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Improving Source Localizationin NLOS Conditions via Ranging Contraction

Giuseppe Destino, Student Member, IEEE andGiuseppe Thadeu Freitas de Abreu, Senior Member, IEEE Centre for Wireless Communications

University of Oulu, FinlandE-mails: destino@ee.oulu.fi, giuseppe@ee.oulu.fi

Abstract—In this paper a novel distance-based source local-ization algorithm is proposed that is effective in minimizing theerror due to biased measurements. In particular, we show howto exploit the knowledge of the feasibility region, constructedvia trilateration, to contract the measured distances such thatthe cost-function of the LS formulation becomes convex andthe global optimum is closer to the true location. The proposedranging contraction source-localization algorithm is shown viasimulations to overperform existing alternatives, such as theunconstrained and the constrained LS approach. The results alsoshow that the localization error performance of the proposedtechnique remains close to the theoretical position error bound.

I. INTRODUCTION

In the last few years, the raise of new radio technologies,such as ultra-wideband (UWB), RF-ID, and the develop-ment of radio-based ranging techniques, allowed location-information to be a key-feature of novel wireless communica-tion systems. In fact, several network optimization function-alities and application services are nowadays developed underthe assumption of location-awareness.

A typical location-aware network consists of few nodes withknown locations (anchors), nodes whose location are to bedetermined (targets) and an server (centralized or distributed)where the localization algorithm runs [1]. This work focuseson the development of a centralized distance-based localizationalgorithm for a star-like network architecture, and we mainlyaddress the problem of non-line-of-sight (NLOS) channel con-ditions typically caused by the blockage or partial obstructionof the direct-path (DP) or by delay propagation of the materialcrossed. [2], [3]. Specifically, we tackle the issue of bias errorsin the ranging measurements, which typically causes poorlocalization accuracy.

In contrast to state-of-the-art algorithms, which are based onconstrained optimization methods [4]–[7], we propose a novelnon-parametric and unconstrained approach that proves moreeffective and practical. To be specific, we exploit the novelconcept of ranging contraction proposed in [8] to compute anew set of distances used in the optimization. The core of thealgorithm exploits the existence of a feasibility region wherethe target lies inside. In addition, by means of the proposedcontraction algorithm, we solve many of the issues tackled inother localization techniques. In short, we are able to modifythe optimization into a convex problem, which in turn canprovide lower localization errors.

The remainder of the article is as follows. In section II thelocalization problem is formulated, in section III we provide anexcerpt of the ranging contraction theory, in sections IV and Vwe introduce the contraction mechanism and the localizationalgorithm, and finally, in sections VI and VII numerical resultsand conclusion remarks are given.

II. LEAST-SQUARE FORMULATIONS OF THE SOURCELOCALZIATION PROBLEM

Consider a wireless network of NA anchors and a targetdeployed in the η-dimensional space. An anchor is a nodewhose position is known a priori, while a target is a nodewhose location is to be determined. Let ai ∈ Rη and x ∈ Rηbe row-vectors whose elements are the coordinates of the i-th anchor and the target, respectively. The Euclidean distancebetween the i-th anchor and the target, denoted by di, is givenby di = ‖ai − x‖F, where ‖ · ‖F is the Frobenius norm.

A measurement sample of di, denoted by di, is given by

di = di + νi + ρi, (1)

where νi ∼ N (0, σ2i ) and ρi ∼ U(0, ρMAX) are random vari-

ables with Gaussian and Uniform distributions, respectively.The variable νi models small variations of the error due to

thermal-noise, while ρi refers to a bias caused by the miss-detection of the DP. Hence, ρi = 0, i.e. ρMAX = 0 correspondsto a distance measurement in LOS conditions, while ρi > 0with ρMAX � σ2

i indicates the presence of a NLOS channel.Given a set of distances {di}, the target’s location x is

computed via the following least-square optimization problem

x = arg minx∈Rη

NA∑i=1

wi

(di − di

)2

, (2)

where di , ‖ai− x‖F and wi is a weighing factor that relatesto the reliability of di [9], [10] (in this paper wi = 1∀i).

In the presence of severe ranging errors, especially due toNLOS, the global minimum of equation (2) is largely shiftedfrom the true point, thus the above problem formulation leadsto large localization errors in the presence of bias. In order toimprove the performance of a LS-based localization algorithm,constrained optimizations can be utilized as those proposedin [6], [11], [12]. For instance, the method described in [6]uses a priori information on LOS/NLOS channel conditionsto formulate the following constrained optimization approach

978-1-4244-7157-7/10/$26.00 ©2010 IEEE 56

2

x = arg minx∈Rη

NA∑i=1

hi

(di − di

)2

, (3)

s.t. di ≤ di, ∀i |hi = 1,

where hi = 1 if the i-th link is in LOS condition or hi = 0otherwise.

Alternatively, considering ρi’s as variables, the LS-problemin equation (2) can be reformulated as [12]

x = arg minx∈Rη,bi∈R+

NA∑i=1

(di − di − hibi

)2

, (4)

s.t. di ≤ di, ∀i.bi ≤ min

j=1,...,NA{di + dj − di,j},

bi ≥ 0

where di,j is the distance between the i-th and the j-th anchorand hi is the complement of hi, i.e. hi = 0 if hi = 1 andvice-versa.

Our approach, instead, relies only on the simple informationthat ρi > 0 and the feasibility region, defined in the sectionIV, exists. These information will be sufficient to design amechanism that adjusts the distance measurements (distancecontraction algorithm), and yields a robust source-localizationalgorithm based on an unconstrained optimization.

III. THEORY OF RANGING CONTRACTION

The theory of ranging contraction was presented for thefirst time in [8] and in what follows, only an excerpt will beprovided. For the proofs of Theorems, Lemmas and Corollarieswe refer to the original paper [8].

Lemma 1 (Convexity): Let di = di,∀i. For all x such thatNA∑i=1

di−didi

≥ 0 then ∇2xf(x) � 0, where f(x) denotes the

objective function in equation (2).Definition 1 (Convex-hull): Let C({ai}) denote the convex-

hull defined by the anchors.Definition 2 (Enclosed point): A point x ∈ C({ai}) is

referred to as enclosed.Corollary 1 (Convexity with Enclosure): Let B ⊆ C({ai})

be a non-empty convex set. Ifx ∈ C({ai}) and di=di,∀i then,

∃B | ∇2xf(x) � 0, ∀x ∈ B. (5)

Definition 3 (Contracted distance): Let di ≤ di, then di isreferred as contracted distance.

Theorem 1 (Convexity with contracted distances): Let x ∈B. If di = di and di ≥ 0 ∀i, then ∇2

xf(x) � 0 ∀x ∈ B.Corollary 2 (Wider convexity): Let B′ be a non-empty con-

vex set such that B ⊆ B′ ⊆ C({ai}). If x ∈ C({ai}) anddi = di, di ≥ 0 ∀i then,

∃B′ | ∇2xf(x) � 0, ∀x ∈ B′. (6)

Theorem 2 (Full convexity with negative contraction): Ifdi = di and di ≤ 0 ∀i, then ∇2

xf(x) � 0 ∀x ∈ Rη .In the sequel, we illustrate the ranging contraction theory

with a typical source-localization study.

Consider a network with NA = 3 and a target, and assumex enclosed. All anchor-target links are assumed in NLOSchannel conditions, therefore ρi > 0 ∀i.

In figure 1 we illustrate the contour levels of f(x) andwe indicate with dots, the points where the Hessian matrix∇2

xf(x) � 0. We consider different conditions in order toexemplify the aforementioned theory.

The first study, illustrated in figure 1(a), consists of theevaluation of LS objective function with exact distance, i.e.di = di. We recognize that f(x) is generally convex in thewhole domain but around the anchors.

Next, we evaluate the LS objective function with the originaldistance measurements di’s, and as shown in figure 1(b), noisymeasurements cause multiple minima and, in addition, modifythe convex area, which is significantly reduced. The intuitionis that concave areas expand with di ≥ di.

From figure 1(c), this phenomenon is more evident. Indeed,in contrast to figure 1(b), the assumption of contracted dis-tances causes a reduction of concave areas and, subsequentlyan expansion of the convex regions. Theorem 1 and Corollary2, in fact, prove this observation.

Finally, figure 1(d) shows that by means of negative con-tracted distances, f(x) is convex in the whole domain, asproved in Theorem 2.

IV. RANGING CONTRACTION ALGORITHM

From the previous section, we established the advantage ofutilizing a set of contracted distances di’s to improve convexityof the objective function inside the convex-hull. In the sequel,we will propose a mechanism that allows for the computationof the contracted distances given a set of measurements {di}.

To begin with, consider the following definition and Lemma.Definition 4 (Feasibility region): Given a set of distance

measurements di’s, the feasibility region I of the LS source-localization problem is defined as

I , {x|di ≤ di ∀i}. (7)

Lemma 2 (Property of the feasibility region): If I is not anempty set, then

i) I is a convex set,ii) x ∈ I.

Proof: See [6], [13], [14].The Lemma above provides fundamental information about

the location of the target. Indeed, by force of property ii), wecan ensure that by confining the search within I, the truetarget’s location can be found. This result, in fact, is oneof the fundaments of all algorithms based on tri-laterationmechanism such those given in equations (3) and (4). In whatfollows, instead, we will show that the knowledge about thefeasibility region I can also be used to correct the distancemeasurements. To this end, consider the following Theoremand Corollary.

Theorem 3 (Unique minimum-distance projection): LetA ⊆ Rn be a closed convex set and let z /∈ A. ThenP · z ∈ A is the unique projection of z on A if and only if

〈z−P · z,y −P · z〉 ≤ 0 ∀y ∈ A, (8)

where P ∈ Rn×n is the projection matrix and 〈, 〉 is the innerproduct.

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3

!0.25 0 0.25 0.5 0.75 1 1.25!0.25

0

0.25

0.5

0.75

1

1.25

Convex area of the LS Cost-FunctionNA = 3, NT = 1, di = di

x-axis

y-a

xis

AnchorTarget

Optimum point

(a) Exact distances

!0.25 0 0.25 0.5 0.75 1 1.25!0.25

0

0.25

0.5

0.75

1

1.25

Convex area of the LS Cost-FunctionNA = 3, NT = 1, di

x-axis

y-a

xis

AnchorTarget

Optimum point

(b) Noisy measurements

!0.25 0 0.25 0.5 0.75 1 1.25!0.25

0

0.25

0.5

0.75

1

1.25

Convex area of the LS Cost-FunctionNA = 3, NT = 1, di = di, di ! 0

x-axis

y-a

xis

AnchorTarget

Optimum point

(c) Contracted distances

!0.25 0 0.25 0.5 0.75 1 1.25!0.25

0

0.25

0.5

0.75

1

1.25

Convex area of the LS Cost-FunctionNA = 3, NT = 1, di = di, di ! 0

x-axis

y-a

xis

AnchorTarget

Optimum point

(d) Negative contracted distances

Fig. 1. Illustrative example of the distance contraction theory.

Proof: See [15] and reference thereby.Corollary 3 (Nearest point): P · z is the point inA nearest

z.Proof: See [15].

From the above, we recognize that ∀ai such that ai /∈ I,the minimum distance separating ai to I can be computed byequation (8). By construction, such a distance is shorter thandi and therefore, it is a contracted distance. Thus, the desireddi can be computed as

di = ‖ai −P · ai‖F, (9)

where P · ai is the unique projection of ai on I.

Invoking Theorem 3, P · ai can be found by an exhaustivesearch within I until equation (8) is satisfied. This approach,however, is not practical due to the large number of trials tobe required.

An alternative method to obtain di is proposed in [15], [16,thm. 2.3]. In this case the minimum distance problem is posedas a maximization problem using hyperplanes. For the sake ofcompletion, we provide the following definitions [17].

Definition 5 (Supporting hyperplane): Let A be a set inRn. The supporting hyperplane H has form

H , {z|cT · z = cT · z0}, (10)

where c 6= 0, cT · z ≤ cT · z0, the superscript T is transpose,z0 ∈ bd(A) and bd() is the boundary.

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Definition 6 (Separating hyperplane): Let A and D twodisjoint convex set in Rn. The separating hyperplane H isdefined as

H , {z|cT · z = b}, (11)such that

cT · z ≤ b, z ∈ A and cT · z ≥ b, z ∈ D. (12)

In light of the definitions above, the minimum distance fromz /∈ A toA can be found by solving the following optimizationproblem [15] [16]

‖z−P · z‖F =1

τmax

aaT · z− γA(a), (13)

s.t. ‖a‖ ≤ τwhere γA(a) is the supporting function of A and τ is anarbitrary positive value [15].

Considering our objective, that is to find the minimumdistance from ai to I, this approach can be difficult to applybecause of the unknown supporting function γI(a). Therefore,we propose an alternative optimization problem.

Theorem 4: Let ai /∈ I. The nearest point xi ∈ I to ai isgiven by

xi = arg maxx∈I

(di − d

)2

, (14)

s.t. di − d ≥ 0, ∀i.Proof: By definition the nearest point xi ∈ I to ai is

xi , arg minx∈I

d. (15)

In order to obtain the formulation in equation (14), do

xi , arg minx∈I

d = arg maxx∈I

(−d), (16)

= arg maxx∈I

(di − d) = arg maxx∈I

(di − d)2,

where the first equality holds due to a sign change, the secondone since di≥0 and the last one becasue di≥ d,∀x∈I.

Using the Theorem above, P · ai can be easily computedfrom equation (14) and, by means of equation (9) we obtainthe desired contracted distance di.

V. LOCALIZATION ALGORITHM

This section will provide a concise description of the op-erations to be executed by the localization algorithm utilizingcontracted distances.

First, we verify the existence of the feasibility region I. Tothis end, it is sufficient that for any η or more anchor-to-targetlinks the following inequalities hold

di,j ≤ di + dj . (17)

If I does not exist, meaning there is no intersection, thenthe measurements di ≤ di ∀i, thus di’s are already contracted.In this case, the following steps will not be considered andequation (2) can be minimized.

On the other hand, if I exists, then we use equation (14)for all i-th anchor-target links. In so doing, we initialize theoptimization with a point x0 ∈ I which, for instance, can becomputed as

x0 = arg minx∈Rη

NA∑i=1

max(

0, di − di)2

, (18)

where the cost-function is convex and 0 in I.

Next, we use equation (9) to compute di. Finally, weminimize equation (2) where di’s are replaced by di’s.

In figure 1, we illustrate the aforementioned idea with anexample. Reconsider the network shown in figure with NA = 3anchors and a target. All links are in NLOS conditions andwe set ρMAX = 0.2 and σ = 0.05.

We indicate in dash-lines the circular traces computed fromthe measurements di > di. Using equation (17) we verify theexistence of I, which is indicated in bold-line. By means ofequation (18), we compute x0. As mentioned above, x0 is usedto initialize the constrained optimization given in equation(14). Next, using equation (14) for all anchor-target links, weobtain the set of contracted distances di’s, which will replacedi’s in equation (2). By construction, the shortest distancefrom the i-th anchor to I corresponds to a circular trace thatis tangent to the feasibility region. Given the convexity of Ithe shortest distance is also smaller than the exact one, thuseach di is a contracted distance. Finally, using equation (2)the target’s location estimate is computed and indicated by x.

VI. SIMULATION RESULTS

In this section, the performance of the proposed distancecontraction (DC) localization algorithm will be evaluated andcompared to state-of-the-art algorithms, namely, the uncon-strained linear-global distance continuation (L-GDC) [18], theconstrained non-linear LS (C-NLS) [13] and the sequentialquadratic programming (SQP) [12] approaches that solveequations (2), (3) and (4), respectively. In addition, the resultsare compared to the position error bound (PEB) [3] given by

PEB ,GDOP2

A, (19)

!0.2 0 0.2 0.4 0.6 0.8 1 1.2

!0.2

0

0.2

0.4

0.6

0.8

1

1.2

Example of Localization with Distance Contraction! = 2, NA = 3, NT = 1

x-axis

y-a

xis

AnchorTargetxCNLSxSQP

xDC

Fig. 2. Illustration of the contraction algorithm

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0.05 0.1 0.15 0.2 0.25 0.30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Source-Localization Algorithm Performanceη = 2, NA = 4, NT = 1, σ = 0.05

ρMAX: maximum bias

ζ:m

ean

squar

eer

ror

L-GDCCNLSSQPDCPEB

Fig. 3. Comparison of the localization MSE performance as function of themaximum bias.

10−1

10−1

100

Source-Localization Algorithm Performance Comparisons (CDF)η = 2, NA = 4, NT = 1, ρMAX = 0.2, σ = 0.05

L-GDCCNLSSQPDC

PEB1/2

Localization Error

CD

F

Fig. 4. Comparison of the CDF localization error.

where

A=1

σπ√

2ρMAX

+∞∫−∞

exp(−y + ρMAX

σ√

2

)− exp

(−y2

)Q(√

2y)−Q(√

2y + ρMAXσ )

dy.

(20)and GDOP, that states for Geometric Dilution of Precision, is

GDOP ,NA

NA∑i=1

cos(θi)2NA∑i=1

sin(θi)2 −(NA∑i=1

cos(θi) sin(θi)

)2

(21)where θi is the angle between the target and the i-th anchormeasured with respect to the horizon.

The performances of each algorithm are evaluated utiliz-ing the mean-square-error (MSE) and the cumulative densityfunction (CDF) of the localization error ζ

ζ , ‖x− x‖F. (22)

4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Source-Localization Algorithm Performanceη = 2, NT = 1, σ = 0.05, ρMAX = 0.2

NA: number of anchors

ζ:m

ean

squar

eer

ror

L-GDCCNLSSQPDC

Fig. 5. Comparison of the localization MSE performance as function of thenumber of anchors.

10−1

10−1

100

Source-Localization Algorithm Performance Comparisons (CDF)η = 2, NA = 6, NT = 1, ρMAX = 0.2, σ = 0.05

L-GDCCNLSSQPDC

Localization Error

CD

F

Fig. 6. Comparison of the CDF localization error.

Specifically, the MSE is given by

MSE ,1

LP

P∑p=1

L∑l=1

ζ2(l,p), (23)

where the indexes p and l indicate the l-th realization of thep-th network configuration and, the CDF of ζ is computed as

CDF , Pr{ζ ≤ ξ}, (24)

where ξ is the localization accuracy.We perform two studies. In the first one, we evaluate the

performances of the algorithms as a function of the rangingerror, and mainly, of the maximum bias ρMAX. In the secondone, instead, we evaluate the impact of the number of anchorson the localization error.

Starting from the first case of study, the typical networkconsists of NA = 4 anchors and a target, which are randomly

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deployed. All anchor-target links are considered in NLOSconditions. The results are shown in figures 3 and 4.

In both figures, the proposed DC localization algorithmoutperforms the alternatives, and more relevantly, performsclose to the PEB. This result is very remarkable becausethe proposed ranging contraction algorithm has not beendesigned to compensate/estimate the bias, as it turns out, itprovides a seem-less compensation in a non-paramtric fashion.In figure 3, we also notice that the MSE achieved with theDC algorithm are slightly below the PEB. The reason is stillunder investigation, but the main intuition is the fact thatby using a contraction algorithm the statistics of di’s are nolonger Gaussian. Thus, the PEB computed in [3] does not holdanymore.

In the second case of study, instead, we consider a networkwhose number of anchors varies from 3 to 10. As in theprevious case, anchors and target are randomly located, allanchor-target links are considered in NLOS conditions, thenoise standard deviation σ = 0.05, and ρMAX = 0.2.

The results of this test are shown in figures 5 and 6. Alsoin this case the DC algorithm is the best performing methodand, in particular from figure 3, it is shown that only littleadvantage is ripped by increasing the number of anchors.

VII. CONCLUSION

In this paper, we proposed a robust distance-based sourcelocalization algorithm that relies on the novel concept ofranging contraction. It is shown that by means of contracteddistances, we can improve the convexity of the LS cost-function and, in turn, we can decrease the localization errordue biased distance measurements.

In the paper, we described in detail both a ranging con-traction algorithm and the localization method. We comparedthe performance of the DC technique to state-of-the-art algo-rithms and, we showed that the novel method outperforms thealternatives.

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This result indicates that utilizing the proposed DC algorithm,a localization system can provide accurate location estimateswith very few anchors, and by product, it will result energyand computational cost efficient.

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