing algorithms

0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2

Figure 5 Comparison of positioning error in north direction withdifferent data fusing algorithms

0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)


Figure 6 Comparison of rmse in positioning result with differentdata fusing algorithms

5 Conclusion

In this paper an adaptive variational Bayesian cubatureKalman filtering is proposed as the CP algorithm Themajor focus of the algorithm is to improve the performanceof absolute position estimation in emerging intelligent

transportation systems In each update step of the pro-posed AVBCKF both the system state and time-variantmeasurement noise are considered as random variables tobe estimated and the variances of measurement noise areapproximated using variational Bayesian (VB) approachThereafter the system states are updated by cubature KalmanfilteringThe AVBCKF-based data fusion algorithm is shownto be robust to the variations in the measurement noiseThe experimental results show that the AVBCKF-based CPmethod outperforms the CKF-based CP method and EKF-based CP method

Conflict of Interests

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

Acknowledgments

The authors would like to thankDr N Alam fromCaterpillarTrimble Control Technologies Dayton OH USA for hisprevious work and kind assistance in this projectThe authorswould also like to thank Associate Professor A Kealy fromthe Department of Infrastructure Engineering University ofMelbourne Parkville Australia and Dr C Hill and Dr SInce from the Nottingham Geospatial Institute University ofNottingham Nottingham UK for their support and collabo-ration in the experiments conducted for this workThis workis supported byNational Nature Science Foundation of Chinaunder Grant nos 61102107 and 61374208 and by the ChinaFundamental Research Funds for the Central Universitiesunder Grant no HEUCFX41310

References

[1] ldquoVehicle Positioning for C-ITS inAustralia Austroads ResearchReportrdquo httpswwwonlinepublicationsaustroadscomauitemsAP-R431-13

[2] R Tatchikou S Biswas and F Dion ldquoCooperative vehiclecollision avoidance using inter-vehicle packet forwardingrdquo inProceedings of the IEEE Global Telecommunications Conference(GLOBECOM rsquo05) pp 2762ndash2766 December 2005

[3] N Patwari J NAsh S Kyperountas AOHero III R LMosesand N S Correal ldquoLocating the nodes cooperative localizationin wireless sensor networksrdquo IEEE Signal Processing Magazinevol 22 no 4 pp 54ndash69 2005

[4] E D Kaplan and C J Hegarty Understanding GPS Principlesand Applications Artech House Norwood NJ USA 2ndedition 2006

[5] B Hofmann-Wellenhof H Lichtenegger and J Collins GlobalPositioning SystemTheory and Practice Springer NewYork NYUSA 5th edition 2001

[6] H Wymeersch J Lien and M Z Win ldquoCooperative localiza-tion in wireless networksrdquo Proceedings of the IEEE vol 97 no2 pp 427ndash450 2009

[7] A Boukerche H A B F Oliveira E F Nakamura and A AF Loureiro ldquoVehicular Ad Hoc networks a new challenge forlocalization-based systemsrdquoComputer Communications vol 31no 12 pp 2838ndash2849 2008


[8] J R I Christie P-Y Ko A Hansen et al ldquoThe effects of localionospheric decorrelation on LAAS theory and experimentalresultsrdquo in Proceedings of the National Technical Meeting of theInstitute of Navigation pp 769ndash777 San Diego Calif USAJanuary 1999

[9] R Parker and S Valaee ldquoCooperative vehicle position estima-tionrdquo in Proceedings of the IEEE International Conference onCommunications (ICC rsquo07) pp 5837ndash5842 June 2007

[10] A Benslimane ldquoLocalization in vehicular Ad Hoc networksrdquoin Proceedings of the Systems Communications pp 19ndash25Montreal Canada August 2005

[11] N Drawil and O Basir ldquoVehicular collaborative technique forlocation estimate correctionrdquo in Proceedings of the IEEE 68thVehicular Tchnology Conference pp 1ndash5 September 2008

[12] R Parker and S Valaee ldquoVehicular node localization usingreceived-signal-strength indicatorrdquo IEEETransactions onVehic-ular Technology vol 56 no 6 I pp 3371ndash3380 2007

[13] N Patwari J N Ash S Kyperountas A O Hero R L Mosesand N S Correal ldquoLocating the nodes cooperative localizationin wireless sensor networksrdquo IEEE Signal Processing Magazinevol 22 no 4 pp 54ndash69 2005

[14] R Parker and S Valaee ldquoRobust min-max localization algo-rithmrdquo in Proceedings of the IEEE Intelligent TransportationSystems Conference (ITSC rsquo06) pp 1000ndash1005 Toronto CanadaSeptember 2006

[15] J Arias J Lazaro A Zuloaga and J Jimenez ldquoDoppler locationalgorithm for wireless sensor networksrdquo in Proceedings of theInternational Conference on Wireless Networks (ICWN rsquo04) pp509ndash514 Las Vegas Nev USA June 2004

[16] B Xu L Shen and F Yan ldquoVehicular node positioning based onDoppler-shifted frequency measurement on highwayrdquo Journalof Electronics vol 26 no 2 pp 265ndash269 2009

[17] N Alam A T Balaei and A G Dempster ldquoA DSRC doppler-based cooperative positioning enhancement for vehicular net-works with GPS availabilityrdquo IEEE Transactions on VehicularTechnology vol 60 no 9 pp 4462ndash4470 2011

[18] N Alam A T Balaei andA G Dempster ldquoRelative positioningenhancement in VANETs a tight integration approachrdquo IEEETransactions on Intelligent Transportation Systems vol 14 no 1pp 47ndash55 2013

[19] N Alam A Kealy and A G Dempster ldquoAn INS-aided tightintegration approach for relative positioning enhancementin VANETsrdquo IEEE Transactions on Intelligent TransportationSystems vol 14 no 4 pp 1992ndash1996 2013

[20] I Arasaratnam and S Haykin ldquoCubature Kalman filtersrdquo IEEETransactions on Automatic Control vol 54 no 6 pp 1254ndash12692009

[21] M Svensen andCM Bishop ldquoRobust Bayesianmixturemodel-lingrdquo Neurocomputing vol 64 pp 235ndash252 2005

[22] S Sarkka and A Nummenmaa ldquoRecursive noise adaptivekalman filtering by variational bayesian approximationsrdquo IEEETransactions on Automatic Control vol 54 no 3 pp 596ndash6002009

[23] G Storvik ldquoParticle filters for state-space models with thepresence of unknown static parametersrdquo IEEE Transactions onSignal Processing vol 50 no 2 pp 281ndash289 2002

[24] I Arasaratnam S Haykin and T R Hurd ldquoCubature Kalmanfiltering for continuous-discrete systems theory and simula-tionsrdquo IEEE Transactions on Signal Processing vol 58 no 10pp 4977ndash4993 2010

[25] E Ozkan V Smıdl S Saha C Lundquist and F GustafssonldquoMarginalized adaptive particle filtering for nonlinear modelswith unknown time-varying noise parametersrdquo Automaticavol 49 no 6 pp 1566ndash1575 2013

[26] T Schon F Gustafsson and P J Nordlund ldquoMarginalizedparticle filters for mixed linearnonlinear state-space modelsrdquoIEEE Transactions on Signal Processing vol 53 no 7 pp 2279ndash2289 2005

[27] M I Jordan Z Ghahramani T S Jaakkola and L K SaulldquoIntroduction to variational methods for graphical modelsrdquoMachine Learning vol 37 no 2 pp 183ndash233 1999

[28] F Shen J W Cheong and A G Dempster ldquoA tight integrationapproach for relative positioning enhancement based on low-cost IMU and DSRC dopplerrdquo IEEE transaction on ITS Sub-mitted

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


that the ranges among vehicles can be estimated using RSSTOA or TDOA and proposed a CP method that uses theranges and GPS-based positions to estimate other GPS-denied positions In [11 12] two range-based CP methodsare proposed to mitigate GPS multipath error and improveaccuracy in a VANET respectively Apart from the range-based CP techniques there is another class of CP techniquesbased on radio-ranging rate which was presented in [13ndash15]In [13] a location estimation algorithm based on the Dopplereffect is presented to estimate the position of a moving targetin a wireless sensor network However the geometry of thetarget over a large distance is not consistentwith the geometryof the vehicular environment in urban streets which is notsuitable for vehicle positioning A Doppler-shift-based CPmethod for VANETs is proposed in [14] which mainly relieson the infrastructure

In the second phase measurements are aggregated andpassed to the fusing algorithm For most of the CP datafusion the system and measurement models are nonlineartherefore the nonlinear filtering techniques are used toachieve better positioning results As a widely used nonlinearfiltering technique extended Kalman filter (EKF) is usuallyused as the core of the CP algorithm [16 17] Howeverthe EKF can result in particularly poor performance if thedynamic systems are highly nonlinear As better alternativesto the EKF some other filters based on Bayesian samplingsuch as unscented Kalman filter (UKF) and particle filter[18 19] can also be used for nonlinear system However aheavy computational load and the curse of the dimension-ality in practical application are major constraints in thesefiltering algorithms The cubature Kalman filter (CKF) [20]has recently received increasing attention which employsdeterministic sampling to evaluate the intractable integralsencountered in filtering problem However the performanceof these filters mentioned above may degrade due to the factthat in practical situations the statistics ofmeasurement noisemight change

In this paper an enhanced CP method by utilizingthe ultra-wideband- (UWB-) based-range measurements isproposed It focuses on the absolute positioning in emergingintelligent transportation systems In this method the GPS-based positions are shared among the participating vehi-cles Then each vehicle fuses the GPS measurements andUWB-based range to obtain the position For data fusionan adaptive variational Bayesian cubature Kalman filtering(AVBCKF) is proposed as the core of the CP algorithmIn each update step of the AVBCKF both the system stateand time-variant measurement noise are considered as ran-dom variables to be estimated Measurement noise varianceis approximated by variational Bayesian (VB) approachthereafter system states are updated by cubature Kalmanfiltering The analytical and experimental results show thatthe AVBCKF can dynamically estimate the measurementnoise and theAVBCKF-basedCPmethodoutperformsCKF-based CP method and EKF-based CP method

The rest of the paper is organized as follows In Section 2the novel AVBCKF is described Section 3 explains theUWB-based-range technique and the estimator of the proposed CPmethod In Section 4 the experimental results are discussed

and the performance of the proposed system is evaluatedSection 5 summarizes the contributions of this work andfuture work

2 Adaptive Variational Bayesian CubatureKalman Filtering

21 Variational Bayesian Approach The VB approach alsoknown as ensemble learning takes its name from Feyn-manrsquos variational free energy method developed in statisticalphysics [21] VB is developed by the machine learningcommunity and has been applied in a variety of statisticaland signal processing domains In this paper VB is used toestimate the noise variance of measurements in the proposedCP algorithm Therefore the following will explain the VBapproach

In the parameter estimation domain after getting theobserved data setZ the posterior probability density function(PDF) of parameter set 120579 (119901(120579 | Z)) can be calculated byBayesrsquo rule as

119901 (120579 | Z) =119901 (Z | 120579) 119901 (120579)

119901 (Z) (1)

where 119901(Z) is the marginal likelihood function and is thekey parameter to obtain the value of 119901(120579 | Z) As a resultof the intractability of the integral in computing 119901(Z) itis difficult to calculate 119901(Z) to obtain 119901(120579 | Z) The keyidea of VB is to use a new distribution 119902(120579) to approximatethe true distribution 119901(120579 | Z) The form of the newdistribution 119902(120579) is selected freely owing to the conjugacyproperty Furthermore if the set 120579 is partitioned into119873 partsas 120579 = 120579

1 1205792 120579

119894 120579

119873 then each 120579

119894is assumed to be

independent of each other Naturally 119902(120579) factorizes into 119873independent 119902(120579

119894) and the joint distribution as

119902 (120579) =

119873

prod119894=1

119902 (120579119894) (2)

The technique used in VB approach is also called mean fieldvariational Bayes

In VB framework the goal is to find a 119902(120579) which isas close as to 119901(120579 | Z) Fortunately Kullback-Leibler (KL)divergence is a nonnegative dissimilarity function measuringthe discrepancy between two distributions 119902(120579) and 119901(120579 | Z)Hence 119902(120579) can be obtained byminimizing the KL divergencebetween 119902(120579) and 119901(120579 | Z) It has also been shown in [22] thatminimizing KL(119902(120579) 119901(120579 | Z)) is equivalent to maximizingthe lower bound of log119901(Z) The lower bound function canbe written as

119865 (119902 (120579)) = int 119902 (120579) log119901 (Z 120579)119902 (120579)

d120579 (3)

Maximizing 119865(119902(120579)) tightens the lower bound and gives riseto the optimal distribution The approximate distribution isoptimal in the sense of KL divergence because KL(119902(120579)

119901(120579 | Z)) equals zero if and only if 119902(120579) = 119901(120579 | Z) It is worthnoting that this optimal sense differs from the counterpart



119894) can



119902 (120579119894) =

exp (119864119902(120579119895 =119894)

(log119901 (Z 120579)))

int exp (119864119902(120579119895 =119894)

(log119901 (Z 120579))) d120579119896

(4)

where 119864119902(120579119895 =119894)




x119896= 119891 (x

119896minus1) + w119896 (5)

z119896= ℎ (x

119896) + v119896 (6)








x119896sim 119901 (x

119896| x119896minus1) = 119873 (119891 (x

119896minus1) q119896)

z119896sim 119901 (z

119896| x119896) = 119873 (ℎ (x

119896) r119896)

(7)


119896and



) = 119873(mminus119896Pminus119896)

and 119901(x119896| z1119896) = 119873(m




120594(119894)



119883(119894)

119896= 119891 (120594

(119894)

119896minus1) 119894 = 1 2119899

mminus119896=

1

2119899

2119899

sum119894=1

119883(119894)

119896

Pminus119896=

1

2119899

2119899

sum119894=1

(119883(119894)

119896minusmminus119896) (119883(119894)

119896minusmminus119896)T+ q119896

(8)


120594minus(119894)


119896

z(119894)119896= ℎ (120594

minus(119894)

119896) 119894 = 1 2119899

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

(9)






119901 (x119896 r119896| z1119896minus1

)



| z1119896minus1

) dx119896minus1

dr119896minus1

(10)



119896and r119896is

119901 (x119896 r119896| z1119896) =

119901 (x119896 r119896 z1119896)

119901 (z1119896)

=119901 (z119896| x119896 r119896) 119901 (x119896 r119896| z1119896minus1

)

119901 (z119896| z1119896minus1

)

(11)




119901 (x119896 r119896| z1119896minus1

) = 119901 (x119896| z1119896minus1

) 119901 (r119896| z1119896minus1

)

= 119873 (x119896| mminus119896Pminus119896)

119889

prod119894=1

IG (1205902119896119894| 120572minus

119896119894 120573minus

119896119894)

(12)






119896 r119896)


119896are






obtained

119902 (x119896) = 119862 exp(minus1


+HT119896rminus1119896H119896) x119896


(13)

119902 (r119896) = 119862 exp(

119889

sum119894=1

[(minus120572minus

119896119894minus1

2minus 1) ln1205902

119896119894]

minus

119889

sum119894=1

1

1205902119896119894

[120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

)])

(14)


119896) and 119902(r



)

and 119901(r119896| z1119896minus1





m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896(1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

(15)

where

r119896= 119864r119896= diag (1198641205902

1198961 1198641205902

1198962 119864120590

2

119896119889) (16)


119896) = prod

119889


120572119896119894= 120572minus

119896119894+1

2 (17)

120573119896119894= 120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

) (18)


1198641205902

119896119894=120573119896119894

120572119896119894

119894 = 1 sim 119889 (19)


119864x119896 [((z119896 minus ℎ119896 (x119896))119894)2

]

= ((z119896minus1

2119899

2119899

sum119894=1


119896))

119894

)

2

(20)




119896



Vehicle k


UWB


Vehicle k


Rl

(x y)

(x y)

(xl yl)

(xl yl)





0 1205730m0 P0






119897 119910119897) of vehicle




119897are used as



119883 (119905 + 120591) = 119865119883 (119905) + 119866119882(119905) (21)

where 119883 = [119909 119910 V119909

V119910119886119909119886119910]







119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)



119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)


119897between


119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)






8 satellites325

7 satellites379

6 satellites21


13











0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894) can



119902 (120579119894) =

exp (119864119902(120579119895 =119894)

(log119901 (Z 120579)))

int exp (119864119902(120579119895 =119894)

(log119901 (Z 120579))) d120579119896

(4)

where 119864119902(120579119895 =119894)




x119896= 119891 (x

119896minus1) + w119896 (5)

z119896= ℎ (x

119896) + v119896 (6)








x119896sim 119901 (x

119896| x119896minus1) = 119873 (119891 (x

119896minus1) q119896)

z119896sim 119901 (z

119896| x119896) = 119873 (ℎ (x

119896) r119896)

(7)


119896and



) = 119873(mminus119896Pminus119896)

and 119901(x119896| z1119896) = 119873(m




120594(119894)



119883(119894)

119896= 119891 (120594

(119894)

119896minus1) 119894 = 1 2119899

mminus119896=

1

2119899

2119899

sum119894=1

119883(119894)

119896

Pminus119896=

1

2119899

2119899

sum119894=1

(119883(119894)

119896minusmminus119896) (119883(119894)

119896minusmminus119896)T+ q119896

(8)


120594minus(119894)


119896

z(119894)119896= ℎ (120594

minus(119894)

119896) 119894 = 1 2119899

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

(9)






119901 (x119896 r119896| z1119896minus1

)



| z1119896minus1

) dx119896minus1

dr119896minus1

(10)



119896and r119896is

119901 (x119896 r119896| z1119896) =

119901 (x119896 r119896 z1119896)

119901 (z1119896)

=119901 (z119896| x119896 r119896) 119901 (x119896 r119896| z1119896minus1

)

119901 (z119896| z1119896minus1

)

(11)




119901 (x119896 r119896| z1119896minus1

) = 119901 (x119896| z1119896minus1

) 119901 (r119896| z1119896minus1

)

= 119873 (x119896| mminus119896Pminus119896)

119889

prod119894=1

IG (1205902119896119894| 120572minus

119896119894 120573minus

119896119894)

(12)






119896 r119896)


119896are






obtained

119902 (x119896) = 119862 exp(minus1


+HT119896rminus1119896H119896) x119896


(13)

119902 (r119896) = 119862 exp(

119889

sum119894=1

[(minus120572minus

119896119894minus1

2minus 1) ln1205902

119896119894]

minus

119889

sum119894=1

1

1205902119896119894

[120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

)])

(14)


119896) and 119902(r



)

and 119901(r119896| z1119896minus1





m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896(1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

(15)

where

r119896= 119864r119896= diag (1198641205902

1198961 1198641205902

1198962 119864120590

2

119896119889) (16)


119896) = prod

119889


120572119896119894= 120572minus

119896119894+1

2 (17)

120573119896119894= 120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

) (18)


1198641205902

119896119894=120573119896119894

120572119896119894

119894 = 1 sim 119889 (19)


119864x119896 [((z119896 minus ℎ119896 (x119896))119894)2

]

= ((z119896minus1

2119899

2119899

sum119894=1


119896))

119894

)

2

(20)




119896



Vehicle k


UWB


Vehicle k


Rl

(x y)

(x y)

(xl yl)

(xl yl)





0 1205730m0 P0






119897 119910119897) of vehicle




119897are used as



119883 (119905 + 120591) = 119865119883 (119905) + 119866119882(119905) (21)

where 119883 = [119909 119910 V119909

V119910119886119909119886119910]







119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)



119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)


119897between


119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)






8 satellites325

7 satellites379

6 satellites21


13











0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896and r119896is

119901 (x119896 r119896| z1119896) =

119901 (x119896 r119896 z1119896)

119901 (z1119896)

=119901 (z119896| x119896 r119896) 119901 (x119896 r119896| z1119896minus1

)

119901 (z119896| z1119896minus1

)

(11)




119901 (x119896 r119896| z1119896minus1

) = 119901 (x119896| z1119896minus1

) 119901 (r119896| z1119896minus1

)

= 119873 (x119896| mminus119896Pminus119896)

119889

prod119894=1

IG (1205902119896119894| 120572minus

119896119894 120573minus

119896119894)

(12)






119896 r119896)


119896are






obtained

119902 (x119896) = 119862 exp(minus1


+HT119896rminus1119896H119896) x119896


(13)

119902 (r119896) = 119862 exp(

119889

sum119894=1

[(minus120572minus

119896119894minus1

2minus 1) ln1205902

119896119894]

minus

119889

sum119894=1

1

1205902119896119894

[120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

)])

(14)


119896) and 119902(r



)

and 119901(r119896| z1119896minus1





m119896= mminus119896+ K119896[z119896minus1

2119899

2119899

sum119894=1

z(119894)119896]

P119896= Pminus119896minus K119896(1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)KT119896

K119896=

1

2119899

2119899

sum119894=1

(120594minus(119894)


2119899

2119899

sum119894=1

z(119894)119896)

T

sdot (1

2119899

2119899

sum119894=1

(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)(z(119894)119896minus1

2119899

2119899

sum119894=1

z(119894)119896)

T

+ r119896)

minus1

(15)

where

r119896= 119864r119896= diag (1198641205902

1198961 1198641205902

1198962 119864120590

2

119896119889) (16)


119896) = prod

119889


120572119896119894= 120572minus

119896119894+1

2 (17)

120573119896119894= 120573minus

119896119894+1

2119864x119896 (((z119896 minus ℎ119896 (x119896))119894)

2

) (18)


1198641205902

119896119894=120573119896119894

120572119896119894

119894 = 1 sim 119889 (19)


119864x119896 [((z119896 minus ℎ119896 (x119896))119894)2

]

= ((z119896minus1

2119899

2119899

sum119894=1


119896))

119894

)

2

(20)




119896



Vehicle k


UWB


Vehicle k


Rl

(x y)

(x y)

(xl yl)

(xl yl)





0 1205730m0 P0






119897 119910119897) of vehicle




119897are used as



119883 (119905 + 120591) = 119865119883 (119905) + 119866119882(119905) (21)

where 119883 = [119909 119910 V119909

V119910119886119909119886119910]







119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)



119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)


119897between


119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)






8 satellites325

7 satellites379

6 satellites21


13











0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Vehicle k


UWB


Vehicle k


Rl

(x y)

(x y)

(xl yl)

(xl yl)





0 1205730m0 P0






119897 119910119897) of vehicle




119897are used as



119883 (119905 + 120591) = 119865119883 (119905) + 119866119882(119905) (21)

where 119883 = [119909 119910 V119909

V119910119886119909119886119910]







119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)



119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)


119897between


119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)






8 satellites325

7 satellites379

6 satellites21


13











0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865 = [

[

119868 120591119868 051205912119868

119874 119868 120591119868

119874 119874 119874

]

]

119866 = [

[

051205912119868

120591119868

119868

]

]

(22)



119885 (119905) = ℎ (119883 (119905)) + 120577 (119905) (23)


119897between


119877119897= radic(119909 minus 119909

119897)2

+ (119910 minus 119910119897)2

(24)






8 satellites325

7 satellites379

6 satellites21


13











0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800

012

Time (s)

Posit

ioni

ng er

ror i

n ea

st di

rect

ion

(m)


minus6

minus5

minus4

minus3

minus2

minus1


0 100 200 300 400 500 600 700 800

0

2

4

6

Time (s)

Posit

ioni

ng er

ror i

n no

rth

dire

ctio

n (m

)


minus2


0 100 200 300 400 500 600 700 800012345678

Time (s)

rmse

of p

ositi

onin

g (m

)



5 Conclusion





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









Documents

Research Article An Enhanced UWB-Based Range/GPS ...downloads.hindawi.com/journals/mpe/2015/843719.pdf · suitable for vehicle positioning. A Doppler-shi -based CP method for VANETs