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A Linear Programming Approach to NLOS Error Mitigationin Sensor Networks

Swaroop Venkatesh and R. Michael BuehrerMobile and Portable Radio Research Group (MPRG), Virginia Tech,

Blacksburg, VA 24061, USA.

{vswaroop, buehrer}@vt.edu

ABSTRACT

In this paper, we propose a linear programming approachto the problem of non-line-of-sight (NLOS) error mitigationin sensor networks. The locations of sensor nodes can beestimated using range or distance estimates from location-aware anchor nodes. In the absence of line-of-sight (LOS)between the sensor and anchor nodes, e.g., in indoor net-works, the NLOS range estimates can be severely biased. If

these biased range estimates are directly incorporated intopractical location estimators such as the Least-Squares (LS)estimator without the mitigation of these bias errors, thiscan potentially lead to degradation in the accuracy of sen-sor location estimates. On the other hand, discarding thebiased range estimates may not be a viable option, since thenumber of range estimates available may b e limited. Wepresent a novel NLOS bias mitigation scheme, based on lin-ear programming, that (i) allows us to incorporate NLOSrange information into sensor location-estimation, but (ii)does not allow NLOS bias errors to degrade sensor localiza-tion accuracy.

Categories and Subject Descriptors

C.2.1 [Network Architecture and Design]: Distributednetworks - Wireless communication; G.1.0 [Mathematicsof Computing]: Numerical Analysis - Numerical algorithms

General Terms

Algorithms, Design, Measurement, Performance

Keywords

line-of-sight, location estimation, NLOS environment, time-of-arrival estimation, wireless sensor networks.

1. INTRODUCTIONThe envisioned applications for ad hoc wireless sensor net-

works often depend on the automatic and accurate location

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of deployed sensors. In numerous sensor networks, partic-ularly for environmental applications [1], [2] such as wa-ter quality monitoring, precision agriculture, and indoor airquality monitoring, the available sensing data may be ren-dered useless by the absence of accurate sensor location esti-mates. The availability of accurate sensor location estimatescan help reduce configuration requirements and device cost.Further, accurate sensor location estimation enables applica-tions such as inventory management [3], intrusion detection

[4], traffic monitoring, and locating emergency workers inbuildings.

The design of ad hoc location-aware sensor networks re-quires the capability of peer-to-peer range or distance mea-surement. A sensor whose location is unknown, can estimateits location based on range measurements from location-aware sensors or anchors, whose locations are known or es-timateda priori. Range estimates from anchor nodes couldbe obtained using received signal strength (RSS) or time-of-arrival (TOA) estimation techniques [5], [6].

ABLOS

B

t = RAB/cx = RAB

BiasError

s(t)

A t = 0x = 0 RAB

Scatterers

TransmitSignal

ReceivedSignal

Figure 1: The bias introduced in the absence of LOSin TOA-based ranging between two sensors A and B .

In sensor networks, especially indoors, the LOS path be-tween sensors may be obstructed as illustrated in Figure 1.As a consequence, TOA-based range estimates are positivelybiased with high probability, since the first multipath com-ponent travels a distance that is in excess of the true LOSdistance. A similar effect is seen in the case of RSS-basedrange estimates, where the received signal power is reduceddue to the obstruction of the LOS path. These effects re-

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sult in range estimates that are often much larger than thetrue distances and as a consequence, in NLOS scenarios, theaccuracy of sensor location estimates is adversely affected.

The problem of location-estimation with biased NLOSrange estimates has been considered before, but mostly inthe context of cellular communications [7], [8], where it wasshown that the NLOS bias errors in the range estimates leadto large errors in the computation of a nodes location. Theliterature on the NLOS problem typically falls in two cat-

egories: NLOS identification and NLOS mitigation. Theformer deals with the problem of distinguishing betweenLOS and NLOS range estimates, whereas the latter typi-cally deals with the reduction of the adverse impact of NLOSrange errors on the accuracy of location-estimates, assumingthe NLOS range estimates have been identified. Several sta-tistical NLOS identification techniques for cellular systemshave been discussed previously [8], [9]. In this work, we fo-cus on the problem of NLOS mitigation in two-dimensionalsensor location estimation, assuming we are able to distin-guish between LOS and NLOS range estimates. It mustbe pointed out that the results in this paper can easily beextended to three-dimensional sensor localization scenarios.

The Cramer-Rao Lower Bound (CRLB) analysis presented

in [10] characterized the performance of the minimum vari-ance unbiased estimator (MVUE) [11] of sensor location,given a mixture of (unbiased) LOS and (biased) NLOS rangeestimates. This analysis showed that the MVUE discardsthe biased NLOS range estimates and utilizes only LOSrange information while estimating sensor locations. How-ever, as will be demonstrated, in the case of practical non-efficient [11] estimators such as the commonly-used Least-Squares (LS) estimator [12], discarding NLOS range infor-mation does not necessarily improve performance. Addi-tionally, for two-dimensional location-estimation, the LS es-timator requires at least three range estimates in order toobtain an unambiguous solution. Consequently, in indoorsensor networks, limited connectivity with anchors may im-ply that we may not have the luxury of discarding any range

estimates. This implies that in general, given a mixture ofLOS and NLOS range estimates, we may be required to usethe entire set of range information in order to compute asensors location, as illustrated in Figure 2.

Anchor Node i

NLOS Link

Obstruction

LOS Link

Ri

Unlocalized node

Figure 2: The NLOS Problem: In general, a mixtureof unbiased LOS and biased NLOS range estimatesneeds to be used to compute a sensors location.

A Semi-Definite Programming (SDP) approach to sensor

localization based on connectivity information was investi-gated in [13], and a quadratic programming approach withNLOS range estimates was discussed in [14], but these ap-proaches result in high computational complexity [15]. TheResidual Weighting Algorithm (Rwgh) was proposed in [16].The main advantage of this algorithm is that NLOS identifi-cation is not requireda priori. However, this algorithm im-plicitly assumes that the range measurement noise is muchsmaller than the bias introduced, in order to distinguish

between the LOS and NLOS range estimates. More impor-tantly, it relies on the availability of a large number of rangeestimates, several of which are LOS, so that the set of rangeestimates finally selected to compute a nodes location re-sults in the smallest residual error. However, in indoor sen-sor networks, situations may arise where only NLOS rangeestimates are available while estimating a sensors location.

In this paper, we present a novel linear programming (LP)approach that effectively incorporates both LOS and NLOSrange information into the estimation of a sensors loca-tion. A linear programming approach was briefly mentionedfor the case of NLOS range estimates in [15], but was notpursued. We demonstrate that this low-complexity LP ap-proach can be generalized to handle a mixture of LOS and

NLOS range estimates (with the only-LOS and only-NLOS range information scenarios as sub-cases) withoutdiscarding any range information.

This paper is organized as follows: In section 2, we discussthe impact of NLOS bias errors on the accuracy of sensorlocation estimates. Section 3 discusses the LP approach toincorporating LOS range estimates, NLOS range estimatesand a mixture of LOS and NLOS range estimates into sensorlocation estimation. We also discuss a series of sub-casesthat need to be addressed in order generalize the proposedapproach. Simulation results are presented in Section 5,where we evaluate the performance of the proposed methodin terms of sensor localization accuracy. We conclude inSection 6.

2. IMPACT OF NLOS BIAS ERRORS

2.1 Notation, Models and AssumptionsSuppose the sensors (unknown) location is x = [x y]T.

Let L denote the set of anchors which provide LOS rangeestimates, with cardinality mL = |L|. The known locationsof the LOS anchors are denoted by {xLj}, j = 1, 2, , mL.Similarly,Nrepresents the set of anchors that provide NLOSrange estimates, with mN = |N |, and the known loca-tions of the NLOS anchors is represented by {xNj }, j =1, 2, , mN.

The LOS range estimates {rLj} are modeled as unbiasedGaussian [2], [17] estimates of the true ranges RLj =x xLj:

rLj =RLj+ nLj , j = 1, 2, , mL, (1)

where nLj represents zero-mean Gaussian range measure-ment noise in the j th LOS range estimate: nLj N(0, 2j ),where the range measurement noise variance2j can be mod-eled as [18]

2j =KER

Lj , (2)

where is the path loss exponent and KE is a proportional-ity constant that determines the accuracy of range estima-tion. This above model for the accuracy of range estimates

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[18] applies to both TOA and RSS-based range estimateswhen = 2. The NLOS range estimates are assumed to bepositively biased Gaussian estimates [8] of the true ranges:

rNj =RNj + nNj + bNj , j = 1, 2, , mN, (3)

where RNj = x x Nj , nNj N(0, KE RNj ) and bNj

are the NLOS bias errors. We assume that the bias errorsare uniformly distributed: bNj U(0, Bmax), where Bmaxrepresents the maximum possible bias. Additionally, we as-sume that the maximum bias is much larger than the rangemeasurement noise Bmax 2j , j = 1, 2, , mN. Finally,without loss of generality, we assume that the coordinateaxes are selected such that x 0.

2.2 NLOS range estimates: To discard or notto discard?

As shall show in a later section, when we have at leastthree range estimates, the LS estimator can be used computean estimate x of the sensors location x. We define thelocalization error, a measure of the accuracy of the location-estimate x, as:

= x x2 meter2 (4)

It must be noted that is a random variable, with dif-ferent instances corresponding to different realizations ofthe range measurement noise, bias errors and anchor loca-tions. Therefore, we characterize the accuracy of location-estimates through the mean and standard deviation of the localization error defined in (4); smaller values of bothand indicate more accurate sensor location-estimates.

When mN = 0 and mL 3, the LS estimator providesaccurate estimates of a nodes location [12]. However, whenmN > 0, we need effective ways of incorporating NLOSinformation into the estimation procedure. Figures 3 and4 show the impact of directly (without mitigation of thebias errors) incorporating NLOS range estimates into theLS solution. For the specific distribution of anchors shownin Figure 3, directly incorporating the NLOS ranges intothe LS solution can degrade localization accuracy. However,in some cases, and in particular for the example shown inFigure 4, introducing the NLOS range estimate directly intoLS location estimation without any mitigation of the bias inthe range estimate can improveperformance in terms ofand .

Generally speaking, discarding the NLOS range estimatesdoes not result in poor performance when the geometry ofLOS anchor nodes has certain properties, best described bythegeometric dilution of precision (GDOP) [2], [19], wherea larger GDOP (as defined in [2]) implies poorer localizationaccuracy. It has been observed that when the GDOP is verylarge, the presence of an additional NLOS range estimateresults in an improvement in performance: the addition of

a NLOS node reduces the GDOP and this compensates forthe inaccuracy of the NLOS range estimate.

These two examples show that (i) directly incorporat-ing NLOS range estimates into existing practical estima-tors without reducing the impact of bias errors can ad-versely affect localization accuracy. However, (ii) we do notwish to discard the NLOS range estimates, since their usecould improve the performance of practical estimators un-der certain conditions. Indeed, in indoor sensor networks,we may have more NLOS range estimates than LOS rangeestimates. Therefore, what is desired is a method that allows

0 2 4 6 8 100

2

4

6

8

10

Yaxis(meters)

Only LOS ranges, mL= 3

0 2 4 6 8 100

5

10

Xaxis(meters)

Yaxis(meters)

Both LOS and NLOS ranges used, mL= 3, m

N= 1

= 1.34,

= 7.27

= 1.07,

= 7.15

LOS Anchors

NLOS Anchor

Figure 3: This example shows several instances ofthe LS location estimate x, one for each realiza-tion of the range estimates, with x = [3 3]T, = 2,KE = 0.1, Bmax= 4 meters, for (i) (top) Only mL = 3

LOS estimates (ii) (bottom) including mL = 3 LOSand mN = 1 NLOS range estimates. The NLOSrange estimate is treated exactly like an LOS rangeestimate and directly incorporated into the LS solu-tion in the bottom figure. In this case, the additionof the biased NLOS range estimate degrades local-ization accuracy with respect to and .

the soft-activation of NLOS range information: the NLOSrange estimates are not incorporated directly, but are usedin conjunction with LOS range estimates when LOS rangeestimates alone do not guarantee accurate sensor locationestimates. In the following section, an LP approach thatachieves this goal is described.

3. A LINEARPROGRAMMINGAPPROACHIn this section, we show that the problem of sensor location-

estimation given LOS range information can be cast in theform of a linear program. We then modify the linear pro-gram to utilize NLOS range information, resulting in a me-thod that utilizes a mixture of LOS and NLOS range esti-mates to compute a sensors location accurately.

3.1 LOS range estimatesThe LOS range estimates, which are modeled as unbi-

ased estimates of the true ranges, can be used to define con-

straints on the unknown sensor location x. We can write fori= 1, 2, , mL:

x xLi = rLi,

(x xLi)2 + (y yLi)

2 = r2Li. (5)

These relations are non-linear equations in x and y and rep-resent the fact that x lies on a circle of radius rLi whosecenter is x Li. These equations can be linearized by extract-ing the difference of each of these equations from the others,

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0 2 4 6 8 100

5

10

Xaxis (meters)

Yaxis(meters)

Only LOS ranges used, mL= 3

0 2 4 6 8 100

5

10

Xaxis (meters)

Yaxis(meters)

Both LOS and NLOS ranges used, mL= 3, m

N= 1

= 16.7

= 22.9

= 5.2

= 5.6

NLOS Anchor

LOS Anchor

Figure 4: This example shows several instances ofthe LS location estimate x, one for each realizationof the range estimates, with x = [3 3]T, mL = 3,mN = 1, KE = 0.1, Bmax = 4 meters. In this case,

the addition of the biased NLOS range estimateim-

proves localization accuracy with respect to and.

forming a system ofM=

mL2

distinctequations:

aij x+ bij y = cij , i, j = 1, 2, 3, , mL, i < j (6)

where a ij = xLi xLj , bij =yLi yLj ,

cij =

x2Li x2Lj

+

y2Li y2Lj

r2Li r2Lj

2

Each of theseMequations can be viewed as representing thelines formed by connecting the intersection points (if any)of pairs of circular constraints defined in (5). From (1),

since the range estimates are noisy, in general rLi = RLi,and solving these equations simultaneously may not yield asolution. Resorting to an error minimization approach, forevery potential solution x, and for every equation, we candefine the residual error as:

eij =aijx+ bijy cij , i, j = 1, 2, 3, , mL, i < j. (7)

The final estimate x can selected such that an objectivefunctionZ, such as the sum of the residual error squares, isminimized:

x= arg minx

Z= arg minx

i

j, j>i

e2ij

This is the equivalent to the LS approach defined in [12].

It is important to note that (i) the objective function Z isnon-linear in x and y, and (ii) we require mL 3 to forman unambiguous solution. The LS solution is given by

x=

AT

A

1

AT

c

where

A=

a12 a13 a1mL a23 a(mL1)mLb12 b13 b1mL b23 b(mL1)mL

T

2M

(8)

and

c=

c12 c13 c1mL c23 c(mL1)mL T

1M

(9)Looking at (7), we see that the set of variables {eij } playsthe role of unconstrained slack variables [20] in the systemof M equations. This linear system of equations can beconverted to a linear program if the objective function Z islinear. Specifically, if we define

Z

i

j, j>i

|eij |,

and then replace the unconstrained variable e ij bye+ij e

ij,

e+ij 0, e

ij 0, we can write an alternative linearizedobjective function, that is to be minimized, as

Z

i

j, j>i

e+ij+ e

ij

. (10)

It must be noted that in the optimal solution that minimizesZ, only one ofe+ij , e

ij will be equal to |eij |, with the otherbeing zero [20]. The constraints are then given by

aij x+ bij y = cij+ e+ij e

ij , i, j = 1, 2, 3, , mL, i < j.(11)

Since there are now 2M non-negative slack variables, the

vector zof (2M+2) variables can be written as z =

x y T T

,where :

=

e+12 e

12 e+13 e

13 e+(mL1)mL

e

(mL1)mL

T

2M1

(12)Thus, the linear program can be formulated in standard form[20] as

min Z = fTL z such that

[A | J] z = c, z 0, (13)

where A and c were respectively defined in (8) and (9),

J=

1 1 0 0 0 00 0 1 1 0 0...

...0 0 0 0 0 0 1 1

M2M

, (14)

and fL = [0T21 1

T2M1]

T. Here 0kl represents a k lmatrix of zeros and 1kl represents a k l matrix of ones.

Figure 5 compares for different values of KE , usingthe (i) LS estimator and (ii) the LP approach, for sensorlocation-estimation with mL = 3 LOS range estimates. Wesee that the linearization of the objective function has neg-ligible impact on performance. Therefore, we now have a

linear program that can be used to solve for a sensors lo-cation given LOS range estimates. In this linear program,the objective function Zdefined in (10) is a function of thedistances of a pointx to the straight lines given in (6). If weuse NLOS range estimates in a similar manner, by incorpo-rating them into the objective function we could potentiallydegrade the accuracy of the location estimate. Instead, asdescribed in the following section, we can use the NLOSrange estimates to constrain the feasible region for x with-out affecting the objective function defined using LOS rangeestimates, thereby limiting the possibility of large errors.

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0 0.002 0.004 0.006 0.008 0.010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

KE

M

eanof

Mean of Location Error, mL= 3, m

N= 0

LP

LS

Figure 5: Mean of the Localization Error, mL= 3,mN = 0 for a specific distribution of anchors.

3.2 NLOS range estimatesAs the bias errors on the NLOS range estimates are always

positive, and are assumed to be much larger than the rangemeasurement noise, we know each NLOS range estimate rNiis, with high probability, larger than the true range RNi , i =1, 2, , mN. Based on this observation, we can convert theNLOS range estimates into inequalities fori = 1, 2, , mN:

x xNi rNi ,

(x xNi )2 + (y yNi )

2 r2Ni . (15)

These inequalities imply that the feasible region for x liesin the interior of each of the circular constraints definedby (15). Note that this assumption cannot be made if thestandard deviation of the zero-mean measurement noise andthe positive bias are comparable in (3). Once again, theseare non-linear constraints on x and y. However, these con-straints can be relaxed to the following linear constraints,as suggested in [15]:

x xNi rNi , x + xNi rNi , y yNi rNi

y+ yNi rNi , i= 1, 2, , mN. (16)

This essentially relaxes the circular constraints to rectan-gular constraints as shown in Figure 6. It is readily seenthat the new rectangular feasible region contains the orig-inal (convex) feasible region formed by the intersection ofthree circular regions. We can now write the above fourconstraints for the ith NLOS range estimate in standardform [20]:

x xNi + u1i = rNi , x + xNi + u2i =rNiy yNi + v1i = rNi , y+ yNi + v2i = rNi

u1i, u2i, v1i, v2i 0, i= 1, 2, , mN. (17)

Definingwi = [ui1 u2i vi1 v2i]T and zi = [x y w

Ti ]

T as thevectors of variables corresponding to the ith NLOS rangeestimate, we can express the above equations in matrix formas

[B1| I44] zi = ri,

zi 0,

Linearized FeasibleRegion

Original FeasibleRegion

Figure 6: Linearization ofmN = 3 NLOS constraints:the NLOS circular constraints are converted to rec-tangular constraints.

where

B1 =

1 01 00 10 1

, ri =

rNi + xNirNi xNirNi + yNirNi yNi

,

and Inn denotes an n n identity matrix. We can nowstack the constraints due to each of the mN NLOS rangeestimates to form a system ofN= 4mNequations as follows:

[B INN] z = r,

z 0,

where

B =

BT1 BT1 B

T1

T

(2N),

r =

rT1 rT2 r

TmN

T

(1N), (18)

w =

wT1 wT2 w

TmN

T

(1N), (19)

with the vector of variables being defined as

z =

x y wT

T

(1(N+2)).

It is important to note that in the above analysis, no ob-jective function was defined based on the NLOS range es-timates, and we only constrained the feasible region for x.

The feasible region can further be constrained by includ-ing the tangents at the intersection points of the circularconstraints defined in (15) to reduce the size of the feasibleregion, but these additional constraints add to the compu-tational complexity of the linear program without provid-ing substantial gains. In the following subsection, we in-tegrate the constraints and objective function obtained us-ing LOS range estimates with the NLOS constraints definedabove, for the problem of sensor location-estimation, givena mixture of LOS and NLOS range estimates with m L 3,mN0.

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3.3 Combining the LOS and NLOS Range In-formation

Based on the above subsections, given mL 3 LOS rangeestimates and m N0 NLOS range estimates, we can com-bine them into a single linear program. We define the vectorof variables as

z = [x y w]T(1(2M+N+2)),

where and w are respectively defined in (12) and (19).The objective function Zis defined as

Z= fTz,

where fT = [0 0 12M1 0N1]1(2M+N+2). The linear pro-gram is formulated as:

min Z = fTz, such that

Dz = g, z 0,

where

D=

A | J 0MNB | 02MN INN

(M+N)(2+2M+N)

,

and

g=

c

r

(2+2M+N)1

.

In the above equations, the matrices A, J and B are re-spectively defined in (8), (14) and (18), and the vectors cand r are defined in (9) and (18) respectively.

It must be pointed out that in the above linear program,LOS range information is used to define both the objectivefunction and the feasible region, whereas the NLOS rangeinformation is used only to define the feasible region. Thisallows the NLOS range estimates to assist in improvingthe accuracy of location estimates by limiting the size ofthe feasible region, but does not allow the NLOS bias er-rors to adversely affect sensor localization accuracy, since

the NLOS range information plays no part in defining theobjective function. The efficacy of the proposed method isdemonstrated through simulations in section 5. The aboveapproach works for any mixture of LOS and NLOS rangeestimates, provided mL 3, mN 0. In the followingsection, we discuss some special sub-cases where there areinsufficient LOS and NLOS range estimates to apply theapproach described above.

4. SPECIAL CASES

4.1 Special Case: mL = 0If no LOS range estimates are available, then the above

LP approach will not be applicable since the NLOS range

estimates are not used to define an objective function, al-though they can be used to define a feasible region. Anexample of this situation with mN= 3 is shown in Figure7. In this case, we could either use the LS estimator (ora constrained LS estimator) without the mitigation of biaserrors, or simply use the center of the rectangular feasibleregion as a location estimate:

x=1

2

mini {xNi + rNi } + maxi {xNi rNi }mini {yNi + rNi } + maxi {yNi rNi }

.

xN1

xN2

xN3

V1 V2

V4 V3

rN3

Figure 7: Special Case with mL = 0and mN= 3. Thevertices of the feasible region are denoted by V1, V2,V3 and V4.

4.2 Special Case: mL = 1, m N2This situation is illustrated in Figures 8(a) and 8(b), which

represent two sub-cases. SincemL = 1, we have a single cir-cular equality constraint. In the first sub-case (Figure 8(a)),

the circle formed using the LOS range passes through thefeasible region formed by the NLOS constraints. In sucha case, there are infinitely many solutions. In the secondsub-case, illustrated in Figure 8(b), the circular constraintformed using the LOS range estimate does not pass throughthe feasible region. In such a case we can pick the vertex ofthe feasible region that is closest to the circle as a potentialsolution.

xN1

xN2

xN3

rN3xL1

rL1

Feasible Region

Possible Solutions

x

(a) mL = 1, mN 2. Sub-case 1: the LOSconstraint cuts throughthe feasible region gener-ated using the NLOS con-straints.

xN1

xN2

xN3

rN3 xL1

rL1

Feasible Region

x

(b) mL = 1, mN 2. Sub-case 1: the LOSconstraint does not passthrough the feasible re-gion generated using theNLOS constraints.

Figure 8: Case: mL = 1, mN2.

4.3 Special Case: mL = 2, m N1In this case, since mL = 2, we have two circular con-

straints, and the linearization procedure is not particularlyuseful since a single linear constraint is generated using thedifference. Therefore, instead of two potential solutions cor-responding to the intersections of the two circles, we havean infinite number of solutions. It is easier to compute thetwo intersections of the circles, and if there are additionalNLOS constraints, three sub-cases arise: (1) Neither of theintersection points lies inside the feasible region formed bythe NLOS constraints, (2) both intersection points lie insidethe feasible region formed by the NLOS constraints (Figure9(a)), and (3) only one of the intersection points lies inside

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xN1

xL1

rL1

x

xL2

rL2

rN1

x True Location

(a) mL = 2, mN 1,Sub-case 1: The feasibleregion contains both in-tersection points of thecircle.

xN1

xL1

rL1

x

xL2

rL2

rN1

x True Location

(b) mL = 2, mN 1,Sub-case 2: The feasibleregion contains one of twointersection points of thecircle.

Figure 9: Case: mL= 2, mN2.

the feasible region formed by the NLOS constraints (Figure9(b)). In the latter sub-case, we simply pick the intersec-tion point that lies within the feasible region. In the firsttwo sub-cases, the ambiguity remains.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

1

2

3

4

5

6

7

8

9

10

KE

(

meter2)and(

meter2)

mL= 3, m

N= 3, B

max= 8 meters

, LP

, LS Pure LOS

, LS (LOS + NLOS)

, LP

, LS Pure LOS

, LS (LOS + NLOS)

Figure 10: The mean and the standard deviation of the localization error are plotted versus KE.Here, mL = 3, mN= 3 and Bmax= 8 meters.

5. SIMULATION RESULTSIn this section, we present simulation results that demon-

strate that the proposed approach mitigates the effect ofNLOS bias errors and utilizes the NLOS range information

to improve sensor localization accuracy. In the following dis-cussion, the anchor nodes are randomly distributed over anL L area where L= 10 meters. The unknown location ofthe sensor node isx = [5 5]T(meters). We compare the per-formance of three location-estimation approaches in termsof the mean and standard deviation of the localization error: (i) the proposed LP approach, (ii) the LS estimator, uti-lizing only LOS range estimates, while discarding the NLOSrange estimates (Pure-LOS), and (iii) the LS estimator,utilizing both LOS and NLOS range estimates, without themitigation of NLOS bias errors (LOS+NLOS).

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

KE

MeanofLocalizationError(

meter2)

mL= 3, m

N= 4

LP, Bias = 4 metersPure LOS LS

LOS+NLOS LS, Bias = 4 metersLP, Bias = 8 metersLOS+NLOS LS, Bias = 8 meters

Figure 11: Mean of the Localization Error ,mL = 3, mN = 4. The maximum bias Bmax is in-creased from 4 meters to 8 meters.

For these three methods, the values of are computedfor a large number of realizations of the measurement noise,bias errors and anchor locations. The mean and thestandard deviation are shown in Figure 10, for differentvalues of the proportionality constant KE defined in (2). Inthis simulation, m L = 3, m N= 3 and Bmax= 8 meters. Asexpected, in all three cases, sensor localization accuracy de-grades as the variance of the range estimates increases (i.e.,KE increases). The proposed LP approach outperforms theother two schemes in terms of both the mean and standarddeviation of the localization error, and therefore, on the av-erage, produces more accurate sensor location estimates. Ingeneral, it was observed that for all three estimation pro-cedures, and follow similar trends in terms of their

variation with KE :,LP < ,LSPure LOS < ,LS(LOS+NLOS)

,LP < ,LSPure LOS < ,LS(LOS+NLOS).

The variation of the mean localization error with KE,while respectively increasing the maximum bias Bmax, andthe number of NLOS range estimates mN, is shown in Fig-ures 11 and 12 for a specific set of anchor locations. Weobserve that (a) the LP approach performs better than boththe LS approaches and is less sensitive to increase in bias er-rors, and (b) the performance of the LP approach improvesasmNincreases, whereas the opposite effect is observed withthe LS estimator that utilizes LOS and NLOS range esti-mates without bias error mitigation. The former effect isdue to the fact that the NLOS ranges do not contribute tothe objective function of the linear program, while the lat-ter is because additional NLOS range estimates reduce thesize of the feasible region, thereby reducing the maximumpossible values of the localization error. The LP approachonce again outperforms the LS case that utilizes only LOSrange estimates.

6. CONCLUSIONSIn this paper, we described a novel linear-programming

approach to the problem of sensor localization in NLOS en-

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8/8

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.41

0.61

0.81

1.01

1.21

1.41

2.01

2.81

KE

(

meter2)

mL= 3, B

max= 8 meters

LP, mN

= 3

LS(LOS+NLOS), mN

= 3

LP, mN

= 4

LS(LOS+NLOS), mN

= 4

LP, mN

= 5

LS(LOS+NLOS), mN

= 5

LS Pure LOS

Figure 12: Mean of the Localization Error ,mL = 3, Bmax = 8 meters. The number of NLOSrange estimates is varied for mN= 3 to mN= 5.

vironments. The main motivation for the development ofthis method was that in typical indoor sensor networks, itis likely that we would be required to compute a sensors lo-cation using a mixture of LOS and NLOS range estimates.Using the LOS range estimates to define the objective func-tion, and the NLOS range estimates to restrict the feasibleregion for the linear program, we showed that NLOS rangeinformation can be used to improve sensor localization ac-curacy without incurring p erformance degradation due tobias errors. This approach was shown to perform as well asthe LS estimator when only LOS range estimates are pro-vided, and better than the LS estimator when a mixture ofLOS and NLOS range estimates is provided. Relative to LSestimation, it was found that the proposed approach is lesssensitive to increase in NLOS bias errors and that increasingthe number of NLOS range estimates improves sensor local-ization accuracy. Further, the proposed approach can beapplied to a general three-dimensional location-estimationproblem.

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