On Quantification of Anchor PlacementYibei Ling

Telcordia TechnologiesScott Alexander

Telcordia TechnologiesRichard Lau

Telcordia Technologies

AbstractThis paper attempts to answer a question: for a giventraversal area, how to quantify the geometric impact of anchorplacement on localization performance. We present a theoreticalframework for quantifying the anchor placement impact. Anexperimental study, as well as the field test using a UWB rangingtechnology, is presented. These experimental results validate thetheoretical analysis. As a byproduct, we propose a two-phaselocalization method (TPLM) and show that TPLM outperformsthe least-square method in localization accuracy by a huge margin.TPLM performs much faster than the gradient descent methodand slightly better than the gradient descent method in localizationaccuracy. Our field test suggests that TPLM is more robust againstnoise than the least-square and gradient descent methods.

1. INTRODUCTION

Accurate localization is essential for a wide range of appli-cations such as mobile ad hoc networking, cognitive radio androbotics. The localization problem has many variants that reflectthe diversity of operational environments. In open areas, GPShas been considered to be the localization choice. Despite itsubiquity and popularity, GPS has its Achilles heel that limitsits application scope under certain circumstances: GPS typicallydoes not work in indoor environments, and the power consump-tion of GPS receiver is a major hindrance that precludes GPSapplications in resource-constrained sensor networks.

To circumvent the limitations of GPS, acoustic/radio-strengthand Ultra-wideband (UWB) ranging technologies have beenproposed [10], [22], [11]. Measurement technologies include(a) RSS-based (received-signal-strength), (b) TOA-based (time-of-arrival), and (c) AoA-based (angle-of-arrival).

RSS-based (received-signal-strength) technologies are mostpopular mainly due to the ubiquity of WiFi and cellularnetworks. The basic idea is to translate RSS into distanceestimates. The performance of RSS-based measurements areenvironmentally dependent, due to shadowing and multipatheffects [22], [25], [3]. TOA-based technologies such as UWBand GPS measures the propagation-induced time delay betweena transmitter and a receiver. The hallmark of TOA-basedtechnologies is the receivers ability to accurately deduce thearrival time of the line-of-sight (LOS) signals. TOA-basedapproaches excel RSS-based ones in both the ranging accuracyand reliability. In particular, UWB-based ranging technologiesappear to be promising for indoor positioning as UWB signalcan penetrate most building materials [10], [22], [11]. AoA-based technologies measure angles of the target node perceivedby anchors by means of an antenna array using either RRS orTOA measurement [22], [11], [2]. Thus AoA-based approachesare viewed as a variant of RRS or TOA technologies.

Rapid advances in IC fabrication and RF technologies makepossible the deployment of large scale power-efficient sensornetworks. The network localization problem arises from suchneeds [26], [6]. The goal is to locate all nodes in the network inwhich only a small number of anchors know their precise posi-tions initially. A sensor node (with initially unknown position)measures its distances to three anchors, and then determinesits position. Once the position of a node is determined, thenthe node becomes a new anchor. The network localization hastwo major challenges: 1) cascading error accumulation and2) insufficient number of initial anchors and initially skewedanchor distribution. The first challenge is addressed by utiliz-ing optimization techniques to smooth out error distribution.The second challenge is addressed by using multihop rangingestimation techniques [28], [27].

Centralized optimization techniques for solving network lo-calization include semidefinite programming by So and Ye[24] and second order cone programming (SOCP) relaxationby Tseng [26]. Local optimization techniques are realistic inpractical settings. However, they could induce flip ambiguitythat deviates significantly from the ground truth [6]. To dealwith flip ambiguity, Eren et. al. [9] ingeniously applied graphrigidity theory to establish the unique localizability condition.They showed that a network can be uniquely localizable iffits grounded graph is globally rigid. The robust quadrilateralsalgorithm by Moore et al. [21] achieves the network localiz-ability by gluing locally obtained quadrilaterals, thus effectivelyreducing the likelihood of flip ambiguities. Kannan et al. [12],[13] formulate flip ambiguity problem to facilitate robust sensornetwork localization.

Lederer et al. [15], [16] and Priyantha et al. [23] revolvearound anchor-free localization problem in which none of thenodes know their positions. The goal is to construct a globalnetwork layout by using network connectivity. They develop analgorithm for constructing a globally rigid Delaunay complexfor localization of a large sensor network with complex shape.Bruck, Gao and Jiang [6] establish the condition of networklocalization using the local angle information. They show thatembedding a unit disk graph is NP-hard even when the anglesbetween adjacent edges are given.

Patwari et al. [22] used the Cramer-Rao bound (CRB) toestablish performance bounds for localizing stationary nodes insensor networks under different path-loss exponents. Dulmanet al. [8] propose an iteration algorithm for the placementof three anchors for a given set of stationary nodes: in eachiteration, the new position of one chosen anchor is computedusing the noise-resilience metric. Bishop et al. [2], [4], [5], [3]

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study the geometric impact of anchor placement with respectto one stationary node. Bulusu et al. proposed adaptive anchorplacement methods [7]. Based on actual localization error atdifferent places in the region, their algorithms can empiricallydetermine good places to deploy additional anchors.

Our work primarily focuses on the geometric impact quan-tification of an anchor placement over a traversal area. As abyproduct, it also forms the basis for optimal anchor selectionfor mitigating the impact of measurement noise. To our bestknowledge, only a few papers [8], [7], [2], [22], [4], [5], [3]mentioned about the geometric effect of anchor placement butunder a different context from this paper. The idea of quantify-ing the geometric impact of anchor placement on localizationaccuracy over a traversal area has not been considered before.

The rest of this paper is structured as follows. Section 2shows examples to illustrate the anchor placement impacton localization accuracy. Section 3 proposes a method forquantifying the anchor placement impact. Section 4 presents atwo-phase localization method Section 5 conducts a simulationstudy to validate the theoretical results. Section 6 presentsthe field study using the UWB ranging technology. Section 7concludes this paper.

2. EFFECT OF ANCHOR PLACEMENT

Throughout this paper we use the term anchor to denotea node with known position. The positions of anchors arestationary and available to each mobile node (MN hereafter)in a traversal area [22]. The goal is to establish the position ofa MN through ranging measurements to available anchors. Theproblem is formulated as follows: Let pi = (xi, yi), (1 i m) denote the known position of the ith anchor, and p the actualposition of the MN of interest. The distance between p and pi isthus expressed as di = d(p, pi) =

(x xi)2 + (y yi)2. In

practical terms, the obtained distance measurement is affectedby measurement noises. Thus the obtained distance is asdi = di+i where di the true distance, and i is widely assumedto be a Gaussian noise with zero mean and variance 2i [30],[29], [4], [14], [8], [22], [11]. Define a function

f(x, y) =

mi=1

((x xi)2 + (y yi)2 di

2)2,

where m is the number of accessible anchors, (xi, yi) is theposition of the ith anchor, and di noisy ranging between the ith

anchor and the MN. The localization problem [22], [8], [14] isformulated as

min(x,y)

for a given AP and noise level, the error statistics in Table 1are obtained by traversing the HT 10 times. Each HT traversalinvolves the establishment of 8190 positions using both LSMand GDM (GDM uses the estimated position derived from LSMas its iteration initial value), with a step size of 0.00001. Thetermination condition of GDM is set as ||p(i+1)p(i)||2

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1.4

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Fig. 2. LVT of anchor placement (0, 100), (0, 0), (100, 0)

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Fig. 3. LVT of anchor placement (0, 100), (7, 50), (3, 40)

relation between gm(p1, , pm)() and gm(p1, , pm)(p)(gm(p1, , pm)(x, y)) becomes

gm(p1, , pm)() =1

||

gm(p1, , pm)(x, y)dx dy

(10)

Similarly, the notion of gm(p1, , pm)(p) can be extendedfrom a point p to a trajectory as

gm(p1, , pm)() =1

||

gm(p1, , pm)(x, y)dl (11)

The discrete form of (11) becomes

gm(p1, , pm)() =

1kn

g(p1, , pm)(x(tk), y(tk))

n.

(12)

(10) provides a means for quantifying the impact of an APover an area. In practice, due to the arbitrariness of the areaboundary and of an anchor placement, it would be impossibleto derive a closed-form expression. In this paper, we useTrapezium rule method to compute (10). It is done by firstsplitting the area into 10, 000 non-overlapping sub-areas, thenapplying the Trapezium rule on each of these sub-areas.

TABLE 2ANCHOR PLACEMENT IMPACT QUANTIFICATION

Anchor Placementp1 = (0, 100), p2 = (0, 0), p3 = (100, 0), p4 = (7, 50), p5 = (3, 40)

Anchor Placement Impact over g3(p1, p2, p3)() g3(p1, p4, p5)()

1.501 2.615Anchor Placement Impact over

g3((p1, p2, p3)() g3(p1, p4, p5)()1.499 2.622

: 100 100 area,: HT

Table 2 provides a numerical calculation of AP impact over and over . The AP impact calculated in Table 2 implies

that the localization accuracy over the AP1 is better than thatunder AP2 as g3(p1, p2, p3)() < g3(p1, p4, p5)(), whichagree with the results obtained in noisy environments in Table 1.Comparing g3(p1, p2, p3)() and g3(p1, p2, p3)() in Table 2shows that the difference between them is very small. However,computation of g3(p1, p2, p3)() takes about 23 seconds, asopposed to 0.2 seconds in computing g3(p1, p2, p3)().

4. TWO-PHASE LOCALIZATION ALGORITHM

It is clear from Fig(2) that gm(p1, , pm)(p) is determinedby an OSAP and the OSAP varies by area. Fig(4) shows theOSAP areas under AP (0, 100), (0, 0), (100, 0). Observe thenoise-free scenario in the left-hand side graph in Fig(4) whereeach colored area represents an anchor pair chosen over its twoalternatives based on (9). The white colored areas represent theregions in which the anchor pair at (0, 100), (0, 0) is chosen.The blue colored areas represent the regions where the anchorpair at (0, 0), (100, 0) has its geometric advantage over itsalternative anchor pairs. The red colored areas represent theregions where the anchor pair at (100, 0), (0, 100) is expectedto produce the most accurate localization. The right-hand sidegraph in Fig(4) plots the OSAP areas under the noise levelof = 1.0. The presence of noise makes the borderlines ondifferent colored areas rough and unsmooth. This is becausethat borderlines correspond to the isolines by two anchor pairsand hence are highly sensitive to noise.

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Fig. 4. a) noise free b) noise level (1.0): white area by anchor pairat (0, 100),(0, 0), black area by anchor pair at (0, 0), (100, 0), red areaby anchor pair at (0, 100),(100, 0)

This observation of Fig(4) leads to a new two-phase localiza-tion algorithm (TPLM) that differs significantly from LSM andGDM. TMPL differs from LSM and GDM. It proceeds in twophases: In the first phase called optimal anchor pair selection,an optimal anchor pair at (pi, pj) is identified based on (9),then the position(s) can be estimated by solving

||p pi|| = di, ||p pj || = dj (13)

Through a lengthy manuplation, two possible solutions to (13)are obtained as follows(

xy

)=

(cos() sin()sin() cos()

)(uv

)+

(xiyi

)(xy

)=

(cos() sin()sin() cos()

)(uv

)+

(xiyi

)(14)

where = arctan( yjyixjxi ), dij = ||pi pj ||, and

u =(d2ij + di

2 dj

2)

2dij, v =

d2i d2ij + di2 dj2

2dij

2(15)

In the second phase called disambiguation, the reference pointobtained from LSM is used to single out one from two possiblepositions (14). It is done by choosing one position that has ashorter distance to the reference point.

Algorithm 1 Two-phase Localization MethodInput: pi = (xi, yi), 1 i m: ith anchors positionInput: di: distance measurement from jth anchor to MN pInput: p = (x, y): a reference point obtained via LSM

1: gmin , p+ (0, 0), p++ (0, 0)2: for i = 1 to m 1 do3: for j = i+ 1 to m do4: if g2(pi, pj)(p) < gmin then5: gmin g(pi, pj)(p), (p+, p++) (pi, pj)6: end if7: end for8: end for9: d = |p+ p++|, d+ = |p p+|, d++ = |p p++|

10: = arctan y++y+x++x+ , u =d2+d

2+++d

2

2d , v =d2+ u2

11: p = (cos() sin()sin() cos()

)(uv

)+(x+

y+

)12: p = (

cos() sin()sin() cos()

)(uv)

+(x+

y+

)13: if |p p| < |p p| then14: return p15: end if16: return p

The code between lines 2-8 examines each anchor pair inturn and chooses an optimal anchor pair, which correspondsto (9). The complexity is

(m2

). Lines 2-12 constitute the basic

block of the optimal anchor selection phase. The code betweenlines 13-16 is used to disambiguate the two possible solutionsusing the reference position p obtained by LSM.

TABLE 3TWO-PHASE LOCALIZATION METHOD

positions of anchors ave std time

(0, 100), (0, 0), (100, 0)0.3 0.40 0.22 3.171.0 1.32 0.73 3.17

(0, 100), (7, 50), (3, 40)0.3 0.70 0.88 3.441.0 2.76 5.67 3.15

Gaussian noise N (0, 2)

We present the results of TPLM in Table 3 using the sameanchor placement setups in Table 1. In all cases, TPLM gives asignificant error reduction over LSM with a minor degradationas TPLM uses the reference point obtained from LSM. It isfairly clear that TPLM is an order of magnitude faster thanGDM, and performs slightly better than GDM in terms oflocalization accuracy.

5. SIMULATION STUDY

The aim of this section is threefold: first, to conduct theperformance comparison between TPLM, LSM, and GDMin noise environments; second, to study the impact of noisemodels; and third, to investigate the random anchor placementimpact.

A. Visualizing Noise-induced Distortion

To visualize the difference among LSM, GDM, and TPLM,we plot the restored HT by LSM, GDM, and TPLM underGaussian noise model in Figs(5)-(7) using AP4 in Table 4.Notice that anchor positions are plotted as solid black circles.

TABLE 4ANCHOR PLACEMENT SETUP

Anchor Positionp1 = (0, 100), p2 = (7, 50), p3 = (0, 0)p4 = (3, 40), p5 = (100, 0), p6 = (1, 98)

Anchor Placement SetupAP1 (0, 100), (0, 0), (100, 0)AP2 (0, 100), (7, 50), (3, 40)AP3 (0, 100), (0, 0), (100, 0), (1, 98)AP4 (0, 100), (7, 50), (3, 40), (1, 98)

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Fig. 6. restored HT by GDM: a) noise (0.02); b) noise (0.2)

Visual inspection reveals the apparent perceived differenceamong the restored HTs by LSM, GDM and TPLM: whenthe noise level is 0.2, the restored HT by LSM becomescompletely unrecognizable in Fig(5)(b). While in Fig(6)(b) theupper right portion of the restored HT by GDM to some degreepreserves the hallmark of the HT, but the most part of therecovered HT is severely distorted and barely recognizable.The restored curve by TPLM contrasts sharply with those

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Fig. 7. restored HT by TPLM: a) noise (0.02); b) noise (0.2)

by LSM and GDM in its preservation of fine details of theHT for the most part, while having minor distortion in thelower- and upper-left corner areas in Fig(7)(b). Such spatiallyuneven localization performance can be explained by examiningFig(3), which shows that the lower- and upper-left areas exactlycorrespond to the LVT peak areas. The perceived differencesbetween GDM and TPLM in Figs(5)-(7) can be quantified asfollows: GDM has the average error of 0.69 per HT traversal,while TPLM produces the average error of 0.465. In addition,GDM takes about 218.48 seconds per HT traversal, in contrastto 2.94 seconds taken by TPLM.

B. Gaussian Noise vs. Non-Gaussian Noise

In this subsection, we will compare the performance of GDMand TPLM under Gaussian and uniform noise models, there aresome scenarios where ranging noise may follow uniform model[7], [20]. In the experimental study, the performance betweenGDM and TPLM is compared under a same noise level with thedifferent noise models. For Gaussian noise model N (0, 2), refers to the noise level in Fig(8), and for a uniform noise modelU(a, a), the noise level in Fig(9) is expressed as a2/3. Thedata presented reflects the average error of GDM and TPLMover 10 HT traversals under the anchor placements in Table 2.

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ter)

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gaussian noise: noise level

AP1AP2AP3AP4

Fig. 8. Accuracy: (top) GDM; (bottom) TPLM (Gaussian noise)

The graphs in Figs(8)-(9) represent the localization errorcurves of GDM and of TPLM, respectively. Under both the

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ete

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Fig. 9. Accuracy: (top) GDM; (bottom) TPLM (uniform noise)

noise models, TPLM outperforms GDM by huge margins underAP2 and AP4. Both GDM and TPLM perform indistinguish-ably under AP1 and AP3. While the impact difference betweenAP1 and AP3 is barely noticed as their performance curvesare overlapped, the performance curves between AP1 and AP2and between AP3 and AP4 are clearly separated, thus theimpact difference AP1 and AP2 and between AP3 and AP4are evident.

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sed

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sed

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AP1AP2AP3AP4

Fig. 10. Elapsed time: (top) GDM; (bottom) TPLM (normal noise)

The results show that the Gaussian noise has more impact onGDM than the uniform one. By comparison, the performance ofTPLM is insensitive to noise models. Fig(10) plots the elapsedtime by GDM and TPLM per HT traversal, showing that TPLMis orders of magnitude faster than GDM.

C. Random Anchor Placement

We study the impact of random anchor placement on theperformance of LSM, GDM and TLPM. To achieve this, thenumber of anchors are randomly placed in 100 100 and50 100 regions. For each randomly generated anchor place-ment (RGAP) in a region, the gm(p1, , pm)() function iscalculated, and the error statistics of LSM, GDM, and TPLMunder Gaussian noise level of 0.3 are gathered and compared.

Fig(11)(a)-(b) show the actual gm(p1, , pm)() (averageLVT elevation over )) distribution by 100 RGAPs. In thetop graph, 3 anchors are randomly placed (r.u.p.) in the entiretraversal area while in the bottom graph 3 anchors are r.u.p. inthe upper half of the traversal area. Fig(11)(a) clearly exhibitpositive skewness. This implies that a RGAP over the entiretraversal area in general yields good localization accuracy.Recall that the smaller gm(p1, , pm)() is, the better local-ization is. The gm(p1, , pm)() distribution in Fig(11)(b)is more dispersed than that in the Fig(11)(a), meaning that aRGAP over the half traversal area in general underperforms aRGAP over the entire traversal area.

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Fig. 11. (a) g3(...)() distribution by 100 RGAPs with 3 anchors in100 100; (b) g3(...)() distribution by 100 RGAPs with 3 anchorsin 50 100

TABLE 5RANDOM ANCHOR PLACEMENT

Random Anchor Placement in The Traversal Area

Anchors gm(p1, ..pm)()TPLM LSM GDM

ave ave ave3 2.05 0.57 1.52 0.874 1.69 0.44 0.79 0.495 1.53 0.40 0.59 0.386 1.48 0.39 0.51 0.33

Random Anchor Placement in the Half Traversal Area

Anchors gm(p1, ..pm)()TPLM LSM GDM

ave ave ave3 2.72 0.97 2.82 1.544 1.93 0.52 1.37 0.695 1.77 0.46 0.81 0.466 1.69 0.44 0.72 0.40

Fig(12) presents the performance curves of LSM, GDM, andTPLM under RGAPs, where the x axis refers to the value ofgm(p1, ..pm)() and the y axis the error deviation and averageerror. As is clear from Fig(12), TPLM outperforms both LSMand GDM in localization accuracy and reliability: both the errordeviation and average error curves of TPLM are consistentlylower than those of LSM and GDM.

Table 5 tabulates the results of RGAPs with different num-ber of anchors, where gm(p1, .., pm)() denotes the averagegm(p1, ..pm)() over 100 RGAPs with a fixed number ofanchors. It shows that the localization accuracy of TPLM overGDM is diminished as the number of anchors increases. Itbecomes evident that under the noise level of 0.3, the actual

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(b)

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Fig. 12. (a) error deviation; (b) average error (100 RGAPs with 3anchors over the 100 100 region, = 0.3)

performance of LSM, GDM, and TPLM deteriorates as thevalue of gm(p1, .., pm)() increases. This solidifies the factthat gm(p1, ..pm)() is an effective discriminator for the anchorplacement impact on localization performance.

6. FIELD EXPERIMENTAL STUDY

This section focuses on the field test of a DARPA-sponsoredresearch project, using the UWB-based ranging technologyfrom Multispectrum Solutions (MSSI) [10]. and Trimble dif-ferential GPS (DGPS). Our field testbed area was a 100 100square meters. It consisted of an outdoor space largely oc-cupied by surface parking lots and an indoor space inside awarehouse. This testbed area was further divided into 10, 000non-overlapping 1 1 grids. Each grid represents the finestpositioning resolution to evaluate RF signal variation.

TABLE 6FIELD TESTBED AREA AND ANCHOR PLACEMENT

x0 y0 xf yf

74.476069 40.537808 84719 111045 0.381583Anchor placement

GPS position transformed position74.475585 40.538468 p4 = (65.345179, 52.75145)74.475287 40.53856 p5 = (92.580022, 52.83239)74.475186 40.538294 p6 = (89.52274, 22.232383)

In the field test, DGPS devices were mainly used for outdoorpositioning while MSSI UWB-based devices were used forindoor positioning. The experimental system was composed offour MNs equipped with both the Trimble DGPS and MSSIUWB-based ranging devices. Three UWB devices were usedas anchors being placed at known fixed positions. Using theknown positions of the UWB anchors and real-time distancemeasurements, each MN could establish the current position aswell as that of other MNs locally, at a rate of approximately 2samples per second. Each MN could also establish the positionvia the DGPS unit at a rate of 1 sample per second whenresiding in the outdoor area. The field test area was dividedinto four slightly overlapping traversal quadrants denoted as(0,1,2,3). A laptop (MN) equipped with DGPS and

UWB ranging devices was placed in a modified stroller asshown in Fig(13), and one tester pushed the stroller, travelingalong a predefined trajectory inside a quadrant (see Fig(14)).Each run lasted about 30 minutes of walk.

GPS Antenna

UWB radio

Laptop running

system s/w

Speedo

Basket w/GPS

unit & batteries

Rain Cover

Fig. 13. Stroller with a laptop attached with DGPS and UWB rangingdevices

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warehouse

3

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1

Fig. 14. Four trajectories (0,1,2,3), black circles: the positionsof anchors, yellow rectangle: warehouse

With repeated trial runs, we found that the UWB signal couldbarely penetrate one cement wall of the warehouse. To establishpositions inside the warehouse, we placed one anchor close tothe main entrance of the warehouse while placing two otheranchors around the center of the field testbed area. Fig(14)showed the four trajectories (0,1,2,3) in the field test,which are translated from the GPS trajectories. A part of 0trajectory was inside the warehouse, thus the part of GPS of0 was not available. Table 7 provides the calculated resultsof g3(p4, p5, p6)() using the GPS trajectory data, indicatingthat 0 would yield the most accurate localization while 1produced the least accurate localization.

TABLE 7g3(p4, p5, p6)() OF TRAJECTORIES IN FIELD TESTBED

g3(p4, p5, p6)(0) g3(p4, p5, p6)(1)1.667025 1.967605

g3(p4, p5, p6)(2) g3(p4, p5, p6)(3)1.787370 1.835048

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(a)

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warehouse

warehouse

Trajectory

Fig. 16. Restored Trajectories by (a) TPLM, (b) GDM, (c) LSM (blackcircles: positions of UWB anchors)

In the field testbed, the trajectory of each MN was controlledby an individual tester during a 30-minute walk. For purposeof the primary experiment being conducted, the strollers werefitted with bicycle speedometers to allow the tester to controlhis speed, in order to produce sufficient reproducibility for theprimary experiment, but not for our testing of localization. Dueto inherent variability in each individual movement, an objec-tive assessment of localization accuracy without a ground truthreference is almost impossible. For a performance comparison,we used GPS trajectories as a reference for visual inspectionof the restored trajectories by LSM, GDM, and TPLM.

Three curves in Fig(15)(a)-(d) represent the field distancemeasurements between a tester and the three UWB anchor de-vices during a 30-minute walk in trajectories (0,1,2,3). itis fairly obvious that measurement noise in different trajectoriesvary widely: The distance measurement curves in trajectories(0,3) are discontinuous and jumpy in Figs(15)(a)&(d), incontrast to the relatively smooth distance curves in trajectories(1,2) in Fig(15)(b)&(c). Such a discontinuity in measure-ments occurred when testers were traveling in an area whereUWB devices on stroller had no direct line of sight to UWBanchors, thus introducing additional noise in 0,3.

Figs(16)(a)-(c) show the restored trajectories by TPLM,GDM, and LSM using the UWB ranging technology in thefield test. A visual inspection suggests that LSM was extremelyprone to noise as its restored trajectories were appreciablydistorted beyond recognition in some parts. As indicated inFig(16)(b), GDM gave an obvious error reduction over LSMbut at the expense of computational cost. The most visuallyperceived difference between LSM and GDM can be seen inthe circled areas in Figs(16)(b)-(c). By contrast, the differencebetween GDM and TPLM can be visualized in the circled areasFigs(16)(a)-(b) where TPLM produced a detail-preserved butslightly distorted contour of the trajectory. A further offlineanalysis showed that on average GDM takes 18.87ms perposition establishment, while TPLM/LSM take 0.4/0.35ms.

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Fig. 15. Distance measurements to three UWB anchors: (a) 0, (b) 1, (c) 2, (d) 3

7. CONCLUSION

This paper studies the geometric effect of anchor placementon localization performance. The proposed approach allows theconstruction of the least vulnerability tomography (LVT) forcomparing the geometric impact of different anchor placements.As a byproduct, we propose a two-phase localization method.

To validate theoretical results, we conduct comprehensivesimulation experiments based on randomly generated anchorplacements and different noise models. The experimental resultsagree with our anchor placement impact analysis. In addition,we show that TPLM outperforms LSM by a huge margin inaccuracy. TPLM performs much faster than GDM and slightlybetter than GDM in localization accuracy. The field study showsthat TPLM is more robust against noise than LSM and GDM.

In this paper, we adopt a widely used assumption thatmeasurement noise is independent of distance. This assumptionis somewhat unrealistic in some practical settings. In our futureresearch, we will perform experiments using UWB rangingtechnology to study the noise-distance relation in indoor andoutdoor environments. The anchor placement impact analysiswill be refined to incorporate more realistic noise-distancemodels.

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1 Introduction2 Effect of Anchor Placement3 Anchor Placement Effect Quantification4 Two-Phase Localization Algorithm5 Simulation Study5-A Visualizing Noise-induced Distortion5-B Gaussian Noise vs. Non-Gaussian Noise5-C Random Anchor Placement

6 Field Experimental Study7 ConclusionReferences