IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 6, DECEMBER 2013 623

Robust Coarse Position Estimation for TDOA LocalizationTianzhu Qiao, Stephen Redfield, Arash Abbasi, Zhenqiang Su, and Huaping Liu, Senior Member, IEEE

Abstract—The non-linear least squares (NLLS) algorithm iswidely used in localization systems. Its performance approachesthe Cramer-Rao lower bound under i.i.d. additive white Gaussiannoise. However, when the initial position chosen is not closeenough to the actual target position, the NLLS algorithm willvery likely diverge. The non-iterative method of moments estima-tor does not have this divergence problem, but it performs worsethan NLLS and requires at least one more anchor to linearize therange measurement equations. In this paper, we develop a coarseposition estimation algorithm based on scaling by majorizing acomplicated function for time-difference-of-arrival localization,which is robust with regard to the initial position and does notrequire redundant receivers.

Index Terms—Location estimation, time-difference-of-arrival(TDOA), iterative algorithms.

I. INTRODUCTION

T IME-OF-ARRIVAL (TOA) [1]–[3] and time-difference-of-arrival (TDOA) [4] techniques are widely used in

high-precision pulsed ultrawideband localization systems [5],[6]. TDOA is in general easier to implement than TOA sinceit only requires synchronization among the receivers. Thenonlinear least-squares (NLLS) algorithm [7] is widely used inTDOA localization systems, as it achieves better performancethan some non-iterative methods (e.g. method of moments(MOM) approaches [8]–[12]), and is easier to implementthan some more advanced methods (e.g. linear programmingmethods [13], [14]).

However, NLLS is not guaranteed to converge [7], [9]; ifthe initial position chosen is far away from the actual targetposition, then the offset from the Taylor expansion will belarge [7], which will likely cause NLLS to diverge.

Divergence becomes a more serious problem when thesystem coverage area is large since it becomes more difficultto fix one initial position for all possible target locations.

In this case, MOM algorithms [8]–[12] can be applied toprovide a coarse position estimate as the initial position forNLLS. However, the MOM algorithms require at least onemore additional anchor to linearize the range measurementequations [8]–[12] by canceling the quadratic items, which iscostly and sometimes not available.

In this paper, we derive a coarse position estimation algo-rithm employing the Scaling by MAjorizing a COmplicatedFunction (SMACOF) strategy [15]–[18] for TDOA localiza-tion. The proposed algorithm does not converge as fast asthe NLLS algorithm, but it is not sensitive to the choice ofthe initial position, and the mean-square error is guaranteed

Manuscript received July 28, 2013. The associate editor coordinating thereview of this letter and approving it for publication was S. Gezici.

The authors are with the School of Electrical Engineering and ComputerScience, Oregon State University, Corvallis, OR 97331 USA (e-mail: {qiaoti,redfiels, abbasia, suzh, hliu}@eecs.oregonstate.edu).

This work is supported by the National Science Foundation under GrantIIS-1118017.

Digital Object Identifier 10.1109/WCL.2013.082813.130543

to decrease at each iteration. More importantly, it does notrequire redundant receivers; when redundancy in the rangemeasurements is unavailable, it can be used to provide theinitial target position estimate for the NLLS algorithm.

II. COARSE POSITION ESTIMATION

A. System Model

Let θ be the unknown target location, which is to beestimated with the known M anchors (v1, · · ·vM ). Withoutloss of generality, let anchor 1 be the reference anchor. ForTDOA system, the range measurements are written as

rm1(i) = dm(θ)− d1(θ) + bm − b1 + nm(i)− n1(i),

i = 1, · · · , N ;m = 2, · · · ,M (1)

where dm(θ) = ‖θ − vm‖ is the true distance between themth anchor and the target, i is the index of the observationset, N is the total number of observation sets, bm is a positiveoffset caused by non-line-of-sight (NLOS) propagation, andnm(i)

iid∼ N (0, σ2m) is the range measurement error.

NLOS propagation often exists between the target andanchors. In practice, NLOS links may be detected [19], [20]and severe NLOS links that do not contain much range infor-mation are discarded. Otherwise, LOS range measurementscan be estimated from raw data [7], [21], [22]. In derivingthe proposed algorithm, bm and nm(i) will be lumped intoa single error term, since this does not affect the algorithmdevelopment.

B. SMACOF Algorithm for TDOA Localization

The stress function δ(θ) to be minimized is defined as [15],[17]:

δ(θ) =

N∑

i=1

M∑

m=2

(rm1(i)− dm(θ) + d1(θ))2. (2)

Expanding the stress function, we have

δ(θ) =N∑i=1

M∑m=2

(rm1(i)− dm(θ) + d1(θ))2

=

N∑i=1

M∑m=2

rm1(i)2 +

N∑i=1

M∑m=2

(dm(θ)2 + d1(θ)

2)−

2N∑i=1

M∑m=2

(rm1(i)dm(θ)− rm1(i)d1(θ) + d1(θ)dm(θ))

= ηr + ηd − 2ρr,d, (3)

where ηr =∑N

i=1

∑Mm=2 rm1(i)

2 is a constant since it isindependent of θ, ηd =

∑Ni=1

∑Mm=2

(dm(θ)2 + d1(θ)

2)

is a quadratic function of θ, but ρr,d =∑Ni=1

∑Mm=2 (rm1(i)dm(θ)− rm1(i)d1(θ) + d1(θ)dm(θ))

is not a quadratic function of θ. Consequently, δ(θ) is nota quadratic function, and the common method to find the

2162-2337/13$31.00 c© 2013 IEEE

624 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 6, DECEMBER 2013

minimum by setting its first partial derivative to zero cannotbe applied here.

However, if a suitable quadratic surrogate function for ρr,dis available, then the minimum of the stress function can bedetermined through iteration. First, we consider the case thatrm1(i) ≥ 0, and examine the individual terms of δ(θ):

1) rm1(i)dm(θ): the same technique used in SMACOFalgorithm [17] can be applied here. From the Cauchy-Schwarz inequality:

dm(θ) = ‖θ − vm‖ ≥⟨θ − vm, θ′ − vm

⟩

‖θ′ − vm‖:= α(vm, θ, θ′), (4)

where θ′ is obtained either from the previous iterationor an initial guess, 〈a, b〉 denotes the inner product ofa and b. And it is easy to see

dm(θ′) = α(vm, θ′, θ′). (5)

2) rm1(i)d1(θ): it is easy to show that

d1(θ) = ‖θ − v1‖ ≤ ‖θ − v1‖2 + ‖θ′ − v1‖22‖θ′ − v1‖

:= β(v1, θ, θ′), (6)

and

d1(θ′) = β(v1, θ

′, θ′). (7)

3) d1(θ)dm(θ): applying the same method used in 1) tod1(θ) and dm(θ), we have

d1(θ)dm(θ) ≥⟨θ − vm, θ′ − vm

⟩

‖θ′ − vm‖

⟨θ − v1, θ

′ − v1

⟩

‖θ′ − v1‖:= γ(v1,vm, θ, θ′), (8)

and

d1(θ′)dm(θ′) = γ(v1,vm, θ′, θ′). (9)

With Eqs. (4), (6) and (8), we can define

τ(θ, θ′) = ηr + ηd − 2

N∑

i=1

M∑

m=2

τm1,i(θ, θ′), (10)

where for the case that rm1(i) ≥ 0,

τm1,i(θ, θ′) = rm1(i)α(vm, θ, θ′) + γ(v1,vm, θ, θ′)

−rm1(i)β(v1, θ, θ′). (11)

And similarly, for the case that rm1(i) < 0,

τm1,i(θ, θ′) = rm1(i)β(vm, θ, θ′) + γ(v1,vm, θ, θ′)

−rm1(i)α(v1, θ, θ′). (12)

τ(θ, θ′) is a quadratic function and easy to verify

δ(θ) ≤ τ(θ, θ′), ∀θ,δ(θ′) = τ(θ′, θ′). (13)

Therefore, τ(θ, θ′) is a valid quadratic surrogate functionfor δ(θ). It is easy to find its minimum by setting its firstderivatives to zeros. For 3-dimensional (3-D) localization, wehave

∂τ(θ, θ′)∂x

=∂ηd∂x

− 2

N∑

i=1

M∑

m=2

∂τm1,i(θ, θ′)

∂x= 0. (14)

θ

f (θ)

δ(θ)τ (θ, θk− 1 )

θk− 1θk

τ (θ, θk )

Fig. 1. Illustration of the SMACOF iteration process for 1-D localization.

And after some algebraic simplifications, we have

a(1, 1)x+ a(1, 2)y + a(1, 3)z = d(1), (15)

where for rm1(i) ≥ 0 case,

a(1, 1) =∑i,m

(2 +

rm1(i)

‖θ′ − v1‖ − 2(x′ − xm)(x′ − x1)

‖θ′ − vm‖‖θ′ − v1‖),

a(1, 2) =∑i,m

(− (x′ − xm)(y′ − y1) + (x′ − x1)(y

′ − ym)

‖θ′ − vm‖‖θ′ − v1‖),

a(1, 3) =∑i,m

(− (x′ − xm)(z′ − z1) + (x′ − x1)(z

′ − zm)

‖θ′ − vm‖‖θ′ − v1‖),

d(1) =∑i,m

(xm + x1 +

rm1(i)(x′ − xm)

‖θ′ − vm‖ +rm1(i)x1

‖θ′ − v1‖−

(x′ − x1) 〈vm,θ′ − vm〉+ (x′ − xm) 〈v1,θ′ − v1〉

‖θ′ − vm‖‖θ′ − v1‖). (16)

The derivation for the case of rm1(i) < 0 is similar.Similarly, by setting ∂τ(θ,θ′)

∂y = 0 and ∂τ(θ,θ′)∂z = 0, we

obtain two more linear equations of x, y and z,

a(2, 1)x+ a(2, 2)y + a(2, 3)z = d(2), (17)

a(3, 1)x+ a(3, 2)y + a(3, 3)z = d(3), (18)

where the coefficients a and d take a similar form as the onesdefined in Eq. (16).

Then, the algorithm is summarized as follows:

1) Initialize θ0;

2) Find the surrogate function τ(θ, θ′) with Eq. (10), where

θ′ = θk−1

;3) Estimate θ

kwith Eqs. (15), (17) and (18);

4) If ‖θk − θk−1‖ < εθ , or the maximum number of

iteration has been reached, then stop;5) Go to step 2.Fig. 1 illustrates how the SMACOF algorithm approaches

the minimum of the stress function through iteration for 1-Dlocalization. The above derivation shows that with the SMA-COF algorithm, the stress function in Eq. (2) is guaranteed todecrease at each iteration.

III. SIMULATION AND EXPERIMENTAL RESULTS

Fig. 2 shows the layout for the simulation. 4 anchors arelocated at {±1m,±1m} and 25 targets are placed on the gridx, y = {0m,±0.3m,±0.6m}. And the total number of rangemeasurements N is set to 50. The anchors and target locationsare chosen in an area small enough to ensure the convergenceof NLLS given one fixed initial position near the center of allthe anchors.

QIAO et al.: ROBUST COARSE POSITION ESTIMATION FOR TDOA LOCALIZATION 625

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

1 2

3 4

TargetAnchor

Fig. 2. Layout of the anchors and the targets for simulation.

0 5 10 15 20 25 30

10−3

10−2

10−1

Iterations

RM

SE

(m)

NLLSSMACOF

σ=0.1

σ=0.01

σ=0.03

Fig. 3. Convergence performance of the proposed algorithm and NLLSalgorithm.

Fig. 3 shows the convergence performance of the NLLSand the proposed approach. It is observed that the proposedmethod does not converge as fast as the NLLS algorithm.Therefore, the proposed scheme is in general not suitable forfine location estimation, but works very well to provide coarseestimates with only a few iterations.

Fig. 4 shows the convergence of the proposed method undervarious NLOS scenarios. The NLOS offset (a%) defined in[21] is adopted here, which is randomly chosen from theuniform distribution bm ∼ U(0, a%× dm). It is observed thatwhen the NLOS offset is large, NLLS becomes unstable andmight diverge (for example, for three of the six cases shown inFig. 4, NLLS diverges easily), whereas the proposed methodconverges for all these cases.

The proposed algorithm has been implemented in a fieldtest as follows. A pulsed ultrawideband TDOA localizationsystem is set up in a large vacant metal-structure building,covering a space of approximately 17m × 22m × 4m. Fig. 5shows the layout of the anchors.

We first simulate NLLS localization with this anchor con-figuration, while target positions are chosen on the grid witha 30 cm step in each of the three spatial dimensions. Withclean range measurements, the target positions where NLLS

0 5 10 15 20 25 30

10−2

10−1

100

Iterations

RM

SE

(m)

NLLSSMACOF

NLOS offset 1%

NLOS offset 3%

NLOS offset 5%

NLOS offset 10%

NLOS offset 30%

NLOS offset 50%

Fig. 4. Comparison of the proposed algorithm and NLLS algorithm in termsof convergence (The NLOS offset (a%) denotes the random positive NLOSerror with uniform distribution bm ∼ U(0, a%× dm)).

0500

10001500

2000

−10000

10002000

30000

100

200

300

400

x(cm)y(cm)

z(cm

)

Anchor locationsTarget locations where NLLS diverges

Fig. 5. Anchor layout for field test, and simulated target positions (withthis anchor layout) for which NLLS diverged with a fixed initial position ataround the center of the contour formed by the anchors.

fails to converge given a fixed initial position (at around thecenter of the contour formed by the anchors) are also shownin Fig. 5. As observed, the NLLS algorithm diverged at 14%of the target locations. This high divergence rate with NLLSis because of the large coverage area, making it difficult tofix one ‘good’ initial position for all target locations. If theinitialization position is randomly chosen within the cubeformed by the anchors, NLLS diverges for 30% of all targetpositions. The divergence ratio is even higher because forthe target positions near the edge or corner, there is higherprobability that the randomly chosen initial position will befurther away from the targets than the fixed initial position (ataround the center of contour formed by the anchors).

For comparison, the proposed algorithm with a maximumof 10 iterations is applied to determine a coarse position asthe initial position for NLLS. With this implementation, theNLLS algorithm now converges for all target positions.

Fig. 6 shows the anchor layout, target positions, and thelocations where NLLS fails to converge of the experimentconducted in the large vacant building. The experimentalsystem uses a commercial pulsed UWB transmitter operatingin the 3.1 − 5 GHz range as the target. Each of the fourreceive antennas is connected to an amplifier/filter board we

626 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 6, DECEMBER 2013

0500

10001500

2000

−1000

0

1000

2000

30000

100

200

300

400

x(cm)y(cm)

z(cm

)Anchor locationsTarget locationsTarget locations where NLLS diverges

Fig. 6. Experimental results obtained in a large building: anchor layout,target positions, and locations where NLLS fails to converge with a fixedinitial position set at the center of the anchor-formed contour.

TABLE ITEST RESULT (THE PROPOSED ALGORITHM)

abs(average error) error stdx 2.9 cm 2.8 cmy 5.7 cm 3.68 cmz 29.9 cm 13.58 cm

designed in our lab to form the anchors placed as shown inFig. 6. The four antenna/amplification/filtering units are thenconnected to a real-time sampling scope operating at 12.5Gsps. The sampled data is transferred to a PC through Ethernetfor range estimation and localization. Each location estimateis generated from 40μs of data, covering an interval of about400 pulses. For NLLS, a fixed location near the center of thecontour formed by the anchors is set as the initial position. Theactual positions of the targets are precisely measured by usinga commercial laser with millimeter accuracy. In contrast tothe simulation results in Fig. 5 where NLLS diverges at 14%of the target locations, here NLLS diverges at only 4 out ofa total of 70 measured locations. This is because the z valuesof all the locations where measurement took place are closeto the z-values of the initial position chosen.

Table I shows the localization results with field test data.To obtain these results, first the proposed algorithm with amaximum of 10 iterations is used to provide a coarse estimatefor each target location. This coarse estimate is then used asthe initial position for NLLS also with a maximum of 10iterations. After this additional data is supplied, the NLLS nowconverges at all 70 target locations. As expected, the accuracyof the z-coordinate is much worse than those of x- and y-coordinates, because the anchor spread in the z-direction isonly 4m, whereas in the x- and y-directions it is about 20m.

IV. CONCLUSIONS

We have developed a coarse position estimation algorithmbased on the SMACOF algorithm for TDOA localization.This algorithm works robustly for any initial position chosen.Additionally, it does not require redundant anchors as in theMOM scheme. However, it might converge slowly, and is thussuitable to provide a coarse position estimate with only a

few iterations. Both simulation results with realistic settings

and field test results are obtained. The proposed algorithmis shown to work much more robustly than NLLS whenNLOS between the target and an anchor is present. When theproposed coarse positioning algorithm is used in conjunctionwith NLLS, among all the simulated and field test cases, nolocation estimation divergences were observed.

REFERENCES

[1] C. Falsi, D. Dardari, L. Mucchi, and M. Z. Win, “Time of arrivalestimation for UWB localizers in realistic environments,” EURASIP J.Advances in Signal Process., vol. 2006, pp. 1–13, Apr. 2006.

[2] J. Shen, A. Molisch, and J. Salmi, “Accurate passive location estimationusing TOA measurements,” IEEE Trans. Wireless Commun., vol. 11,no. 6, pp. 2182–2192, Jun. 2012.

[3] Y. Shen and M. Win, “On the accuracy of localization systems usingwideband antenna arrays,” IEEE Trans. Commun., vol. 58, no. 1, pp.270–280, Jan. 2010.

[4] B. Yang and J. Scheuing, “Cramer-Rao bound and optimum sensor arrayfor source localization from time differences of arrival,” in Proc. 2005ICASSP, vol. 4, pp. 961–964.

[5] R. Ye, S. Redfield, and H. Liu, “High-precision indoor UWB localiza-tion: technical challenges and method,” in Proc. 2012 IEEE ICUWB.

[6] Z. Sahinoglu, S. Gezici, and I. Guvenc, Ultra-Wideband PositioningSystems: Theoretical Limits, Ranging Algorithms, and Protocols. Cam-bridge University Press, 2008.

[7] W. Kim, J. Lee, and G. Jee, “The interior-point method for an optimaltreatment of bias in trilateration location,” IEEE Trans. Veh. Technol.,vol. 55, no. 4, pp. 1291–1301, Jul. 2006.

[8] T. Qiao and H. Liu, “An improved method of moments estimator forTOA based localization,” to appear in IEEE Commun. Lett., 2013.

[9] Y. T. Chan, H. Yau Chin Hang, and P. chung Ching, “Exact andapproximate maximum likelihood localization algorithms,” IEEE Trans.Veh. Technol., vol. 55, no. 1, pp. 10–16, Jan. 2006.

[10] K. Cheung, H. So, W.-K. Ma, and Y. Chan, “Least squares algorithmsfor time-of-arrival-based mobile location,” IEEE Trans. Signal Process.,vol. 52, no. 4, pp. 1121–1130, Apr. 2004.

[11] X. Huibin and W. Ying, “A linear algorithm based on TDOA techniquefor UWB localization,” in Proc. 2011 Electric Information and ControlEngineering, pp. 1013–1015.

[12] S. Colonnese, S. Rinauro, and G. Scarano, “Generalized method ofmoments estimation of location parameters: application to blind phaseacquisition,” IEEE Trans. Signal Process., vol. 58, no. 9, pp. 4735–4749,Sep. 2010.

[13] M. R. Gholami, S. Gezici, E. G. Strom, “A concave-convex procedurefor TDOA based positioning,” IEEE Commun. Lett., vol. 17, no. 4, pp.765–768, Apr. 2013.

[14] S. Venkatesh and R. Buehrer, “NLOS mitigation using linear pro-gramming in ultrawideband location-aware networks,” IEEE Trans. Veh.Technol., vol. 56, no. 5, pp. 3182–3198, Sep. 2007.

[15] P. Groenen and M. v. d. Velden, “Multidimensional scaling,” ErasmusSchool of Economics (ESE), Tech. Rep., May 2004.

[16] J. A. Costa, N. Patwari, and A. O. H. III, “Distributed weighted-multidimensional scaling for node localization in sensor networks,”ACM Trans. Sensor Networking, vol. 2, pp. 39–64, 2005.

[17] J. de Leeuw and P. Mair, “Multidimensional scaling using majorization:SMACOF in R,” J. Statistical Software, vol. 31, no. 3, pp. 1–30, 8,2009.

[18] P. Oguz-Ekim, J. P. Gomes, J. Xavier, and P. Oliveira, “Robust local-ization of nodes and time-recursive tracking in sensor networks usingnoisy range measurements,” IEEE Trans. Signal Process., vol. 59, no.8, pp. 3930–3942, Aug. 2011.

[19] I. Guvenc and C.-C. Chong, “A survey on TOA based wireless localiza-tion and NLOS mitigation techniques,” IEEE Commun. Surveys Tuts.,vol. 11, pp. 107–124, 3rd Quarter 2009.

[20] S. Marano, W. Gifford, H. Wymeersch, and M. Win, “NLOS identifica-tion and mitigation for localization based on UWB experimental data,”IEEE J. Sel. Areas Commun., vol. 28, no. 7, pp. 1026–1035, Sep. 2010.

[21] E. Casas, A. Marco, J. Guerrero, and Falco, “Robust estimator fornon-line-of-sight error mitigation in indoor localization,” EURASIP J.Applied Signal Process., pp. 1–8, 2006.

[22] N. Decarli, D. Dardari, S. Gezici, and A. A. D’Amico, “LOS/NLOSdetection for UWB signals: a comparative study using experimentaldata,” in Proc. 2010 IEEE Wireless Pervasive Computing, pp. 169–173.