[22] Li, J., Stoica, P., and Wang, Z.On robust Capon beamforming and diagonal loading.IEEE Transactions on Signal Processing, 51, 7 (July2003), 1702—1715.

[23] Li, J., Stoica, P., and Wang, Z.Doubly constrained robust Capon beamformer.IEEE Transactions on Signal Processing, 52, 9 (Sept.2004), 2407—2423.

[24] Lorenz, R., and Boyd, S. P.Robust minimum variance beamforming.IEEE Transactions on Signal Processing, 53, 5 (May2005), 1684—1696.

[25] Besson, O., Scharf, L. L., and Kraut, S.Adaptive detection of a a signal known only to lie on aline in a known subspace, when primary and secondarydata are partially homogeneous.IEEE Transactions on Signal Processing, to be published.

[26] De Maio, A.Robust adaptive radar detection in the presence ofsteering vector mismatches.IEEE Transactions on Aerospace and Electronic Systems,41, 4 (Oct. 2005), 1322—1337.

[27] Ramprashad, S., Parks, T. W., and Shenoy, R.Signal modeling and detection using cone classes.IEEE Transactions on Signal Processing, 44, 2 (Feb.1996), 329—338.

[28] Boyd, S., and Vandenberghe, L.Convex Optimization.London: Cambridge University Press, 2004.

[29] Horn, R., and Johnson, C.Matrix Analysis.London: Cambridge University Press, 1990.

[30] Moon, T. K., and Stirling, W. C.Mathematical Methods and Algorithms for SignalProcessing.Upper Saddle River, NJ: Prentice-Hall, 2000.

[31] Bandiera, F., De Maio, A., and Ricci, G.Adaptive CFAR radar detection with conic rejection.IEEE Transactions on Signal Processing, submitted forpublication.

[32] Gradshteyn, I. S., and Ryzhik, I. M.In A. Jeffrey (Ed.), Table of Integrals, Series and Products(5th ed.).New York: Academic Press, 1994.

Multisensor Target Tracking Performance with BiasCompensation

In this paper, multisensor-multitarget tracking performance

with bias estimation and compensation is investigated when only

moving targets of opportunity are available. First, we discuss

the tracking performance improvement with bias estimation and

compensation for synchronous biased sensors, and then a novel

bias estimation method is proposed for asynchronous sensors with

time-varying biases. The performance analysis and simulations

show that asynchronous sensors have a slightly degraded

performance compared with the “equivalent” synchronous ones.

The bias estimates as well as the corresponding Cramer-Rao

lower bound (CRLB) on the covariance of the bias estimates,

i.e., the quantification of the available information on the sensor

biases in any scenario are also given. Tracking performance

evaluations with different sources of biases–offset biases, scale

biases, and sensor location uncertainties, are also presented and

we show that tracking performance is significantly improved

with bias estimation and compensation compared with the

target tracking using the original biased measurements. The

performance is also close to the lower bound obtained in the

absence of biases.

I. INTRODUCTION

Registration error compensation is vital inmultiple sensor systems in order to carry out datafusion. This requires estimation of the unknownsensor measurement biases. It is important to correctfor these bias errors so that the multiple sensormeasurements and/or tracks can be referencedas accurately as possible to a common trackingcoordinate system (frame). If uncorrected, registrationerror can lead to large tracking errors and potentiallyto the formation of multiple tracks (ghosts) for thesame target.To estimate the bias vector, the classical approach

is to augment the system state to include the biasvector as part of the state, then implement anaugmented state Kalman filter (ASKF) by stackingthe state of all the targets and the sensor biases intoa single vector. The problem with this approachis that the implementation of this ASKF can becomputationally infeasible. In addition, numerical

Manuscript received September 24, 2004; revised October 31, 2005;released for publication February 23, 2006.

IEEE Log No. T-AES/42/3/884477.

Refereeing of this contribution was handled by B. La Scala.

0018-9251/06/$17.00 c° 2006 IEEE

CORRESPONDENCE 1139

problems may arise during the implementation,mainly, for ill-conditioned systems. Friedland [15]proposed the idea of implementing two parallel,reduced-order filters instead of the use of an ASKF.Ignagni [19] generalized the two-stage method of[15]. Alouani, Rice, and Blair [1] showed that undera restrictive algebraic constraint, the optimal estimateof the state can be obtained as a linear combinationof the outputs of the local bias-ignorant estimateand the bias estimate. They claimed that, since thealgebraic constraint can be too restrictive in practice,all practical two-stage filters are suboptimal. VanDoorn and Blom [41] gave the exact solution forthe augmented Kalman filter problem but thendecoupled the equations using an approximationin order to make the implementation feasible. Asimilar approach was used in [23] and [39] to separatethe tracker from the bias filter and it undoubtedlyimposed some (uncharacterized) loss in estimationperformance. Okello and Ristic’s recent work in[35] is a batch algorithm that estimates biases byiterating on the inverse of the measurement equationlinearized around the latest central estimate as if itwere perfect, which leads to a calculated Cramer-Raolower bound (CRLB) that is too small. The recursiveperformance bound in [16] is realistic, but it isobtained using ASKF, which is not computationallyefficient.Lin, Kirubarajan, and Bar-Shalom [26] presented

bias estimation based on the local unassociated trackestimates at a single time, i.e., based on a singleframe. In [27], the authors extended the work of[26] to include dynamic bias estimation based onthe local track estimates at different times, i.e., basedon multiple frames. The technique in [27] has beenshown to yield absolute registration for the biasessubject to an observability condition. The work of[28] extended [26] and [27] to a bias model whichconsiders both offset biases and scale biases, andshowed some preliminary results.However, in reality, the target data reported

by the sensors are usually not time coincidentor synchronous, due to different data rates. Thebias estimation of asynchronous sensors does notseem to be understood or even widely recognized.All the above papers as well as other well-knownpapers on sensor bias estimation, such as [10, 11,13, 34], consider only synchronous sensors. In[18], the authors extended the work of [17] toasynchronous sensors by using one-step fixed-laginteracting multiple model (IMM) predictor totranslate the estimates to common time, and onlyrelative registration was achieved under a relativelyrestrictive assumption that the distance betweentwo sensors is small. The situation of asynchronoussensors can be handled by the ASKF, but for a largenumber of targets it becomes infeasible.

The work by Lin, Kirubarajan, and Bar-Shalomin [30] extended the work of [26], [27], and [28] to abias model which includes scale biases and unknownlocations of the sensors, as well as the usual offsetbiases. It also extended the method to asynchronoussensors, which makes it applicable for practicalsystems. The asynchronous sensors considered in[30] have the same sampling rate but with a phaseoffset. A “proper time slot” for asynchronous sensorbias estimation was introduced for the first timein [30] (definition given in Section IIIB). A novelscheme is proposed to transform the measurementsfrom a proper time slot but obtained at differenttimes into bias pseudomeasurements such that thepseudomeasurement noises of the sensor biasesare zero-mean, white and with easily calculatedcovariances. In [31], the bias estimation for the moregeneral asynchronous sensors which have differentsampling rates and phase offsets was discussed.Another significant benefit of this scheme is that thebias estimation is decoupled from the state estimationwithout ignoring or approximating the crosscovariancebetween the state estimate and the bias estimate. Inthis paper, the track-to-track association is assumedknown.The purpose of the present paper is to investigate

the multisensor tracking performance with biascompensation after bias estimation when only movingtargets of opportunity are available, that is, withunknown target positions and unknown bias withoutother prior information. For the asynchronous sensorbias estimation, [30] and [31] assume the biases to beconstant within a proper time slot. However, whenthe sensor location uncertainty is considered, thesensor bias vector is usually a time-varying vector.This is because sensor location error is, typically, timevarying. In this paper, we extend the asynchronoussensor bias estimation methods proposed in [30]and [31] to include the time-varying sensor biaseswithin a proper time slot. Performance analysisand simulations are also given to show the trackingperformance of the asynchronous sensors and thisis also compared with the “equivalent” synchronoussensors. Then the tracking performance evaluationswith the different sources of biases–offset biases,scale biases, and sensor location uncertainties forasynchronous sensors, are presented. We demonstratethe significant tracking performance improvementwith the bias estimation and compensation comparedwith no bias compensation, and the performance isalso compared with the lower bound which is obtainedin the absence of biases.This paper is organized as follows. The bias

model and the assumptions for bias estimation arediscussed in Section II. In Section III, a review of thedynamic bias estimation for synchronous sensors isgiven and the asynchronous sensor bias estimationwith sensor location uncertainty is proposed. The

1140 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 3 JULY 2006

bias compensation for asynchronous sensors withtime-varying biases is discussed in Section IV.Section V presents the tracking performances withbias compensation for some typical scenarios.Conclusions are in Section VI.

II. BIAS MODEL

Consider M sensors which measure the range andazimuth for N common targets in the surveillanceregion. The model for the biased measurements of acommon target at time k in polar coordinates (denotedby superscript p) for sensor s is

zps (k) =·rps (k)

µms (k)

¸=·[1+ ²rs(k)]rs(k)+ b

rs(k) +w

rs(k)

[1+ ²µs (k)]µs(k)+ bµs (k) +w

µs (k)

¸(1)

where rs(k) and µs(k) are the true range and azimuth;brs(k) and b

µs (k) are the offset biases for the range and

azimuth; ²rs(k) and ²µs (k) are the scale biases of the

range and azimuth, respectively; the measurementnoises wrs(k) and w

µs (k) are zero-mean, white with

corresponding variances ¾2r and ¾2µ , and are assumed

mutually independent of each other. It is based onthese N common targets of opportunity that the sensorbias estimation will be carried out.The bias vector ¯s(k) = [b

rs(k) b

µs (k) ²

rs(k) ²

µs (k)]

T

can be modeled as a nonrandom variable (a constant,in which case the maximum likelihood estimator orweighted least squares can be used to estimate it)or Gauss-Markov random variable (in which case aKalman filter can be used).Then,

zps (k) =·rs(k)

µs(k)

¸+Cs(k)¯s(k)+

·wrs(k)

wµs (k)

¸(2)

where

Cs(k)¢=·1 0 rs(k) 0

0 1 0 µs(k)

¸(3)

is assumed known–the observed azimuth µms (k)and range rms (k) can be used in the above without asignificant loss of performance.The problem is to estimate the bias vectors for all

sensors. After the bias estimation is complete, the biasestimate can be used to compensate the state estimatesof the targets. After this, the track-to-track fusion canbe performed.In addition to this, for mobile sensors, their

location might not be perfectly known, so theseshould also be estimated. However since only thedifferences of the sensor locations are observable,1

1This follows from the fact that a translation of all the sensorsand all targets results in exactly the same observations, i.e., such atranslation is not observable. Thus the exact location of at least onesensor has to be known.

we assume the location of the first sensor is knownexactly, and the location of other sensors (Xs,Ys) ismodeled as a known nominal location (Xs0,Ys0) plus azero-mean stationary first-order Markov process withstandard deviation ¾s in each coordinate. That is,

Xs = Xs0 +¸sx, Ys = Ys0 +¸sy, s= 2, : : : ,M:

(4)The stationary first-order Markov process is driven byzero-mean white noises as follows:

¸sx(k+1) = ®¸sx(k)+ vsx(k) (5)

¸sy(k+1) = ®¸sy(k)+ vsy(k) (6)

with ® assumed known2 and the process noisestandard deviation in (5) and (6) is ¾.After transforming the polar measurements (1) into

Cartesian coordinates, the measurement equation forsensor s is

zs(k) =H(k)x(k) +Bs(k)Cs(k)¯s(k) +ws(k) (7)

where the state vector is assumed to be x(k)¢=

[x(k) _x(k) y(k) _y(k)]0, H(k) is the measurementmatrix, defined as

H(k) =·1 0 0 0

0 0 1 0

¸¢=H: (8)

The matrix Bs(k) is a nonlinear function of the truerange and azimuth. If Bs(k)Cs(k) is constant, onlythe difference of the biases at different sensors isobservable, which means “relative registration” or“incomplete observability.” Using the observedazimuth µms (k) and range r

ms (k) from sensor s, one has

the matrix Bs(k) in (7) as (see [9, 27] for details)

Bs(k) =·cosµms (k) ¡rms (k) sinµms (k)sinµms (k) rms (k)cosµ

ms (k)

¸: (9)

III. DYNAMIC BIAS ESTIMATION

A. A Review of Dynamic Bias Estimation forSynchronous Sensors

Assume the dynamic equation of the targets is

x(k+1)¢=x(tk+1) = F(tk+1, tk)x(k) + v(tk+1, tk)

(10)

where F(tk+1, tk) is the transition matrix and v(tk+1, tk)is a zero-mean white process noise with covarianceQ(tk+1, tk). The measurement equation in Cartesiancoordinates is defined in (7).The local trackers can not estimate the sensor

biases based on their own local measurements.

2Higher order processes can be used as long as their models areknown.

CORRESPONDENCE 1141

Therefore, the local trackers assume there are nobiases in the measurements and the local estimates arebiased estimates. The measurement model assumed bylocal trackers is

z0s (k) =Hx(k)+ws(k): (11)

Note that the measurement equation (11) is differentfrom (7), with no bias term in (11). The localestimates are bias-ignorant state estimates,3 and,consequently, there is a model mismatch problem. Themeasurement in model (11) is denoted by superscriptzero since it assumes the sensor biases to be zero.1) The Pseudomeasurement of the Bias Vector: As

discussed in [27], we construct a pseudomeasurementbased on their local bias-ignorant Kalman filters. Weassume there are M = 2 sensors. This can be easilyextended to the case where M is larger than two.Here we assume that the locations of the sensors areknown. The local track estimate at sensor s is x̂s(k j k).We define, for s= 1,

zb1 (k+1)¢=W†1 (k+1)[x̂1(k+1 j k+1)

¡ (I¡W1(k+1)H(k+1))F(k)x̂1(k j k)]=H(k+1)F(k)x(k) +H(k+1)v(k)

+B1(k+1)C1(k+1)¯1(k+1)+w1(k+1) (12)

where the pseudoinverse of the Kalman filter gain4 atthe local tracker s is

W†s¢=(W0

s Ws)¡1W0

s : (13)

Similarly, we define5 for s= 2

zb2 (k+1)¢=W†2 (k+1)[x̂2(k+1 j k+1)

¡ (I¡W2(k+1)H(k+1))F(k)x̂2(k j k)]=H(k+1)F(k)x(k) +H(k+1)v(k)

+B2(k+1)C2(k+1)¯2(k+1)+w2(k+1): (14)

Note that the true state vector x(k) and the processnoise v(k) in (12) and (14) are the same and theycancel if we subtract (12) and (14). Consequently, wedefine a pseudomeasurement of the bias vector as6

zb(k+1)¢=zb1 (k+1)¡ zb2 (k+1) (15)

which is independent of the target state. Then, wehave the pseudomeasurement equation of the bias

3In the literature (e.g., [1]) these local estimators are called“bias-free,” but this is a misnomer because they are biased but onlythe fusion center can estimate the biases.4Since the gain is, in general, not a square matrix, it is necessary touse the pseudoinverse.5If the measurements are available at the center, there is no need togo through the process of reconstructing them as in (12) and (14)from the local estimates. However, some established communicationlinks transmit only local state estimates.6Differencing of measurements was proposed in [14] but under therather restrictive assumption that one of the sensors has no bias.

vector

zb(k+1) =H(k+1)b(k+1)+ w̃(k+1) (16)

where the pseudomeasurement matrix H(k+1), thebias vector b(k+1) and the pseudomeasurement noisew̃(k+1) are defined as

H(k+1) ¢=[B1(k+1)C1(k+1) ¡B2(k+1)C2(k+1)]

(17)

b(k+1)¢=·¯1(k+1)

¯2(k+1)

¸(18)

w̃(k+1)¢=w1(k+1)¡w2(k+1): (19)

The bias pseudomeasurement noises w̃ are zero-meanand white, with covariance

R(k+1) = R1(k+1)+R2(k+1): (20)

The whiteness property of (19) is the key to theexact solution for the bias estimate. This method,unlike [41], [23], and [39], achieves the decouplingbetween the state estimation and the bias estimationwithout ignoring or approximating the crosscovariancebetween the state estimate and the bias estimate. Thismethod was called in [27] as the EX method andwas generalized for synchronous sensors with sensorlocation uncertainty in [29].

B. Asynchronous Sensor Bias Estimation with SensorLocation Uncertainty

By indicating arbitrary sampling the times as ti andtj , the dynamic equation (10) and the measurementequation (7) from sensor s are rewritten as

x(tj) = F(tj , ti)x(ti) + v(tj , ti) (21)

zs(tj) =H(tj)x(tj)+Bs(tj)Cs(tj)¯s(tj)+ws(tj)

(22)

where we assume7 the state vector at time tj isx(tj)

¢=[x(tj) _x(tj) y(tj) _y(tj)]

0, F(tj , ti) is the transitionmatrix from time ti to time tj , and the processnoise v(tj , ti)

¢=[vx(tj , ti) v _x(tj , ti) vy(tj , ti) v _y(tj , ti)]

0 iszero-mean white with covariance Q(tj , ti).Consider M=2 asynchronous sensors, as shown

in Fig. 1. The circles represent the measurement setswhich are defined as the measurements of all targetsof opportunity at a specific time. The data rate of aspecific sensor is not necessarily constant, as shownin Fig. 1. A proper time slot is defined in [30] and[31] as the minimum interval starting at the time of

7While the discussion is done in this context, the method can begeneralized at the expense of much more complicated notation.

1142 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 3 JULY 2006

Fig. 1. Asynchronous measurement sets and proper time slots.

a particular measurement such that there is a linearcombination of all the measurements (from the sametarget) within the time slot, which is independent ofthe target state. The proper time slots are shown inFig. 1 with dashed boxes.A proper time slot has the following two

properties.

1) The number of the measurement sets in aproper time slot should be as small as possible. This isbecause with the smaller number of the measurementsets, the earlier the bias estimation can be performedand the updated bias estimates are available earlier.Therefore, nearly real-time bias estimation can beachieved.2) There should be at least one measurement set

from each sensor and at least two measurement setsfrom one sensor.

A state-independent pseudomeasurement of the biasesrequires three measurement sets. For n measurementsets in a proper time slot, n¡2 state-independentpseudomeasurements of the biases will be constructed.Since only the differences of the sensor locations

are observable, we assume the location of the firstsensor is known exactly. The location of the secondsensor is modeled using (4)—(6). In this subsection,we assume there are exactly three measurements ina proper slot and the relationship among the threetimes is ti+1 < ti+2 < ti+3. It can be easily extendedto a more general slot. We distingush two casesbelow.Case 1 There is one measurement from sensor

2 and, consequently, there are two measurementsfrom sensor 1. We can define the asynchronouspseudomeasurement of the biases as a linearcombination of the measurements in a proper timeslot. Since the exact location of the sensor 2 is notavailable, the Cartesian measurement zb2 from sensor2 is not available. However, a Cartesian measurementbased on the nominal location of sensor 2, (X0,Y0),which is denoted as z02, is available. The followingequation describes the relationship between zb2and z02 ,

zb2 (ti) = z02(ti) +¸2(ti) (23)

with

¸2(ti)¢=·¸2x(ti)

¸2y(ti)

¸: (24)

We define the pseudomeasurement of the biasesusing z02 as (see [31] for the selection of ®1,®2)

zb,0(ti+1, ti+2, ti+3) = z02(ti+2)¡ [®1zb1 (ti+1)+®2zb1 (ti+3)]:

(25)

Assuming the dynamic equation of ¸2 between time tiand time tj is

¸2(ti) = F̧ (tj , ti)¸2(tj)+ v¸(tj , ti) (26)

then,

zb,0(ti+1, ti+2, ti+3) =H0(ti+1, ti+2, ti+3)b0(ti+3)

¡ v¸(ti+3, ti+1)+ w̃(ti+1, ti+2, ti+3)(27)

where the asynchronous pseudomeasurement matrixH0(ti+1, ti+2, ti+3) and the asynchronous bias vectorat time ti+3 (including sensor location uncertainty)b0(ti+3) are defined as

H0(ti+1, ti+2, ti+3)

=

264¡®1B1(ti+1)C1(ti+1)¡®2B1(ti+3)C1(ti+3)B2(ti+2)C2(ti+2)

¡I2£2

375(28)

b0(ti+3)¢=

264 ¯1

¯2

¸2(ti+3)

375 : (29)

Case 2 If there is one measurement from sensor1 in a proper time slot and, consequently, there aretwo measurements from sensor 2, we can define thepseudomeasurement of the biases using z02 as

zb,0(ti+1, ti+2, ti+3) = zb1 (ti+2)¡ [®1z02(ti+1)+®2z02(ti+3)]

=H0(ti+1, ti+2, ti+3)b0(ti+3)

¡®1v¸(ti+3, ti+1)+ w̃(ti+1, ti+2, ti+3)(30)

where the asynchronous pseudomeasurement matrixH0(ti+1, ti+2, ti+3) is

H0(ti+1, ti+2, ti+3)

=

264 B1(ti+2)C1(ti+2)

¡®1B2(ti+1)C2(ti+1)¡B2(ti+3)C2(ti+3)¡®1F̧ (ti+3, ti+1)¡®2

375 :(31)

Note that the variance of the pseudomeasurement hasincreased due to the process noise of the dynamicequation of the sensor location uncertainty.Assuming the updated bias estimate vector and its

corresponding variance including the sensor location

CORRESPONDENCE 1143

uncertainty based on up to the last slot are b̂0(ti) and§0(ti), a Kalman filter can be applied to update theestimate and its variance to b̂0(ti+3) and §

0(ti+3),respectively, based on the pseudomeasurementequation using zb,0 from (27) or (30).The above method is referred to as the EXX

method.

IV. THE COMPENSATION OF THE STATE ESTIMATESFOR ASYNCHRONOUS SENSORS

There are two options when it comes to using thebias estimates for compensation.Approach 1 (centralized). The measurements

are compensated with the bias estimates as theyare obtained and the centralized state estimationalgorithm will use these compensated measurements(whose variance is increased by the variance of theresidual bias errors). However, the residual bias errorsare not white and thus, one of the assumptions instate estimation–the whiteness of the measurementerrors–is not completely satisfied. However, if thebias estimates are accurate enough–the bias errorstandard deviations are one half of the measurementnoise standard deviations, then the lack of whitenessof the bias errors can be ignored. Since the biasestimates were obtained from measurements on a largenumber of targets, it is reasonable to assume that theresidual bias errors are negligibly correlated with anysingle target measurement noise.Approach 2 (decentralized). The local filter

outputs (state estimates) are compensated with the biasestimates and their variances are increased becauseof the imperfectness of this correction. The followingtwo assumptions can be made: the impact of the biason the state estimates is assumed to be approximatelyconstant on the position estimates (the same as in (7))and negligible on the velocity estimates; similarlyto Approach 1, the errors in the bias estimates,which are obtained from a large number of targets ofopportunity, can be assumed negligibly correlated withthe state estimates of any single target. The correctedstate (position) variance will have to account for theeffect of the residual bias errors.Following a decentralized approach, these local

state estimates should be fused. However, for thisone needs not only their (corrected) covariances butalso their crosscovariance [7, 12]. Furthermore, thecrosscovariance have to be also adjusted in view ofthe bias compensation. This fusion-with-compensationapproach is a topic being investigated. Consequently,we pursue in the sequel Approach 1.In this section, we assume there are exactly three

measurements in a proper slot and the relationshipamong the three times is ti+1 < ti+2 < ti+3. It can beeasily extended to a more general slot.

The stacked true asynchronous measurements inthe proper time slot are264z

b1 (ti+1)

zb2 (ti+2)

zb1 (ti+3)

375=264 zb1 (ti+1)

z02(ti+2)+¸2(ti+2)

zb1 (ti+3)

375 : (32)

However, since the exact sensor locations arenot available, zb2 (ti+2) is not available. We have thestacked asynchronous measurements based on thefixed location as

Zc¢=

264zb1 (ti+1)

z02(ti+2)

zb1 (ti+3)

375=H ¢264x(ti+1)x(ti+2)

x(ti+3)

375

+

264 B1(ti+1)C1(ti+1)¯1B2(ti+2)C2(ti+2)¯2¡¸2(ti+2)

B1(ti+3)C1(ti+3)¯1

375+264w1(ti+1)w2(ti+2)

w1(ti+3)

375(33)

where the dynamic equation of ¸2, given in (26), isapplied.Using the state dynamic equation (21), omitting

the time indices in Hc, Hb, and w̃ for simplicity, yields

Zc =Hcx(ti+3)+Hbb0(ti+3)+ w̃ (34)

where

Hc¢=

24HF(ti+1, ti+2)F(ti+2, ti+3)HF(ti+2, ti+3)

H

35 (35)

Hb¢=

24B1(ti+1)C1(ti+1) 0 0

0 B2(ti+2)C2(ti+2) ¡F̧ (ti+2, ti+3)¸2(ti+3)B1(ti+3)C1(ti+3) 0 0

35(36)

w̃¢=

24F(ti+1, ti+2)I

0

35v(ti+2, ti+3)+24 I00

35v(ti+1, ti+2)+

240I0

35v¸(ti+3, ti+2)+24w1(ti+1)w2(ti+2)

w1(ti+3)

35 (37)

with b0(ti+3) defined in (29).Assuming the corrected state estimate and the

corresponding variance based on the last proper timeslot is x̂(ti j ti) and P(ti j ti), a Kalman filter is usedto obtain the state estimate at time ti+3 based on thecompact stacked measurement equation (34) with thebias estimate b̂0(ti+3) and variance §

0(ti+3) obtainedusing the EXX method of Section IIIB.Note that in (34), the error in b0 in not white and

its variance can not just be added to the varianceof the measurement noise. However, its effect isminor based on the consistency evaluation after

1144 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 3 JULY 2006

Fig. 2. Geometry of targets and sensors.

compensation. The Schmidt-Kalman approach[21, 38] can be used to evaluate the effect of theresidual biases–the filtered state covariance (actuallythe mean square error matrix) accounting for theresidual biases; however this does not appear to benecessary for the cases studied.

V. SIMULATION RESULTS

In this section, we compare the performanceof the state estimates after the bias estimation andcompensation with two asynchronous sensors.We investigate the tracking performance with biascompensation for different bias combinations, theoffset bias only, the offset and scale biases, and finallythe offset biases, scale biases and sensor locationuncertainty.Consider a scenario with 32 targets and two

asynchronous sensors with the same sampling ratebut offset by half a period. Both sensors are reportingmeasurements at 1 s intervals, but there is a 0.5 s timeoffset between the reporting times of the two sensors.The offset biases for the two sensors are 20 m

and 2 mrad for range and azimuth, respectively. Thescale biases are 3£10¡5 and 2£ 10¡4 for range andazimuth measurements, respectively. The geometryof the targets and sensors is shown in Fig. 2 andthe targets are moving at nearly constant velocitywith _x= _y = 20 m/s. The standard deviations of themeasurement noise variances are ¾r = 10 m and ¾µ =1 mrad for the range and the azimuth measurements,respectively.The dynamics of the target are modeled using

discretized continuous white noise acceleration(DCWNA) models [9] with the power spectraldensities q̃x = q̃y = 6 m

2=s3. The initial bias estimateof sensor s is zero with the initial bias covariance

§s(0 j 0) = diag[(100 m)2,(200 mrad)2,(0:01)2, (0:1)2],s= 1,2.We investigate the tracking performance with

bias compensation for three different bias scenarios.In scenario A, the two asynchronous sensors haveoffset bias only. In the second scenario, scenario B,the sensors have offset and scale biases, and in thelast scenario, scenario C, the sensors have offsetbiases, scale biases, and (for sensor 2) sensor locationuncertainty. The location of the first sensor is assumedknown. The location of the second sensor is modeledusing (4)—(6) with (X20,Y20) = (100 km,0), ¾ = 10 m,and ®= 0:99.The bias estimation errors using the EXX

method of Section IIIB in these three scenarios areconsistent with their corresponding CRLB. The NEES(normalized estimation error squared [9]) of the stateestimates with no compensation is well above theupper limit of the probability interval, while NEESof the state estimates with EXX Compensation ofSection IV is in its acceptance region for everyestimation time, which means the estimator isconsistent with the CRLB. Similar trends are alsoobserved for scenarios B and C.Fig. 3 compares the combined position rms errors

of these three compensation methods in scenario A.The rms position error with EXX compensation issignificantly reduced and it is very close to the rmsposition error with perfect compensation, while,without the compensation, the rms position error isalmost three times the value of perfect compensation.For scenario B and scenario C, the rms positionerrors are shown in Fig. 4 and Fig. 5. The biasestimation and compensation of the state estimatesusing EXX compensation also improve the positionestimates significantly. The difference between EXXcompensation and perfect compensation is larger

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Fig. 3. Combined-position rms errors in scenario A (offset bias only).

Fig. 4. Combined-position rms errors in scenario B (offset and scale biases).

in scenarios B and C than in scenario A. This isbecause more biases bring more uncertainties in biasestimation. The variances of the bias estimates becomelarger because these extra uncertainties require moreparameters to be estimated.In Table I, the medians of the position rms errors

for these three scenarios are listed for comparison.The tracking performance deteriorates when thereare more bias terms in the measurements. It seemsthat the scale bias is an important source affectingthe tracking performance. The scale biases make theposition rms error from about 11.9 m in scenario Arise 0.8 m to about 12.7 m in scenario B. Thelocation uncertainty brings an extra 0.5 m increasein position rms error to 13.2 m as that in scenario C.The deviations of EXX compensation from perfect

compensation are 0.5 m, 1.5 m, and 1.6 m for thesethree scenarios, respectively. The deviation is largerwhen more biases are involved.

VI. CONCLUSIONS

In this paper, the multisensor tracking performancewith bias estimation and compensation is investigated.The tracking performance improvement with biasestimation and compensation for synchronous biasedsensors is discussed. We proposed a solution forthe multisensor bias estimation and compensationproblem for asynchronous sensors with sensorlocation uncertainty. This algorithm can also beapplied to a general time-varying bias estimationand compensation scenario. Performance analysisand simulations are also given to show the tracking

1146 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 3 JULY 2006

Fig. 5. Combined-position rms errors in scenario C (offset, scale bases and location uncertainty).

TABLE IState Position RMS Errors for Three Scenarios

ScenarioA

ScenarioB

ScenarioC

Median of EXX Compensation 11.9 m 12.7 m 13.2 mMedian of Perfect Compensation 11.4 m 11.2 m 11.6 mMedian of No Compensation 29.8 m 35.3 m 36.6 mEXX Compensation at t = 21 s 12.3 m 13.1 m 13.0 mPerfect Compensation at t= 21 s 11.9 m 11.8 m 11.7 mNo Compensation at t = 21 s 28.2 m 34.6 m 35.7 m

performance of the asynchronous sensors. Thetracking performance with different sources of biasesis also evaluated and our simulation shows that thescale bias is one of the dominant factors in trackingperformance.

XIANGDON LINSiRF Technology, Inc.Santa Ana, CA

YAAKOV BAR-SHALOMDept. of Electrical and Computer EngineeringUniversity of Connecticut371 Fairfield Rd.U2157Storrs, CT 06269-2157E-mail: (ybs@ee.uconn.edu)

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RRC Unnecessary for DGPS Messages

The range rate correction (RRC) was useful in reducing the

latency error of pseudo-range correction (PRC), and the data

baud rate. With selective availability (SA) removed, the temporal

variations in the corrections are much smaller, so the issue has

been raised as to whether the RRC term is still necessary. We

provide results on PRC and RRC variation that account for

seasonal, diurnal, and regional differences in the atmosphere

and receiver noise statistics. We concluded that setting RRC

to zero would help to reduce latency error, and supported this

conclusion by static and dynamic tests using commercial receivers

and the Radio Technical Commission for Marine Services

(RTCM) correction message. The results will be used for defining

differential correction messages in the new Version 3 Standard.

The U.S. Coast Guard is conducting tests to address these issues

based on results to date because of the potential benefits of

reducing the data requirements for differential corrections and

thus making available new services over the existing radio-beacon

broadcast links. The results of this paper are expected to provide

a thorough understanding of the factors affecting the temporal

variations of the corrections, and will support the development of

standards worldwide.

I. INTRODUCTION

The DGPS (differential global positioningsystem) infrastructure should contain areference station, a data link, and user applications.The reference station generates corrections by

Manuscript received August 23, 2005; revised November 22, 2005;released for publication April 21, 2006.

IEEE Log No. T-AES/42/3/884478.

Refereeing of this contribution was handled by G. LaChapelle.

This work was supported in part by the Brain Korea 21 (BK-21)Program for Mechanical and Aerospace Engineering Research, theInstitute of Advanced Machinery and Design, Institute of AdvnacedAerospace Technology at Seoul National University, the Ministryof Maritime Affairs and Fisheries of Korea, and John J. McMullenAssociates (JJMA).

0018-9251/06/$17.00 c° 2006 IEEE

measured pseudo-range or carrier range, the calculateddistance from a reference station to each satellite andestimated receiver clock bias, and then it broadcaststhem to the user application. The correction messagescontain the PRC (pseudo-range correction) for DGPS,the CPC (carrier phase correction) for CDGPS(Carrier-phase DGPS), and their time rates, the RRC(range rate correction)

PRC=¡(¡b+ I+T+ ±R) = d¡ ½+ B̂ (1)

CPC=¡(¡b¡ I+T+ ±R+N¸) = d¡Á+ B̂ (2)

where

½ pseudo-range measurement' carrier phase measurement¸ wavelength of carrier phaseN integer ambiguityd distance from reference station to satelliteb satellite clock bias errorB̂ estimated clock bias of receiverI ionospheric delayT tropospheric delay±R orbit error.

The user is generally distant from the referencestation, so it should correct measurement based ona previous or “old” message. To reduce the problemcaused by this time latency, the reference stations havegenerated and sent the corrections with range ratecorrection (RRC). This has compensated the PRC atthe previous time tk¡1, for the time latency tk ¡ tk¡1,lineally as shown in (3) and Fig. 1gPRC(tk)¼ PRC(tk¡1)+RRC(tk¡1) ¢¢t

where ¢t = (tk ¡ tk¡1): (3)

Before the removal of selective availability (SA),the GPS signal had a fluctuating error such thatDGPS users had to receive the PRC as frequently aspossible at a high rate of measurement. RRC has beenuseful in reducing the rate at which the differentialcorrection is broadcast. Even though the PRC did notvary linearly, the linear compensation by the RRCwas valid because the fast-moving SA effect on thePRC was so large that the nonlinearity of the PRCand some errors in the RRC were easily ignored.With SA removed, the temporal variation in the PRCbecame much smaller than before as shown in Fig. 2and primarily dependent on atmospheric variations,consequently the requirement is to focus on thedifference of its effect within various RRC generationtechniques.The RRC is originally the time rate in the PRC,

but the raw PRC is too noisy to use directly. There area number of RRC generation techniques; these involvethe time difference of CPC, that of filtered PRC, and

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