Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/3156191

TheInterior-PointMethodforanOptimalTreatmentofBiasinTrilaterationLocation

ARTICLEinIEEETRANSACTIONSONVEHICULARTECHNOLOGY·AUGUST2006

ImpactFactor:1.98·DOI:10.1109/TVT.2006.877760·Source:IEEEXplore

CITATIONS

58

READS

23

3AUTHORS,INCLUDING:

Gyu-InJee

KonkukUniversity

188PUBLICATIONS2,034CITATIONS

SEEPROFILE

Availablefrom:Gyu-InJee

Retrievedon:11March2016

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006 1291

The Interior-Point Method for an Optimal Treatmentof Bias in Trilateration Location

Wuk Kim, Jang Gyu Lee, Member, IEEE, and Gyu-In Jee, Member, IEEE

Abstract—This paper presents a new position-determinationestimator for trilateration location. The proposed estimator takesthe measurement bias into consideration and improves the loca-tion accuracy of a mobile location system. In case that a mobilestation (MS) utilizes signals from a set of base stations for its loca-tion, the computed location is largely affected by nonline-of-sight(NLOS) error in signal propagation. A constrained optimizationmethod in a three-stage estimation structure is proposed to es-timate and eliminate the measurement bias contained in eachpseudorange and mainly caused by the NLOS error. A linearobservation model of the bias is formulated, and the interior-pointoptimization technique optimally estimates the bias by introducinga feasible range of the measurement bias. It is demonstrated thatthe new three-stage estimator successfully computes an accuratelocation of an MS in a realistic environment setting. The locationaccuracy of the proposed estimator is analyzed and compared withthe existing methods through mathematical formulations and sim-ulations. The proposed estimator efficiently mitigates the effect ofa measurement bias and shows that the iterated least square (ILS)accuracy of 118 m [67% distance root-mean-square (DRMS)] canbe improved to about 17 m in a typical urban environment.

Index Terms—Enhanced-911 (E-911), interior-point optimiza-tion method, location-based services, path-delay error, radio navi-gation, time of arrival (TOA), trilateration, wireless location.

I. INTRODUCTION

R ECENT developments in Enhanced-911 (E-911),location-based business, and electronic commerce call for

accurate location of cellular subscribers [3], [13]. Wireless loc-ation techniques that use the signals between the base station(BS) and the mobile station (MS) in a cellular network toperform trilateration are widely sought to meet the need [4], [8],[10]. The path-delay error due to reflections, diffractions, andscatterings of signals is the major problem in trilateration,which brings nonline-of-sight (NLOS) propagation and multi-path interference errors. This paper proposes an efficient andpractical method that estimates and eliminates the path-delayerror to facilitate an accurate MS location.

Manuscript received October 15, 2001; revised February 18, 2003, March19, 2005, September 6, 2005, and February 3, 2006. This work was supportedin part by the Automatic Control Research Center, Seoul National Universityand in part by Samsung Electronics. The review of this paper was coordinatedby Dr. E. Molnar.

W. Kim was with Seoul National University, Seoul 151-742, Korea. He isnow with the IT Center, Samsung Electronics, Gyeonggi-do 442-600, Korea(e-mail: wuk.kim@samsung.com).

J. G. Lee is with the Automatic Control Research Center and the School ofElectrical Engineering and Computer Science, Seoul National University, Seoul151-742, Korea (e-mail: jgl@snu.ac.kr).

G.-I. Jee is with the Department of Electronic Engineering, Konkuk Univer-sity, Seoul 143-701, Korea (e-mail: gijee@konkuk.ac.kr).

Digital Object Identifier 10.1109/TVT.2006.877760

The path-delay error that occurs along a propagation pathintroduces a measurement bias to a measured pseudorangebetween a BS and an MS. The measurement bias in wirelesslocation is generally very large and must somehow be removed.In addition, NLOS errors, which occur when the line of sight(LOS) between a BS and an MS is blocked, are the maincause of the bias error. Even though it is difficult to separatethose NLOS and multipath interference errors from the path-delay error, the bias error is largely contributed by the NLOS,whereas the noise error is influenced by multipath interference.The path-delay error always makes a measured time of arrival(TOA) longer than the true distance. To mitigate the effect ofthe bias error, some methods for averaging the measurementsequence, such as LOS reconstruction [8] and residual weight-ing method [10], have been proposed. The LOS reconstructionmethod regards the NLOS error as a positive random variableand subtracts an estimated NLOS error from a mean of a samplemeasurement sequence of pseudoranges to restore a LOS dis-tance. The residual weighting filter outputs the weighted linearcombination of each estimate from a selected BS subset. Boththe LOS reconstruction and the residual weighting methodsreduce the effect of the fast fading noise error mainly causedby multipath errors by utilizing the timing information forrange estimates, but they are less effective for the slow fadingbias error mainly caused by an NLOS error since they do notdirectly estimate NLOS errors. With a priori statistics or anempirical database of NLOS error, location accuracy could beimproved [6].

We are proposing a new method that employs an optimiza-tion technique and a three-stage structure of an estimator toeffectively eliminate slow fading bias error. The employed op-timization technique is the interior-point optimization methoddeveloped by Karmarker, which solves a linear equation withlinear constraints [9]. This method consists of a Lagrangemultiplier for the optimal solution of a linear equation and abarrier function by Fiacco and McCormick [1] that converts aninequality constraint into a corresponding equality constraint.This paper introduces the interior-point optimization methodto pseudorange positioning, where it considers the restrictedcondition of a bias error, linearizes a nonlinear measurementequation, and optimally determines the position of an MS.

This paper also presents a new structure of the three-stageestimator for position determination using measured pseudor-anges that are obtained by applying the interior-point optimiza-tion method and shows the validity of the proposed estimator.First, it uses an iterated least square (ILS) estimator to solvea measurement equation and analyzes the effect of a mea-surement bias on positioning. The interior-point optimization

0018-9545/$20.00 © 2006 IEEE

1292 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006

will be applied to the constrained bias estimation problem, andthe algorithm of the three-stage estimator for position deter-mination using pseudorange is presented in Section III. Thelocation accuracy of the newly proposed method is presented inSection IV. This paper also compares the proposed methodwith the conventional ILS, Wylie and Holtzmann’s LOS recon-struction, and Chen’s residual weighting method through math-ematical formulations and simulations. Finally, we present anaccuracy analysis based on some results from simulation tests.

II. LS ESTIMATE AND BIAS OBSERVATION EQUATION

The wireless location problem may be formulated by thefollowing nonlinear measurement equation:

r = h(x) + b + v (1)

wherex ∈ �n MS position to be estimated;r ∈ �m measured pseudoranges (m ≥ n);h(x) true distance between a BS and an MS (hi(x) =

‖x − xi‖, where xi is ith BS’s position);b measurement bias included in measured pseu-

doranges;v measurement noise with zero mean and covariance

R.Equation (1) can be solved for x using an LS. Define the fol-

lowing quadratic cost function assuming that the measurementbias b is known:

min J(x) = (r − h(x) − b)T R−1 (r − h(x) − b) . (2)

The nonlinear function h(x) is linearized about a referencepoint x0. Ignoring high-order terms,

h(x) ≈ h(x0) + H0(x − x0) (3)

where H0 is the Jacobian matrix of h(x) at x0, viz.,

H0 =

∂h1∂x1

· · · ∂h1∂xn

.... . .

...∂hm

∂x1· · · ∂hm

∂xn

∣∣∣∣∣∣∣x=x0

.

For (3) to be valid, the reference point x0 should be chosenclose enough to the true position. It may be obtained byemploying an algebraic method that solves for an intersection oftwo circular lines-of-position (LOPs) drawn from the locationof the BS and the measured pseudorange, where the location ofthe BS is the center of the LOP and the measured pseudorangeis a radius of the LOP. This initial position contains an NLOSbias error. However, the NLOS bias is estimated to recalculatethe MS position in later steps, which is explained in the nextsection.

Granted that the presumption is met, (1) and (2) can beexpressed as

y =H0x + b + v (4)

min J(x) = (y −H0x − b)TR−1(y −H0x − b) (5)

where y is a linearized measurement, i.e.,

y = r − (h(x0) −H0x0) . (6)

To minimize J(x), a position x that satisfies its first-orderdifferential ∇xJ(x) = 0 is to be solved. The optimal estimatex that minimizes J(x) is given by

x =(HT

0 R−1H0

)−1HT

0 R−1y − (HT

0 R−1H0

)−1HT

0 R−1b.(7)

Suppose for a moment that the measurement bias b can beignored. Then, the position estimate in (7) may be written as

x =(HT

0 R−1H0

)−1HT

0 R−1y (8)

where x is the bias-free estimate that is computed as if no biaswere present. It should be pointed out that both x and x arerandom variables since they are computed using an LS method.

Now, taking the bias error b into consideration, (7) can beexpressed by x and b as

x = x + V b (9)

where V is the correction matrix for the bias error b. Accordingto (7), V holds and is given by

V = − (HT

0 R−1H0

)−1HT

0 R−1. (10)

So far, we have assumed that the bias error b is known.However, in reality, b is not measurable but can be estimatedfrom an observed quantity. The observed quantity used toestimate b is expressed by

z = y −H0x = (H0x + b + v) −H0(x − V b)

= (I + H0V )b + H0(x − x) + v

=S · b + w (11)

where S is defined as S = I + H0V . z is readily availableif y and x are given. y is computed according to (6) fromthe pseudorange measurement r detected by a receiver of anMS. x was computed by (8). In (11), w is a measurementerror with a covariance of E(w · wT ) = H0PxH

T0 + R, where

Px = E[(x − x) · (x − x)T ] = (H0R−1HT

0 )−1.Before we estimate b from (11), we need to check the

observability of the equation. Examine the matrix S defined in(11). It can be observed that S is orthogonal to H0 since

S ·H0 = H0 −H0

(HT

0 R−1H0

)−1HT

0 R−1H0 = 0. (12)

Therefore, the determinant of S = I + H0V is not fullrank according to Sylvester’s inequality theorem [2]. That is,det(S) = 0. Since the matrix S is not full rank, b is notobservable. To estimate b, we need additional information onthe bias error. The upper and lower bounds of the bias errorare employed in this paper. The formulation of the problem toestimate the bias error b is explained in the sequel.

KIM et al.: INTERIOR-POINT METHOD FOR AN OPTIMAL TREATMENT OF BIAS IN TRILATERATION LOCATION 1293

III. INTERIOR-POINT OPTIMIZATION METHOD APPLIED

TO ESTIMATION OF CONSTRAINED MEASUREMENT BIAS

Assume that the bounds of a bias error between a BS andan MS may be guessed based on past experiments or on amodel. Then, a constrained optimization problem that wouldbest estimate the bias error b can be formulated from (11) by

min J(b) = (z − S · b)TQ−1w (z − S · b)

subject to bi ∈ Bi, i = 1, 2, . . . ,m (13)

where Qw is a covariance of the error w, i.e., E(w · wT )and Bi = (li, ui) are sets in which each bias error bi betweenan MS and the ith BS lies. For the TOA measurement, thelower bound li always satisfies li ≥ 0, and the upper boundui satisfies ui ≤ K < ∞. There are several methods to de-termine the lower and upper bounds. The simplest way is tofix the bounds based on the layout of the BSs. For an urbanenvironment, the upper bound may be set to a constant in therange of 250–500 m, whereas the lower bound is set to zero.Venkatraman et al. [12] proposed a simple yet effective way todetermine the bounds using the geometry of the BSs’ layoutand range measurements. The lower bound li is set to zero,and the upper bound ui is set to the minimum of {ri + rj −Li,j | all j = i, Li,j = ‖xi − xj‖} at each instance, where ri

is the measured pseudorange between the MS and the ith BS.Li,j is the distance between the ith BS and the jth BS. Whenthe signal strength or the digital map is available, the path-lossmodel or the simplified ray-tracing method provides the rangeof the lower and upper bounds. Considering the geometry ofurban environments, some imaginary scatters may be appliedto add constraint to the measured pseudorange [11]. We utilizedthe Venkatraman et al. method to simulate our algorithm [12].

A. Estimation of Bias by Interior-Point Optimization Method

To solve (13), we employ the interior-point method devel-oped by Karmarker [9] for two reasons: First, it generatesiterations starting from a point inside a given feasible spaceand is suitable to the estimation problem in case the boundarypoint is difficult to define. The iterative procedure is related tosolving a set of nonlinear equations of (13) with a quadraticcost function. Second, it efficiently considers the restrictedinequality condition of the measurement bias while requiringonly a little additional computation.

In the following, we explain how the interior-point methodis applied to the constrained measurement bias. Introducing abarrier function gi(bi) and a slack variable si, the optimizationproblem of (13) is modified equivalently as

min J(b) = (z − S · b)TQ−1w (z − S · b)

subject to

{gi(bi) − si = 0si > 0

, i = 1, 2, . . . ,m (14)

where the barrier function gi(bi) is defined to satisfy gi(bi) >0 ∀bi ∈ (li, ui). Generally, gi(bi) is given by a smooth second-

order function gi(bi) = (ui − bi)(bi − li) using both lowerand upper bounds. Equation (14) is solved by employ-ing a Lagrangian optimization technique. The correspondingLagrange equation is defined as

L(b,λ,s) = (z − S · b)TQ−1w (z − S · b) − µ

·m∑

i=1

ln si − λT (g(b) − s) (15)

where µ is a positive barrier parameter, g(b) = [g1(b1) g1(b2)· · · gm(bm)]T , and s = [s1 s2 · · · sm]T . Compared to thegeneral form of a Lagrangian equation, (15) includes a loga-rithmic barrier function µ · ∑m

i=1 ln si. The logarithmic barrierfunction is included in the Lagrange equation to satisfy si > 0so that an estimate of the bias error is forced to stay in theinterior of a feasible region of Bi = (li, ui).

The constrained optimization problem described in (13) and(14) has been reduced to a problem of minimizing (15). Thesolution to (15) is obtained by solving the following partialdifferential equations:

∂L

∂b=STQ−1

w (z − Sb) +12GT λ = 0

∂L

∂λ=g(b) − s = 0

∂L

∂s=

∑·λ − µ · 1 = 0 (16)

where G= ∂g(b)/∂b,∑

= diag{s1, s2, . . . , sm}, and 1 =[1, 1, . . . , 1]T . Equation (16) may be solved by the Newton–Raphson method. A detailed description of the Newton–Raphson method is explained in Appendix A. This solutionprovides an optimal estimate of the measurement bias b, sat-isfying the given constrained sets originally defined in (13).

B. Three-Stage Constrained Position-Determination Estimator

Thus far, we have shown that the interior-point method givesan optimal estimate of a measurement bias b contained insidean admissible range of the measurement bias. Once b is esti-mated, the position of an MS is determined using the estimatedb and the pseudorange measurements received by the MS. Thecomplete procedure may be summarized as follows: First, weuse an ILS to obtain the bias-free estimate x of the MS, as if nobias were present. Then, the interior-point optimization methodsolves the Lagrangian equation to estimate the measurementbias error b, while considering some constrained conditions.Finally, the bias-free estimated position is corrected by a mea-surement bias estimate to obtain an optimal estimation x of theposition of the MS. The procedure can be arranged by thefollowing algorithmic format:

Step 1)

x =(HT

0 R−1H0

)−1HT

0 R−1 (r − h(x0) + H0x0) . (17)

1294 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006

Fig. 1. Algorithm of the proposed three-stage estimator of pseudorange positioning using ILS and interior-point optimization method.

Step 2)

b = argmin

[L(b,λ, s) = (z − S · b)TQ−1

w (z − S · b) − µ

·m∑

i=1

ln si − λT · (g(b) − s)

]. (18)

Step 3)

x = x + V b. (19)

When some feasible ranges of the measurement bias are welldefined, the employed interior-point optimization optimallyestimates the measurement bias b. The estimator, however, mayhave limited location accuracy for two reasons: First, the ILShas the linearization error resulting from the inaccuracy of areference point during linearization. Second, the feasible rangeset of the bias error is not easily determined. Given an adequatebarrier function, the estimated bias taken by the interior-pointoptimization method corrects a bias-free estimate. This correc-tion makes a position estimate closer to the true position. An-other iteration takes the new presumed position as a referencepoint and repeats the preceding three steps. Consequently, thisiteration will eventually lead to a converged position x.

The iterative structure of the proposed three-stage [(17)–(19)and their iterations] estimator is illustrated in Fig. 1. A re-ceiver of an MS takes a pseudorange measurement, which iscorrupted by measurement bias and noise. Based on a prior es-timate x(k−1), the measured pseudoranges are linearized aroundx(k−1) and expressed by y(k). Then, an LS method gives abias-free estimate x(k) from y(k). A bias-observation equation

described in (11) and the related variables are computed.Assuming a constrained set on the measurement bias b, theinterior-point method solves a Lagrangian equation consistingof a quadratic cost function and a barrier function to obtainan optimal estimate b(k) of the measurement bias. Equation(19) computes a corrected position x(k), combining a bias-freeestimate x(k) and a bias estimate b(k). The iteration is stoppedwhen the corrected position and the estimate of the bias errorconverge to∥∥∥x(k) − x(k−1)

∥∥∥ < κ3,∥∥∥b(k) − b(k−1)

∥∥∥ < κ4. (20)

Fig. 2 shows a typical convergence path of both bias-freeestimates x(k) and the corresponding bias-corrected positionsx(k). x(1) is computed according to (6) and (8), starting fromx(0), which is obtained by directly solving the functional rela-tionship between the pseudorange r and the location vector x,i.e., r = h(x). x(1) is then obtained from (19), and the iterationrepeats. As illustrated in Fig. 2, x(k) is, in general, convergedto the true position x∗. Our experience tells us that a propertydefining the range of the measurement bias b described in (13)and (14) is the most important factor for x(k) to converge to x∗.A way of defining the range is further explained in the next sec-tion, where the proposed method is implemented and tested.

IV. PROPERTIES OF THE PROPOSED

THREE-STAGE ESTIMATOR

Two important issues for applying the proposed estimatorare the location accuracy of the corrected position estimatex and the convergence of x to true position. In this section,

KIM et al.: INTERIOR-POINT METHOD FOR AN OPTIMAL TREATMENT OF BIAS IN TRILATERATION LOCATION 1295

Fig. 2. Convergence path of x(k) and x(k) by iteration.

the location accuracy and the convergence are analyzed incomparison with other estimation methods.

A. Location Accuracy and Convergence

Consider the difference between the position estimate xand the true position x, and define its l2 norm as locationaccuracy Λ.

Λ = ‖x − x‖2. (21)

To get x, we rewrite (8) and (10) as

x =(HT

0 R−1H0

)−1HT

0 R−1 (r − (h(x0) −H0x0)) (22)

V = − (HT

0 R−1H0

)−1HT

0 R−1. (23)

Substituting r of (1) and V of (23) into (22), we get

x =x +(HT

0 R−1H0

)−1HT

0 R−1

× [b + v + (h(x) − h(x0) −H0(x − x0))]

=x − V · b − V · v − V · ε (24)

where ε is a linearization error given by

ε = h(x) − h(x0) −H0(x − x0). (25)

We have shown that (16) may be approximately solved byemploying the Newton–Rhapson method. Equation (16) maybe rewritten with the approximated solutions b, λ, s, and µk as

STQ−1w (z − Sb) +

12G(b)tλ = 0

g(b) − s = 0∑(s) · λ − µk · 1 = 0 (26)

where {µk} is a decreasing sequence of positive barrier para-meters such that limk→∞ µk = 0.

Equation (26) can be simplified by

STQ−1w z − STQ−1

w Sb

+µk

2

[g1(b1)

g1(b1)

g2(b2)

g2(b2)· · · gm(bm)

gm(bm)

]T

= 0 (27)

where z is a residual defined by (11) and can be expressedaccording to (6) and (24) by the following equation:

z = S(b + v + ε). (28)

Inserting (28) into (27), the solution of (27) b can be ex-pressed by a combination of (b + v + ε) and U + L, whichrepresents the bracket term

b = θ · U + L2

+ (I − θ) · (b + v + ε) (29)

where θ is an arbitrary dividing ratio θ = diag(θ1, θ2, . . . , θm),U is an upper bound set of a range of b, U= [u1, u2, . . . , um]T ,and L is a lower bound set of a range of b, L= [l1, l2, . . . , lm]T .

Now, combining (24) and (28), x can be written as

x = x + V b

=x + V · θ ·[U + L

2− (b + v + ε)

]. (30)

Inserting (30) into (21), the location accuracy Λ is written as

Λ = ‖x − x‖2 =∥∥∥∥V · θ ·

[U + L

2− (b + v + ε)

]∥∥∥∥2

. (31)

1296 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006

We can find a condition in applying the three-stage estimatorto make Λ bounded. Suppose b satisfies li < bi < ui for allmeasurements i = 1, 2, . . . ,m. If we can find ui and li thatsatisfy the following two conditions, where Mi is a positiveconstant larger than 1, i.e., Mi > 1 ∀i = 1, 2, . . . ,m:{

li < bi + vi + εi < ui

ui − li < 2(1 − 1

Mi

)‖bi + vi + εi‖ , ∀i = 1, 2, . . . ,m

(32)

then the location accuracy Λ is bounded (see Appendix B forproof). Both the condition of (32) and the location accuracyof (B10) indicate that the lower and upper bounds shouldbe selected so that the range of the lower and upper boundsincludes the true bias error and the length difference of thelower and upper bounds is not too wide.

B. Comparison With Existing Approaches

The three-stage estimator proposed in this paper is comparedwith the existing methods of the ILS [5], Chen’s residualweighting method [10], and the LOS reconstruction method [8].Each method may be summarized as follows:

ILS

xi+1 =(HT

i R−1Hi

)−1HT

i R−1 (r − h(xi)) + xi (33)

Residual Weighting

x =∑N

k=1 xk · ‖rk − h(xk)‖−12∑N

k=1 · ‖rk − h(xk)‖−12

(34)

LOS Reconstruction

x =(HT

0 R−1H0

)−1HT

0 R−1 (rW − h(x0)) + x0 (35)

where

rW = SW (r) − c · σW (r).

In the residual weighting method, rk is an arbitrarily chosensubset of r and contains m pseudoranges from m BSs. xk

is a bias-free estimate corresponding to rk. N is the numberof subsets obtainable from r. The residual weighting methodestimates x using a weighted linear combination of all estimatesobtained from the subset of BSs.

In the LOS reconstruction method, a number of pseudorangevectors r(t1), r(t2), . . . , r(tW ) are preprocessed before x iscalculated. For the signals of the m BSs, a smoothed rangefrom previous w samples is sW (r) = [s1(r1), s2(r2), . . . ,sm(rm)]T , where sk(rk) indicates a smoothed value of rk(t1),rk(t2), . . . , rk(tW ) at time tW by an N th-order polynomial.σW (r) is a standard deviation of r(t1), r(t2), . . . , r(tW ), andc is an arbitrary positive constant. According to [8], σW (r) canbe given a priori value, and c is generally set to 1. The methodregards an NLOS error as a positive random variable, and itsubtracts an estimated NLOS error from a smoothed value of asample sequence from pseudorange measurements to restore aLOS distance. LOS reconstruction should be performed only onmeasurements from NLOS BSs if the variation test of σW (r)

Fig. 3. Simulation layout of an experimental area and an estimation result bythe three-stage estimator.

is possible to distinguish between LOS measurements andNLOS measurements. In addition, the LOS reconstruction iscorrectly applied to environments where there are intermittentLOS/NLOS measurements from each BS. To compare the loca-tion accuracy of each method, Monte Carlo trials are performedin a realistic situation, as shown in Figs. 3–7. This simulationassumes a microcellular environment in a city where an NLOSsituation occurs more frequently than in the rural area. There-fore, the distances between BSs are assumed to be less than1 km. Four BSs (BS#1, BS#2, BS#3, and BS#4) are located atthe endpoint of roads. A vehicle (which is the MS in the figure)moves along the road from BS#1, which is located at (−500, 0),to BS#3, which is located at (500, 0), with a constant velocityof 5 m/s. In this simulation, 1.2288-MHz pilot sequences areused as the position signals. A 1% measurement error of thechip length of the sequence is assumed, and an analog-to-digital (A/D) converter having 16 times of the chip frequencyas its sampling rate is also assumed. Accordingly, the Gaussianmeasurement noise of the receiver has a variance of 2.442 m2,and the error caused by the sampling rate of A/D converteris uniformly distributed on [−15.26 m, 15.26 m]. In addition,10% of the NLOS path-delay error is added to the pseudorangemeasurement as additive white Gaussian noise. Assuming thatthe MS gets the signals from the neighboring four BSs, four sce-narios happen based on the deployment of BSs, namely 1) morethan three LOS signals; 2) two LOSs and other NLOSs; 3) oneLOS and other NLOSs; and 4) all NLOS signals. In case ofthree or more LOS signals, ILS will be enough to get theposition of the traveling MS. Hence, the other three cases areinvestigated here. In the first place, it is assumed that BS#1and BS#3 transmit LOS signals without NLOS errors, whereasthe signals from BS#2 and BS#4 include NLOS errors ofabout 50–400 m due to shaded building blocks. Given thebuilding blocks that model the neighboring district in which thetargeted MS is located, applying the building-block approachintroduced by Lee [14] gives an idea in generating the travelingdistance between a BS and an MS, including reflections anddiffractions [15]. It causes the NLOS error proportional to the

KIM et al.: INTERIOR-POINT METHOD FOR AN OPTIMAL TREATMENT OF BIAS IN TRILATERATION LOCATION 1297

Fig. 4. Performance comparison of the four methods. (a) Position error.(b) Empirical CDF.

direct distance between a BS and an MS when the signal isblocked by the intermediate buildings, whose proportional rateis locally dependent on the distribution of buildings.

A set of range measurements is taken every second. Forthe LOS reconstruction method, 10 recent samples are usedfor the smoothed estimate, employing the first-order polyno-mial sW (r), i.e., W = 10. The other methods, including thethree-stage estimator, estimate the MS location directly usingthe range measurements. In addition, the three-stage estimatoremploys the Venkatraman et al. method [12] for selecting thelower and upper bounds of NLOS bias errors. For the LOSreconstruction method, the constant c is chosen to be 1.5because it provides the best result in this simulation.

In Fig. 3, the cross-marked (+) trajectory represents positionestimates of the MS obtained by the three-stage estimator. Itshows that the estimator performs well and provides a positionestimation of 17-m, with 1-σ error bound in the presence of50–400-m NLOS errors.

For comparison of the four methods, 50 Monte Carlo trialsare run for the given scenario and depicted in Fig. 4. When the

Fig. 5. Performance comparison of three methods using three BS signals withthe best signal strength. (a) Position error. (b) Empirical CDF.

MS crosses the side streets in Fig. 3, then there are three LOSBSs. Hence, the errors are reduced at the intersection locations,as shown in Fig. 4. Since ILS does not consider a measurementbias, its accuracy is low. In addition, the positioning error of theresidual weighting method is larger than that of the three-stageestimator because three pure LOS signals cannot be chosen andNLOS signals are always included in the estimation of any xk.Moreover, the residual weighting method requires the signalsfrom more than four BSs. In the case of three BSs, it showsthe same result as the ILS method. The other three algorithmswork well for only three BSs. Therefore, the residual weight-ing method is not desirable in an area having the hearabilityproblem due to a shortage of received signals. The LOS recon-struction method exhibits better accuracy than ILS by restoringLOS signals.

Compared to the other methods, the three-stage estimatorshows good accuracy. Fig. 4(b) represents the empirical cumu-lative distribution function (CDF) of the four methods. Con-cerning 67% distance root-mean-square (DRMS) positioningaccuracy, the three-stage estimator has a 67% DRMS of 17 m,

1298 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006

Fig. 6. Performance comparison of four methods with one LOS and threeNLOSs signals. (a) Simulation layout. (b) Empirical CDF of position error.

the LOS reconstruction method has 94 m, the ILS has 118 m,and the residual weighting method has 69 m. Whereas Fig. 4shows the results using all four measurements, Fig. 5 shows theresults using three chosen signals with the best signal strength.Since two BSs are always LOS, the ILS would perform muchbetter in that case, as would the LOS reconstruction method.Concerning 67% DRMS positioning accuracy, the three-stageestimator has a 67% DRMS of 24 m, the LOS reconstructionmethod has 51 m, and the ILS has 72 m. In case that allthree BS signals are LOS, the ILS performs better because theproposed three-stage estimator assumes that each signal has abounded NLOS error. Under NLOS environments, the proposedthree-stage estimator performs better. Identification of LOSsignals will enhance the location accuracy when the three-stageestimator is implemented.

Another Monte Carlo trial is run for the given scenario anddepicted in Fig. 6. It is assumed that only BS#1 transmits LOSsignals, whereas the other three BSs include NLOS errors. Theperformance is degraded compared to the previous experiment,which has two LOSs and two NLOSs signals. Concerning 67%DRMS positioning accuracy, the three-stage estimator has a67% DRMS of 80 m, the LOS reconstruction method has

Fig. 7. Performance comparison of four methods when all signals are NLOS.(a) Simulation layout. (b) Empirical CDF of position error.

133 m, the ILS has 160 m, and the residual weighting has205 m. In addition, the other Monte Carlo trial is run for thegiven scenario and depicted in Fig. 7. It is assumed that there isno LOS signal. Concerning 67% DRMS positioning accuracy,the three-stage estimator has a 67% DRMS of 44 m, the LOSreconstruction method has 181 m, the ILS has 214 m, and theresidual weighting has 245 m.

These results reveal that consideration of a feasible rangeof measurement bias, as done by the three-stage estimator,effectively enhances location accuracy.

V. CONCLUSION

A new position-determination estimator to minimize theeffect of a measurement bias due to NLOS errors has beenproposed for calculating a wireless location. The problem ofwireless location is formulated to solve pseudorange equationsincluding bias errors. Pseudoranges obtained by the signalsfrom multiple BSs are first solved for an MS location assumingfixed bias errors by means of an ILS method. This solution iscalled a bias-free estimate. Bias errors resulting from NLOS er-rors are then formulated by a Lagrangian equation as a form of

KIM et al.: INTERIOR-POINT METHOD FOR AN OPTIMAL TREATMENT OF BIAS IN TRILATERATION LOCATION 1299

a constrained optimization problem. The problem is solved byemploying the interior-point and the Newton–Raphson meth-ods. The solution is called a bias estimate. A combination ofthe bias estimate and the bias-free estimate results in an optimalestimate of the MS location.

It is shown that the three-stage estimator guarantees a con-vergence to a true position if a proper boundary of bias error ischosen. Its convergence and location accuracy are demonstratedby realistic simulations. The performance is compared withsome existing methods, such as the ILS method, the LOSreconstruction method, and the residual weighting method. Thethree-stage estimator exhibits better performance over the othermethods. Simulations under a typical urban environment haveshown that the ILS method’s accuracy of 118 m (67% DRMS)can be improved to about 17 m by using the three-stage estima-tor. The simulations are sufficiently realistic so that they includemicrocellular environments with one or no LOS signal. We areconfident that the three-stage estimator will demonstrate similarimprovements when it is implemented in the field.

The estimator may be extended to other location determina-tion problems using pseudoranges corrupted by a measurementbias, e.g., global positioning system (GPS) and a radar-trackingproblem. The estimator can be further improved if the measure-ment bias including NLOS errors is realistically modeled and ifthe recent development to improve hearability is added [7].

APPENDIX AIMPLEMENTATION OF THE NEWTON–RAPHSON METHOD

TO SOLVE THE LAGRANGIAN EQUATION

The Newton–Rhapson method gives

STQ−1w (z − Sb) +

12GT λ = 0

g(b) − s = 0∑·λ − µ · 1 = 0. (A1)

The Newton–Rhapson’s method makes the increment of eachvariable [δbk δλk δsk]T computed by−STQ−1

w S + 12

δGk

δb

∣∣∣b=bk

·Λk 12 (Gk)T 0

(Gk)T 0 −I

0∑k Λk

δbk

δλk

δsk

= −STQ−1

w (z − Sbk) + 12 (Gk)T λk

g(bk) − sk∑k ·λ0 − µ · 1

(A2)

where Λ = diag{λ1, λ2, . . . , λm}, Gk = G(bk),∑k =∑

(sk), and Λk = Λ(λk).The Newton steps [bk + δbk λk + δλk sk + δsk] may not

exist inside the admissible set, so we update the variables ateach iteration with

bk+1 =bk + αP δbk (A3)

λk+1 = λk + αDδλk (A4)

sk+1 = sk + αP δsk (A5)

where the step lengths of the primal αP and the dual αD pre-vent the estimate of the bias error from being too close to theboundary of the feasible set, as follows:

αP =0.995minj

{− sk

j

δskj

| δskj < 0

}(A6)

αD =0.995minj

{− λk

j

δλkj

| δλkj < 0

}. (A7)

To apply the interior-point optimization method to a bias esti-mation, the initial estimate of the bias error must be determinedin advance. Most importantly, b0 takes the previous estimate ora value satisfying the given conditional inequality of the biaserror. The slack variable is set to qualify s0 = g(b0) > 0, andthe dual variable has the initial value of λ0 = µ · Σ−1 · 1. AsFiacco and McCormick showed, a barrier constant should beµ → 0 to get the required solution, and therefore, the barrierconstant is renewed at each iteration by

µ =λT s

(n + m)2. (A8)

The repeated solution of (A5) is stopped when the followingtwo conditions are satisfied:

‖δbk‖ < κ1,

∥∥∥∥∥ (λk)T sk

m

∥∥∥∥∥ < κ2. (A9)

APPENDIX BBOUNDED PROPERTY OF LOCATION ESTIMATE

Considering the Lagrangian equation and its solution givenby (27) and (28)

STQ−1w z − STQ−1

w S b

+µk

2

[g1(b1)

g1(b1)

g2(b2)

g2(b2)· · · gm(bm)

gm(bm)

]T

= 0 (B1)

b = θ · U + L2

+ (I − θ) · (b + v + ε). (B2)

If the upper and lower bounds of bi, ui, and li satisfy thefollowing two conditions:

li <bi + vi + εi < ui, for all i = 1, 2, . . . ,m (B3)

ui − li < 2 ·(

1 − 1Mi

)· ‖bi + vi + εi‖,

for all i = 1, 2, . . . ,m (B4)

then the location accuracy Λ defined by (31)

Λ = ‖x − x‖2 =∥∥∥∥V · θ ·

[U + L

2− (b + v + ε)

]∥∥∥∥2

(B5)

is bounded.

1300 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 4, JULY 2006

Proof: Equations (B3) and (B4) can be expressed by thefollowing equations, which employ arbitrary Mu

i and M li such

that Mi > M li > Mu

i > 1 is satisfied:

ui = bi + vi + εi + 2 ·(

1 − 1Mu

i

)· ‖bi + vi + εi‖ (B6)

Ii =ui − 2 ·(

1 − 1M l

i

)· ‖bi + vi + εi‖. (B7)

Suppose that bi lies between ui and li. Then, (B6) and (B7)give the following inequality by applying (B2) to bi:

‖1 − θi‖ ·∥∥∥∥ui + li

2− (bi + vi + εi)

∥∥∥∥= ‖1 − θi‖ ·

∥∥∥∥1 − 2Mu

i

+1M l

i

∥∥∥∥ · ‖bi + vi + εi‖

<

∥∥∥∥1 − 1M l

i

∥∥∥∥ · ‖bi + vi + εi‖. (B8)

By the triangle inequality, for each i

∥∥∥∥θi ·[ui + li

2− (bi + vi + εi)

]∥∥∥∥≤ (‖1 − θi‖ + 1) ·

∥∥∥∥ui + li2

− (bi + vi + εi)∥∥∥∥

≤(∥∥∥∥1 − 1

M li

∥∥∥∥ +∥∥∥∥1 − 2

Mui

+1M l

i

∥∥∥∥)· ‖bi + vi + εi‖

< 2 ·∥∥∥∥1 − 1

Mi

∥∥∥∥ · ‖bi + vi + εi‖. (B9)

Now, visit (B5). Equation (B9) and the Schwarz inequalitygive

Λ = ‖x − x‖2 = ‖V ‖2 ·∥∥∥∥θ ·

[U + L

2− (b + v + ε)

]∥∥∥∥2

< 2 ·√(

1 − 1M1

)2

+ · · · +(

1 − 1Mm

)2

· ‖V ‖2 · ‖b + v + ε‖2. (B10)

Λ is bounded according to the equation.

ACKNOWLEDGMENT

The authors would like to thank the Automation and SystemsResearch Institute and the Automatic Control Research Centerat Seoul National University and Samsung Electronics Com-pany, Ltd. for their support.

REFERENCES

[1] A. V. Fiacco and G. P. McCormick, Nonlinear Programming: SequentialUnconstrained Minimization Techniques. New York: Wiley, 1968.

[2] C.-T. Chen, Linear System Theory and Design. New York: Holt,Rinehart and Winston, 1984.

[3] H.-L. Song, “Automatic vehicle location in cellular communicationssystems,” IEEE Trans. Veh. Technol., vol. 43, no. 4, pp. 902–908,Nov. 1994.

[4] J. Caffery, Jr. and G. L. Stüber, “Subscriber location in CDMA cellu-lar networks,” IEEE Trans. Veh. Technol., vol. 47, no. 2, pp. 406–416,Nov. 1998.

[5] J. Caffery, Jr., Wireless Location in CDMA Cellular Radio Systems.Norwell, MA: Kluwer, 2000.

[6] L. Cong and W. Zhuang, “Nonline-of-sight error mitigation in mobilelocation,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 560–573,Mar. 2005.

[7] M. F. Madkour, S. C. Gupta, and Y.-P. E. Wang, “Successive interferencecancellation algorithms for downlink W-CDMA communications,” IEEETrans. Wireless Commun., vol. 1, no. 1, pp. 169–177, Jan. 2002.

[8] M. P. Wylie and J. Holtzmann, “The non-line-of-sight problem in mobilelocation estimation,” in Proc. 1996 5th IEEE Int. Conf. Universal Pers.Commun., 1996, vol. 2, pp. 827–831.

[9] N. Karmarker, “A new polynomial-time algorithm for linear program-ming,” Combinatorica, vol. 4, no. 4, pp. 373–395, Dec. 1984.

[10] P.-C. Chen, “Mobile position location estimation in cellular systems,”Ph.D. dissertation, Univ. New Brunswick, Saint John, NB, Canada, 1999.

[11] S. Al-Jazzar and J. Caffery, Jr., “NLOS mitigation method for urbanenvironments,” in Proc. IEEE VTC Fall Veh. Technol. Conf., 2004,pp. 5112–5115.

[12] S. Venkatraman, J. Caffery, Jr., and H.-R. You, “Location using LOSrange estimation in NLOS environments,” in Proc. IEEE VTC Spring Veh.Technol. Conf., 2002, pp. 856–860.

[13] T. S. Rappaport, J. H. Reed, and B. D. Woerner, “Position location usingwireless communications on highways of the future,” IEEE Commun.Mag., vol. 34, no. 10, pp. 33–41, Oct. 1996.

[14] W. C. Y. Lee,Mobile Communications Engineering: Theory and Practice.Englewood Cliffs, NJ: Prentice-Hall, 1998.

[15] W. Kim, J. G. Lee, and G. I. Jee, “Increase of positioning robustness byintegration of CDMA and GPS,” in Proc. Int. Symp. GPS/GNSS. Korea,Nov. 2001, pp. 124–129.

Wuk Kim was born in Korea in 1971. He receivedthe B.S., M.S., and Ph.D. degrees in electrical engi-neering and computer science from Seoul NationalUniversity, Seoul, Korea, in 1995, 1997, and 2003,respectively.

In 2002, he joined the Telecommunication R&DCenter, Samsung Electronics, Gyeonggi-do, Korea,where he has been a Senior Engineer in the areaof mobile standardization since 2003. His researchinterests include service requirements and architec-ture, mobile applications of multimedia messaging,

mobile location in the Third Generation Partnership Project (3GPP), and OpenMobile Alliance (OMA) standard fora.

KIM et al.: INTERIOR-POINT METHOD FOR AN OPTIMAL TREATMENT OF BIAS IN TRILATERATION LOCATION 1301

Jang Gyu Lee (S’76–M’77) received the B.S. de-gree in electrical engineering from Seoul NationalUniversity, Seoul, Korea, in 1971, and the M.S. andPh.D. degrees from the University of Pittsburgh,Pittsburgh, PA, in 1974 and 1977, respectively.

In 1977, he joined The Analytic Sciences Corpo-ration (TASC), Reading, MA, where he worked onmissile parameter identification, missile guidance,and security assessment of power plants. From 1981to 1982, he was with the Charles Stark DraperLaboratory, Cambridge, MA, where he worked on

inertial navigation systems and optimal control of underwater vehicle. In 1982,he joined the faculty of the School of Electrical Engineering and ComputerScience, Seoul National University, where he is currently teaching and doingresearch on navigation, guidance, and control, and has been named Director ofthe Automatic Control Research Center since 1995. In 1998 and 1999, he wasa Visiting Professor with the Bradley Department of Electrical and ComputerEngineering and the Center for the Study of Science in Society, VirginiaPolytechnic Institute and State University, Blacksburg. He has also publishedmore than 150 journal papers and 350 conference papers on the subject ofnavigation, guidance, and control. His current research interests include theoryand applications of linear and nonlinear filtering, navigation technologies, andmicroelectromechanical-systems-based inertial sensors.

Dr. Lee is a member of the National Academy of Engineering of Korea,Sigma Xi, the Korean Institute of Electrical Engineers, the IFAC TechnicalCommittee on Aerospace, and the IFAC Technical Committee on Education.

Gyu-In Jee (S’82–M’90) received the B.S. and M.S.degrees in control and instrumentation engineeringfrom Seoul National University, Seoul, Korea, in1982 and 1984, respectively, and the Ph.D. degreefrom Case Western Reserve University, Cleveland,OH, in 1989.

He joined the faculty of the Department ofElectronics Engineering, Konkuk University, Seoul,in 1992. In 2000 and 2001, he was a Visiting Profes-sor with the Department of Geomatics Engineering,University of Calgary, Calgary, AB, Canada. His

previous research interests include global positioning systems and wirelesslocation. His current research interests include the software Global NavigationSatellite System (GNSS) receiver, antijamming for GNSS, and IEEE 802.16eand ultrawideband-based wireless location systems.

Dr. Jee is a Chair of the Korean GNSS Technology Council and the WirelessLocation Working Group in the Korean LBS standard forum.