robust least squares approach to passive target localization using ultrasonic receiver array

Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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1

Robust Least Squares Approach to Passive TargetLocalization Using Ultrasonic Receiver Array

Ka Hyung Choi, Won-Sang Ra, So-Young Park, and Jin Bae Park, Senior Member, IEEE

Abstract—A precise range difference (RD) based passive targetlocalization algorithm is proposed for mobile robot applications.To effectively solve the real-time issue, the nonlinear relationbetween the RD information and the target location is removedby introducing the target range as an auxiliary variable tobe estimated. And then, the RD based localization problem isformulated in the setting of linear estimation. The resultantlinear measurement model contains the stochastic parametricuncertainty which causes the severe performance degradation ofthe conventional linear least squares (LS) method when the RDmeasurement noise is not negligible. To cope with this problem,the recently developed linear robust LS (RoLS) estimation theoryis applied for the passive target localization problem. Using thegeometric relation among the ultrasonic receivers, a systematicway to determine the design parameters of the RoLS estimatoris suggested. It is shown that the proposed method can providethe nearly unbiased target location estimates for the wholelocation area. The proposed solution is very practical becauseit is preferable for real-time robot applications owing to itslinear recursive structure. Through the computer simulationsand actual experiments, it is shown that the proposed algorithmguarantees the superior localization performance and the fastconvergence compared to the existing one.

Index Terms—Passive target localization, range difference,ultrasonic receiver array, robust least squares estimation

I. INTRODUCTION

The problem of passive target localization often arises inmany applications such as radar, sonar, and so on [1]-[8].Absolute target localization using passive sensors has beenalso received much interest in the field of robotics becauseit is one of core technologies for developing advanced robotswith autonomous capabilities [9], [10]. Conventional passivetarget localization algorithms are usually based on the bearing,Doppler frequency or range difference (RD) information.Since these information could be obtained from relativelycheap sensors, a passive target localization system might bevery attractive solution for various robot applications. In thispaper, we focus on the RD based passive target localizationalgorithm using the cruciform ultrasonic receiver array shown

Manuscript received December 26, 2011. Accepted for publication March22, 2013. This research was supported by Agency for Defense Develop-ment(Project No.: ADD-11-02-03-01).

Copyright c⃝ 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purpose must beobtained from the IEEE by sending a request to [email protected].

Ka Hyung Choi and Jin Bae Park are with the School of Electrical andElectronic Engineering, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul, 120-749, Republic of Korea.

Won-Sang Ra and So-Young Park are with the School of Mechanicaland Control Engineering, Handong Global University, 558 Handong-ro,Heunghae-eup, Buk-gu, Pohang, Gyeongbuk, 791-708, Republic of Korea (e-mail: [email protected]).

Fig. 1. Passive target localization using an RD measuring system

in Fig. 1 which provides the RD measurement by comparingthe phase difference of the measured target signals.

It is well-known that RD based passive target localizationcan be characterized as a nonlinear parameter estimationproblem [11], [12]. Assuming that the RD measurement noiseis zero mean white Gaussian, the RD based passive targetlocalization problem is reduced to the minimization of aquadratic cost in terms of maximum likelihood or equivalentlynonlinear least squares estimation. Even though this approachcan be used to formulate the RD based passive target lo-calization problem in a mathematically rigorous way, it hasa few limitations that should not be overlooked in practicalapplications. The RD based passive target location shouldrely on numerical methods so that it requires a computationalburden and thus be unsuitable for real-time implementation[13], [14]. In addition, its localization performance tends to besensitive to the initial guess hence it cannot guarantee globaloptimality in many cases.

The above mentioned flaws of existing nonlinear local-ization schemes are inherent to the nonlinear relationshipbetween the source location and the RD measurement. Basedon this observation, a number of researchers have attempted tosolve the RD based passive target localization problem usingframework of a linear estimation. To alleviate the nonlinearityof the problem, the distance from the reference sensor to thetarget is set as an auxiliary parameter to be estimated, and theRD information function is expressed as a linear combinationof passive source location and distance [15], [16]. Based onthis relation, a linear regression model for the passive target



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Fig. 2. Block diagram for RD measuring circuit

localization could be readily obtained by replacing the trueRD and sensor position with their measurements. However,the resultant linear regression model has somewhat differentcharacteristics from the conventional model, the measurementmatrix of which is perfectly known without parametric uncer-tainties. In other words, the unexpected correlations betweenthe errors contained both in the measurement and measurementmatrix should be addressed for the development of an RDbased linear passive target location estimator.

In order to mitigate the complexity of the problem, theapproximate linear estimation scheme based on the so-calledpseudo linear (PL) regression model was taken into consider-ation for passive target localization [15]-[18]. This approachrequires the basic assumptions that the magnitudes of the RDerrors are small and that the performance degradation due tothese errors is negligible. In such a case, the RD based passivetarget localization problem could be solved by applying thenominal least squares (NoLS) estimator for the PL regressionmodel. The linear recursive structure of the passive targetlocation estimate is expected to have fast and consistentconvergence properties regardless of the initial location infor-mation of the source. In this respect, the approximate linearestimation scheme based on the PL regression model might bea feasible alternative to nonlinear passive target localizationmethodologies. However, if the standing assumptions are vio-lated under low SNR conditions, its localization performanceis severely deteriorated. This problem is more prominent atlong-range target engagement scenarios because intentionallyignored errors are multiplied by the target location accordingto the explicit linear regression model.

Meanwhile, the total least squares (TLS) estimation theory

has been attempted to improve passive target localizationperformance [19], [20]. The motivation of this approach camefrom the idea that the RD based passive target localizationproblem can be described by the conventional error-in-variable(EIV) model if the correlation between errors appears in themeasurement and the measurement matrix is ignored. Thisapproach could reduce the localization error in comparisonwith the NoLS-based localization algorithm. Nevertheless,since both of the measurement and the measurement matrixare expressed as a function of RD measurement errors, thecorrelation between them leads to limitations on performanceimprovement by the TLS passive target localization algorithm[21].

To overcome the limitations of the existing localizationmethods, this paper re-interprets the RD based passive targetlocalization problem through the recently developed RoLSframework [22]. Based on the basic observation that theestimate of the NoLS estimator for the uncertain linear regres-sion model contains the auto-correlation of the measurementmatrix error and cross-correlation between the measurementmatrix error and the measurement error, the RoLS estimatorcould remove these correlations using the predefined stochasticinformation and provide nearly unbiased estimates.

Therefore, in this paper, we devise a RD based passivetarget localization algorithm using the RoLS estimator. Thecruciform ultrasonic receiver array placed orthogonally alongthe coordinate system is used for developing a low-cost RDmeasuring sensor. In Section II, the operational principle of ourRD measuring sensor is explained, and the RD measurementequation is derived using the geometrical relation between thereceiver array and the transmitter(or target). In addition, theRD based localization problem is presented with the derivationof the estimation error in the NoLS localization algorithm. InSection III, the RoLS localization algorithm, which is designedto compensate for the NoLS estimation error, is proposed. InSection IV, the proposition is verified with simulations andexperimental results.

II. PASSIVE TARGET LOCALIZATION PROBLEM USINGRANGE DIFFERENCE MEASUREMENTS

A. RD measuring system using an ultrasonic receiver array

In this subsection, our RD measuring system for passivetarget localization is briefly introduced. As shown in Fig. 1,the RD measuring system is constructed of five ultrasonicreceivers disposed in a cruciform arrangement, which canprovide RD measurements for x and y directions. The systemis installed in a roof with height h, and acquires acousticsignals transmitted by an ultrasonic transceiver mounted on amobile robot. The signal processing unit of our RD measuringsystem is depicted in Fig. 2. Receiver #0, located at the center,is the reference for measuring RD information. At the firststage, the acoustic signal sj obtained from jth receiver isamplified and is converted to the rectangular signal zj . Sincereceivers #1 ∼ #4 are separated from the reference by adistance d, the acquired signals z1 ∼ z4 are shifted from thereference signal z0. It is clear that, comparing the phase ofzj with that of z0, the RD measurements can be obtained.



3

To do this, the logic circuit performs the operation z0 · zj forj = 1∼ 4 at the second stage. Then, the logic output passesthrough the low-pass filter to produce voltage proportional tothe RD measurement.

The proposed RD measuring system is advantageous be-cause it can be implemented with a compact size (the actualdimensions of our prototype are 50 × 50[mm]) as well aswith low power consumption. However, since the receiversare close, the localization performance tends to be sensitive toRD measurement noises. Therefore, it is necessary to develop aprecise localization algorithm for real-time robot applications.

B. Linear uncertain measurement model

Before designing a precise localization algorithm, the rela-tionship between RD measurement and target position shouldbe described. The origin of this system is set to be verticallybelow the reference as shown in Fig. 1. Then the RD obtainedfrom the j th receiver is defined as

rj , dt,j − dt,0 (1)

where dt,j denotes the distance between the j th receiver andthe target. If the j th receiver and the target are located at(xj , yj , zj) and (xt, yt, zt) respectively, the distance dt,j isdefined as follows:

dt,j ,√(xt − xj)2 + (yt − yj)2 + (zt − zj)2 (2)

Because the relation between the target position and the avail-able RD information is described by the nonlinear equation(1), the RD based target localization is generally classified asa nonlinear estimation problem. However, to avoid the compu-tational complexity of nonlinear target localization schemes, inthis paper, we try to solve this problem within the frameworkof linear estimation. To this end, the linear relation between afunction of RD information and the target position should bederived.Since dt,j = rj + dt,0 from (1), squaring both sides gives usthe following result.

r2j + d2t,0 − d2t,j = −2rjdt,0 (3)

From the known geometry of the receivers, it is clear thatzt = 0 and

(x0, y0, z0) = ( 0, 0, h),(x1, y1, z1) = ( d, 0, h),(x2, y2, z2) = ( 0, d, h),(x3, y3, z3) = (−d, 0, h),(x4, y4, z4) = ( 0,−d, h).

(4)

By the definition of dt,j in (2), the equation (3) can bemodified as

r2j+ (x20+ y20)− (x2

j+ y2j )+ 2(xj− x0)xt+ 2(yj− y0)yt= −2rjdt,0.

Thus, one can obtain the linear relation between the functionof RD, fRD(rj), and the target location (xt, yt, dt,0).

fRD(rj)= −2(xj − x0)xt − 2(yj − y0)yt − 2rjdt,0 (5)

where

fRD(rj) , r2j + (x20 + y20)− (x2

j + y2j ) = r2j − d2

Applying the above relation for the receivers #1∼#4 resultsin the following linear regression model.

φ = Hξ (6)

In (6), it was defined that

φ ,

r21 − d2

r22 − d2

r23 − d2

r24 − d2

, H , −2

d 0 r10 d r2

−d 0 r30 −d r4

, ξ ,

xt

ytdt,0

.

In practice, the available information is not the true RD rj butrather its measurement contaminated by additive noises. Thenoise corrupted RD measurements are defined as follows:

rj , rj + δrj (7)

where ϑ is the measurement of the true parameter ϑ and δϑindicates its measurement error. Without loss of generality, itcan be also assumed that the RD measurement noises are whiteand uncorrelated each other. Moreover, it is also assumed thattheir statistics are known as follows:

E[δrj ] = 0, var[δrj ] = σ2j (8)

By using the RD measurement equation at time instance k,the linear regression model (6) can be written as the followinguncertain linear regression model.

φk = Hkξ + νk

= (Hk −∆Hk)ξ + νk (9)

where

φk,

r21(k)−d2

r22(k)−d2

r23(k)−d2

r24(k)−d2

, Hk,−2

d 0 r1(k)0 −d r2(k)

−d 0 r3(k)0 −d r4(k)

,

ξ ,

xt

ytdt,0

, νk,

2r1(k)δr1(k)−δr21(k)2r2(k)δr2(k)−δr22(k)2r3(k)δr3(k)−δr23(k)2r4(k)δr4(k)−δr24(k)

,

Hk,−2

d 0 r1(k)0 d r2(k)

−d 0 r3(k)0 −d r4(k)

, ∆Hk,−2

0 0 δr1(k)0 0 δr2(k)0 0 δr3(k)0 0 δr4(k)

C. RD based passive target localization problem

To design a target location estimator based on the linearLS technique, it is necessary to effectively handle the linearuncertain measurement model (9). However, the existence ofthe stochastic parametric uncertainty ∆Hk makes the problemdifficult. Furthermore, since both the measurement matrixuncertainty ∆Hk and the measurement noise νk are functionsof the RD measurement error δrj(k), their correlation shouldalso be taken into consideration for target localization. Todesign a new localization algorithm, the estimation error of



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a nominal LS estimator should be quantitatively analyzed. Fornotational convenience, let us consider the following vector-valued regression model.

φk = Hkξ + νk (10)

= (Hk −∆Hk)ξ + νk (11)

where

φk ,

φ1

φ2...φk

, Hk ,

H1

H2

...Hk

, νk ,

ν1

ν2

...νk

,

Hk ,

H1

H2

...Hk

, ∆Hk ,

∆H1

∆H2

...∆Hk

In the above, the superscript k indicates the number ofaccumulated data. Using this vector-valued representation, theLS solution can be derived in a straightforward manner. From(10), the optimal least squares (OLS) solution is defined as

ξOLS

={(Hk)T (Hk)

}−1{(Hk)T φk

}. (12)

Although the OLS solution provides the unbiased performanceE[ξ

OLS− ξ] = 0, it is not implementable because, in this

case, the available information is not the true measurementmatrix Hk but the error contaminated measurement matrixHk. This means that the LS solution is constructed with onlythe erroneous measurement matrix Hk.

If ∆Hk is small enough, the approximated linear regressionmodel, also known as the pseudo linear (PL) regression model,will be taken into consideration for the RD based targetlocalization problem [15]-[18].

φk ≈ Hkξ + νk (13)

For the PL regression model, the following NoLS solution,which uses on the available information Hk, can be derived

ξNoLS

={(Hk)T (Hk)

}−1{(Hk)T φk

}. (14)

It is well known that this NoLS solution shows a fast conver-gence property and that it can also be realized using a recursiveformula, which imposes little computational burden. However,if the standing assumption of the small size of the uncertaintyis violated under the low SNR conditions, its localizationperformance is severely deteriorated due to the intentionallyneglected term ∆Hk in (9).

The performance degradation of the NoLS solution can beshown by replacing φk in the NoLS solution (14) with (11)as follows:

ξNoLS

= ξ−αkξ + βk︸︷︷︸estimation errors

(15)

where

αk , PNoLSk (Hk)T∆Hk, βk , PNoLS

k (Hk)Tνk,

and

PNoLSk ,

{(Hk)T (Hk)

}−1

.

Since the variables (Hk,∆Hk,νk) in (15) are functions of theRD measurement error δrj(k) as defined in (9), the followingcorrelations exist.

Wk = E{(Hk)T (∆Hk)} = 0,

Vk = E{(Hk)T (νk)} = 0

(16)

If the above correlations cannot be ignored, the localizationperformance of the NoLS solution might be severely degradedespecially in a low SNR environment. Moreover, from (15),the estimation error tends to be proportional to the targetlocation, (ξ

NoLS− ξ) ∝ ξ. As a result, the performance

deterioration could be serious, especially in long-range targetscenarios.

Remark 1: The total least squares (TLS) estimation algo-rithm has been attempted for (9) to achieve the enhancedtarget localization performance [19], [20]. The motivation forthis approach is based on the idea that, if the correlationsbetween ∆Hk and νk are ignored, the measurement model(9) is reduced to the error-in-variable (EIV) model. Thus, theperformance of the TLS-based target localization is superior tothat based on NoLS, but the performance improvement mightbe restrictive due to the unavoidable correlation (16).

The NoLS solution suffers from estimation error (−αkξ +βk), which is caused by the uncertainty in the measurementmatrix. The simplest way to enhance the target localizationalgorithm is to properly compensate for the derived estimationerror of NoLS. Recently, in order to compensate for the NoLSestimation error, the RoLS algorithm was proposed, whichuses the a priori stochastic information of ∆Hk and νk toapproximate the estimation error [22]. The required stochasticinformation is in regards to the correlations between Hk

and ∆Hk and between Hk and νk. The RoLS algorithm issummarized in Table I. This algorithm is designed for thelinear uncertain measurement model, which has same form as(9) in our problem. Therefore, the target location estimatorcan be designed by applying the RoLS estimation theory for(9). According to the RoLS estimation theory, the componentscharacterizing the NoLS estimation error in (15) could beapproximated by using the additional statistical knowledge,Wk and Vk.

αk = PNoLSk (Hk)T∆Hk (17)

≈ PNoLSk E[(Hk)T∆Hk]︸︷︷︸

,Wk

= PNoLSk

k∑i=1

Wi , αk,

βk = PNoLSk (Hk)Tνk (18)

≈ PNoLSk E[(Hk)Tνk]︸︷︷︸

,V k

= PNoLSk

k∑i=1

Vi , βk

Replacing αk and βk with their approximations αk and βk,from (15), the following error-compensating solution can be



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TABLE IROBUST LEAST SQUARES ESTIMATOR [22]

linear uncertainmeasurement model φk=(Hk−∆Hk)ξ+νk, Hk is available

known statisticalinformation

E[νk] = 0, E[∆Hk] = 0,

E{(Hk)T (∆Hk)} = Wk,

E{(Hk)T (νk)} = V k

estimateξRoLS

= PRoLSk

{(Hk)T (φk)−V k

}PRoLSk =

{(Hk)T (Hk)−Wk

}−1

estimation errorproperty E[(PRoLS

k )−1(ξRoLS − ξ)] = 0

considered.

ξRoLS

= (I − αk)−1

(ξNoLS

− β)

(19)

Therefore, the RoLS estimator can be understood as a possibleerror compensating strategy for the NoLS scheme. The follow-ing RoLS recursion can also be derived. For more details onthe derivation of RoLS recursion, see [22] and the referencestherein.

ξRoLS

k = (I + PRoLSk Wk)ξ

RoLS

k−1 (20)

+ PRoLSk Hk(φk − Hkξ

RoLS

k−1 )− PRoLSk Vk,

PRoLSk = {(PRoLS

k−1 )−1 + (Hk)T Hk −Wk}−1 (21)

Therefore, only Wk and Vk need to be defined for thelocalization algorithm for the cruciform ultrasonic receiverarray. Inserting Hk, ∆Hk and νk defined in (9) into (16),the design parameters are calculated as follows:

Wk =

0 0 00 0 0

0 0 4∑4

j=1 σ2j

, Vk =

−4d(σ21 − σ2

3)−4d(σ2

2 − σ24)

−6∑4

j=1 rjσ2j

. (22)

Remark 2: As mentioned before, the design parameters Wk

and Vk should be properly chosen for the RoLS locationestimator. While Wk is composed of the known stochasticinformation of the RD measurement noise variances, the(3, 1) element of Vk in (22) contains the unavailable true RDrj . Fortunately, the geometric relations among the cruciformultrasonic receivers allows us to solve this problem. If d ≪ hor d ≪ dt,0, the transmitted acoustic signal is assumed toarrive at the ultrasonic receivers in parallel. In such a case,the following approximation is appropriate.

r1 + r3 ≈ 0, r2 + r4 ≈ 0

In addition, under the assumption that the zero-mean RDmeasurement noises are uncorrelated and the variances arenearly the same, the design parameter can be set as Vk ≈ 0.This is a major reason for the use of a cruciform ultrasonicreceiver array for our RD measuring system.

III. PERFORMANCE EVALUATION

In this section, the performance of the proposed RoLSpassive target localization algorithm is demonstrated throughsimulation and experiment. In the performance analysis, thereceiver span d is 16[mm] and its height h is 2500[mm].

To verify the performance of proposed method, it is com-pared with the NoLS passive target localization algorithm. Asshown in (15), the RD measurement noise variance and thetarget position are the main factors affecting the localizationperformance. Therefore, the analysis is performed under twoscenarios: a short-range case, where

√x2t + y2t = 250[mm],

and a long-range case, where√x2t + y2t = 750[mm].

A. Simulation results

To check the performance variation along the SNR condi-tion, the variance of the RD measurement error is changedfrom σ2 = 10−5 to σ2 = 10−3. The simulation results for theentire azimuth are shown in Figs. 3 and 4 for the short-rangeand long-range cases, respectively. The performance analysisis carried out with 100 Monte-Carlo trials.

Figures 3 and 4 show the mean errors of the target positionestimates obtained by NoLS and the proposed schemes. Asexpected, the localization performance of the NoLS methodtends to be severely distorted as the RD measurement noisevariance increases. This tendency is intensified in the long-range scenario. When the variance of the RD measurementerror is σ2 = 10−3[mm2], as in the short-range case, themaximum error of the NoLS estimates for both x and ypositions is 68.93[mm]. On the other hand, the proposedalgorithm shows excellent localization performance, and thelocalization error is near to zero. Mathematically, the estimateof the RoLS passive target localization algorithm converges tothe true value in probability. The convergence property of ourRoLS localization algorithm is proved in Appendix.

The localization error of the NoLS is determined by theazimuth angle of the target. For example, for an azimuthangle of zero, the estimation error of the x-position has amaximum value, but that of y-position becomes zero. Thisresult can be explained with the compensation term used inthe proposed RoLS localization algorithm. As mentioned inSection II, the RoLS compensates for the estimation errorof the NoLS. This suggests that the compensation term canbe utilized to determine the stochastic behavior of the NoLSestimation error (see Lemma 1 in Appendix). For brevity ofanalysis, assuming the target is on the x-axis, the matrix HTHbecomes

HTH = −2

8d2 0 4d(r1 − r3)0 8d2 0

4d(r1 − r3) 0 4∑4

j=1 r2j

. (23)

In such a case, using (43) and (44) in Lemma 1, the estimationerrors can be approximated as

αkξ ≈(HTH −W

)−1Wξ =

−16dR13dt,0Sσ

8d2(4Sr−4Sσ)−16d2R13

032d2dt,0Sσ

8d2(4Sr−4Sσ)−16d2R13

,

βk ≈(HTH −W

)−1V =

v1(Sr−Sσ)−v34dR13

8d2(4Sr−4Sσ)−16d2R13v28d2

v1(Sr−Sσ)−v38d2

8d2(4Sr−4Sσ)−16d2R13

where Sr ,

∑4j=1 r

2j , Sσ ,

∑4j=1 σ

2j , R13 , r1−r3, and the

components of correlation V are denoted as v1, v2, and v3.



6

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

(a) x position error of NoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

σ2=10-3

σ2=10-4

σ2=10-5

(b) y position error of NoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

(c) x position error of RoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

σ2=10-3

σ2=10-4

σ2=10-5

(d) y position error of RoLS

Fig. 3. Means of target localization errors according to RD measurementnoise variance(short-range case)

In this simulation, since the variances of the RD measure-ment error are equal, the first and the second components ofV become zero by the definition of V in (22). As a result, theNoLS estimation error becomes

−αkξ + βk ≈

16dR13dt,0Sσ−v34dR13

8d2(4Sr−4Sσ)−16d2R13

0−32d2dt,0Sσ−v38d

2

8d2(4Sr−4Sσ)−16d2R13

. (24)

Therefore, the localization error of the y-axis is nearly zerowhen the target azimuth angle is zero. This theoretical analysisresult coincides with the results of our simulation.

B. Experimental results

To show the effectiveness of the proposed method in realscenarios, experiments are carried out at several test points.Since the cruciform ultrasonic receiver array is symmetric,the test points are restricted to the upper-right quadrant. Testpoints at every 15 degrees are used as shown in Fig. 5. Whiletest points #1 to #7 are for the short-range scenario, thosefrom #8 to #14 are for the long-range scenario.

In practice, it is difficult to know the true variances of thewhole localization region. Therefore, we approximated themusing the averaged values of the noise variances modeled forthe area of interest. The approximated variances are σ2

1 =0.6665 × 10−3, σ2

2 = 0.2266 × 10−3, σ23 = 0.6605 × 10−3,

and σ24 = 0.2231 × 10−3. Since σ2

1 ≈ σ23 and σ2

2 ≈ σ24 ,

relying on Remark 2, the design parameters of the proposed

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

(a) x position error of NoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

σ2=10-3

σ2=10-4

σ2=10-5

(b) y position error of NoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

(c) x position error of RoLS

-40 0 40 80 120[mm]

30

-150

60

-120

90

-90

120

-60

150

-30

180 0°

σ2=10-3

σ2=10-4

σ2=10-5

(d) y position error of RoLS

Fig. 4. Means of target localization errors according to RD measurementnoise variance(long-range case)

250 500 750[mm]

30

60

90

0°#1

#2

#7

#8

#14test point

Fig. 5. Test points used for evaluating localization performance

target localization algorithm can be set as

Wk =

0 0 00 0 00 0 0.0018

, Vk =

000

. (25)

The experimental results are depicted in Figs. 6 and 7 forshort- and long-range cases, respectively.

The RoLS passive target localization algorithm shows bet-ter performance than the NoLS algorithm. The localizationerror of the proposed method is maintained under 26[mm].However, the NoLS estimation error rapidly increases alongthe target range. In the long-range case, the maximum errorof the NoLS is −136[mm], even for small RD measurementerror.

Next, to test the convergence property of the proposition,the experiment is performed using a fixed point (test point#2). From Fig. 8, it can be seen that the proposed algorithm



7

1 2 3 4 5 6 7-40

-30

-20

-10

0

10

index of test point

x po

sitio

n er

ror

[mm

]

RoLSNoLS

1 2 3 4 5 6 7-40

-30

-20

-10

0

10

index of test point

y po

sitio

n er

ror

[mm

]

Fig. 6. Experimental results along test points #1−#7 (short-range case)

8 9 10 11 12 13 14-150

-100

-50

0

50

index of test point

x po

sitio

n er

ror

[mm

]

RoLSNoLS

8 9 10 11 12 13 14-150

-100

-50

0

50

index of test point

y po

sitio

n er

ror

[mm

]

Fig. 7. Experimental results along test points #8−#14 (long-range case)

shows a fast convergence rate and guarantees unbiasedness atlong ranges.100 number of samples are enough to converge to the true

location. Due to its superior performance and fast convergenceproperty, our method is a possible practical solution to real-time robot localization problems.

IV. CONCLUSIONS

A RoLS passive target localization algorithm has beenproposed for robot applications. From the geometrical relationbetween the ultrasonic receiver array and a target, the un-certain linear regression model was derived, the measurementmatrix of which was contaminated with stochastic parameteruncertainties. In order to drastically enhance the localizationperformance, the estimation errors of the NoLS scheme wereanalyzed and successfully eliminated by using the recentlydeveloped RoLS estimation theory. In addition, a systematicmethod to determine the design parameters of the RoLSlocation estimator was suggested. Through both the simulationand experimental results, it was shown that the proposedmethod was able to provide excellent performance under

0 50 100 150 200 250180

200

220

240

260

280

300

step

x po

sitio

n er

ror[

mm

] TrueRoLSNoLS

0 50 100 150 200 25050

55

60

65

70

75

80

step

y po

sitio

n er

ror[

mm

]

Fig. 8. Convergence property of the proposed algorithm at test point #2

various SNR conditions and target ranges. Moreover, due to itslinear recursive structure, the proposed method is suitable forreal-time implementation on microprocessors. Consequently,the proposed method could be a good starting point for thedevelopment of a precise robot localization system.

V. APPENDIX

It can be shown that the target location estimate of theproposed algorithm converges to the true states as follows:

Theorem 1: (Convergence of the RoLS passive target loca-tion estimate) Under the assumption that (PRoLS

k )−1 > 0,the unbiased measurement errors of RD are independentidentically distributed (i.i.d.) for time, and their stochasticproperties are known exactly, the estimate of the RoLS basedpassive target localization for the linear regression model (9)converges to the true position of the source in probability.

ξRoLS p−−−−→ ξ (26)

�

proof. For the case that the measurement sequences of RD areassumed to be i.i.d. for time instant k and its RD measurementerror is zero mean,

E[∆Hk] = E[∆H] = 0,

E[Hk] = E[Hk +∆Hk] = H,(27)

and the correlations (22) are known exactly as

Wk = W, Vk = V. (28)

By using the above relations, one gets

E[(Hk)THk] = E[(Hk)THk + (∆Hk)THk]

= E

[k∑

j=1

HTH +∆HTj H

]= kHTH.

(29)



8

With the similar way, the correlation matrices W k and V k

defined in Table I become

W k =E[(Hk)T∆Hk]=E

[k∑

i=1

HTi ∆Hi

]=kW, (30)

V k =E[(Hk)Tνk]=E

[k∑

i=1

HTi νi

]=kV. (31)

Since

E[(Hk)T Hk −W k] = E[(Hk)T Hk − (Hk)T∆Hk]

= E[(Hk)THk] (32)= kHTH,

and

E[(Hk)Tνk − V k] = E[(Hk)Tνk − (Hk)Tνk] = 0, (33)

from the definition of PRoLSk in Table I, one can conclude the

following convergence in probability.

1

k

{(Hk)T Hk −W k

}︸︷︷︸

=(kPRoLSk )−1

p−−−−→ HTH,

1

k

{(Hk)Tνk − V k

}p−−−−→ 0

At this point, it is noteworthy that the RoLS target locationestimate exists when (PRoLS

k )−1 > 0. This implies thatPRoLSk can be regarded as a continuous function [26]. Thus,

(kPRoLSk )

p−−−−→ (HTH)−1. (34)

From Table I, the RoLS estimate of target location is rewrittenas

ξRoLS

= PRoLSk {(Hk)T φk−V k}

= PRoLSk {(Hk)T (Hkξ+νk)−V k} (35)

= PRoLSk (Hk)THkξ+PRoLS

k {(Hk)Tνk−V k}.

Using Slutsky’s theorem [27] and applying (29) and (34), onecan derive the stochastic behavior of the first term as follows:

plimk→∞

[kPRoLS

k

]×[1k (H

k)THkξ]

=[plimk→∞

kPRoLSk

][plimk→∞

1k (H

k)THk]ξ = ξ.

(36)

In the same way, using (33), the stochastic behavior of thesecond term is

plimk→∞

[kPRoLS

k

]×[1k{(H

k)Tνk−V k}]

=[plimk→∞

kPRoLSk

][plimk→∞

1k{(H

k)Tνk−V k}]=0.

(37)

From the results of (36) and (37), one can see that the RoLSestimate converges to the true target position in probability.

ξRoLS p−−−−→ ξ

�

From (19), the RoLS solution can be regarded as a errorcompensated version of NoLS solution.

ξNoLS

+ (εs − εb)︸︷︷︸compensation terms

= ξRoLS

(38)

where εs , αkξRoLS

and εb , βk. Theorem 1 implies thateach compensation term converges to the true estimation errorεs = αkξ or εb = βk. This result allows us to analyze theNoLS estimation error quantitatively using the approximatedcompensation terms.

Lemma 1: (Stochastic behavior of the error correctionterms of the RoLS estimator) Under the i.i.d. assumption of theRD measurements for time instant k, the differences betweenthe error correction terms of the RoLS estimator (εs, εb) andthe estimation errors of the NoLS estimator (εs, εb) convergeto zero in probability.

εs − εsp−−−−→ 0, εb − εb

p−−−−→ 0 (39)

�proof. From Theorem 1,

E[(Hk)T Hk] = kHTH + kW, (40)

E[(Hk)T∆Hk] = kW. (41)

As shown in [22], the existence condition of the RoLS targetlocalization solution guarantees the existence of the NoLSsolution as (PNoLS

k )−1 , (Hk)T Hk > 0. Thus, from (40),the inverse of (kPNoLS

k )−1 also satisfies

kPNoLSk

p−−−−→ (HTH +W )−1. (42)

By Slutsky’s theorem and the results of (41) and (42), thestochastic behavior of εs can be characterized as follows:

plimk→∞

εs = plimk→∞

PNoLSk (Hk)T∆Hkξ

=[plimk→∞

kPNoLSk

][plimk→∞

1k (H

k)T∆Hk]ξ

=(HTH +W

)−1Wξ.

(43)

Using (33), the stochastic behavior of εb is

plimk→∞

εb = plimk→∞

PNoLSk (Hk)Tνk

=[plimk→∞

kPNoLSk

][plimk→∞

1k (H

k)T (νk)]

=(HTH +W

)−1V.

(44)

Using (26), (30), (31) and (42), the stochastic behavior of theerror correction terms of the RoLS estimator, εs and εb canbe easily shown as follows:

plimk→∞

εs = plimk→∞

PNoLSk W kξRoLS (45)

=[plimk→∞

kPNoLSk

][plimk→∞

1

kW k

][plimk→∞

ξRoLS]

=(HTH +W

)−1Wξ,

plimk→∞

εb = plimk→∞

PNoLSk V k (46)

=[plimk→∞

kPNoLSk

][plimk→∞

1

kV k

]=

(HTH +W

)−1V



9

These results are same with (43) and (44), hence (39) issatisfied.

�

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[1] M. Wax and T. Kailath, “Optimum localization of multiple sources bypassive array,” IEEE Trans. Acoust., Speech, Signal Process., vol. 31,no. 5, pp. 1210-1217, 1983.

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[15] J.O. Smith and J.S. Abel, “Closed-form least squares source locationestimation from range-difference measurements,” IEEE Trans. Acoust.,Speech, Signal Process., pp. 1661-1669, 1987.

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Ka Hyung Choi received his B.S. and M.S. de-grees in Electrical and Electronic Engineering fromYonsei University, Seoul, Korea in 2006 and 2008,respectively. Currently, he is studying for his Ph.D.degree with the Dept. of Electrical and ElectronicEngineering at Yonsei University, Seoul, Korea. Hisresearch interests include frequency estimation andlocalization.

Won-Sang Ra received his B.S. degree in ElectricalEngineering and his M.S. degree in Electrical andComputer Engineering, as well as his Ph.D. degreein Electrical and Electronics Engineering from Yon-sei University, Seoul, Korea, in 1998, 2000, and2009, respectively. From March 2000 to February2009, he was with the Guidance and Control De-partment of the Agency for Defense Development,Daejeon, as a Senior Researcher. Since 2009, he hasbeen with the School of Mechanical and ControlEngineering, Handong Global University, where he

is currently an Assistant Professor. His main research topics are related to therobust filtering theory and its applications to autonomous vehicle guidanceand control.

So-Young Park received her B.S. degree in theSchool of Mechanical and Control Engineering fromHandong Global University, Pohang, Korea, in 2013.She is currently pursuing her Ph.D. degree in theSchool of Mechanical and Aerospace Engineeringat Seoul National University. Her research interest ismainly focused on the personal navigation systems.

Jin Bae Park received his B.S. degree in Elec-trical Engineering from Yonsei University, Seoul,Korea, in 1977 and his M. S. and Ph.D. degrees inElectrical Engineering from Kansas State University,Manhattan, in 1985 and 1990, respectively. Since1992, he has been with the Department of Electri-cal and Electronic Engineering, Yonsei University,Seoul, Korea, where he is currently a Professor.His research interests include robust control andfiltering, nonlinear control, mobile robot, fuzzy logiccontrol, neural networks and genetic algorithms. He

is serving as vice-president for the Institute of Control, Robotics and Systems.He had served as an Editor-in-chief for the International Journal of Control,Automation and Systems from 2006 to 2010.

robust least squares approach to passive target localization using ultrasonic receiver array

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