On Quantification of Anchor Placement

3.1 MB·English

by  Yibei Ling

On Quantification of Anchor Placement
Yibei Ling Scott Alexander Richard Lau
Telcordia Technologies Telcordia Technologies Telcordia Technologies
Abstract—Thispaperattemptstoansweraquestion:foragiven Rapid advances in IC fabrication and RF technologies make
traversal area, how to quantify the geometric impact of anchor possible the deployment of large scale power-efficient sensor
placement on localization performance. We present a theoretical
networks. The network localization problem arises from such
framework for quantifying the anchor placement impact. An
needs[26],[6].Thegoalistolocateallnodesinthenetworkin
experimentalstudy,aswellasthefieldtestusingaUWBranging
technology, is presented. These experimental results validate the whichonlyasmallnumberofanchorsknowtheirpreciseposi-
theoretical analysis. As a byproduct, we propose a two-phase tions initially. A sensor node (with initially unknown position)
2
1 localization method (TPLM) and show that TPLM outperforms measures its distances to three anchors, and then determines
theleast-squaremethodinlocalizationaccuracybyahugemargin.
0 its position. Once the position of a node is determined, then
TPLM performs much faster than the gradient descent method
2 the node becomes a new anchor. The network localization has
andslightlybetterthanthegradientdescentmethodinlocalization

n accuracy.OurfieldtestsuggeststhatTPLMismorerobustagainst two major challenges: 1) cascading error accumulation and
a noise than the least-square and gradient descent methods. 2) insufficient number of initial anchors and initially skewed
J
anchor distribution. The first challenge is addressed by utiliz-

2 1. INTRODUCTION ing optimization techniques to smooth out error distribution.
1
The second challenge is addressed by using multihop ranging

Accurate localization is essential for a wide range of appli- estimation techniques [28], [27].
]
I cations such as mobile ad hoc networking, cognitive radio and Centralized optimization techniques for solving network lo-
N
robotics.Thelocalizationproblemhasmanyvariantsthatreflect calization include semidefinite programming by So and Ye
.
s the diversity of operational environments. In open areas, GPS [24] and second order cone programming (SOCP) relaxation
c has been considered to be the localization choice. Despite its by Tseng [26]. Local optimization techniques are realistic in
[
ubiquity and popularity, GPS has its Achilles’ heel that limits practical settings. However, they could induce flip ambiguity

1 itsapplicationscopeundercertaincircumstances:GPStypically that deviates significantly from the ground truth [6]. To deal
v doesnotworkinindoorenvironments,andthepowerconsump- with flip ambiguity, Eren et. al. [9] ingeniously applied graph
4
tion of GPS receiver is a major hindrance that precludes GPS rigidity theory to establish the unique localizability condition.
7
4 applications in resource-constrained sensor networks. They showed that a network can be uniquely localizable iff
2 TocircumventthelimitationsofGPS,acoustic/radio-strength its grounded graph is globally rigid. The robust quadrilaterals
1. and Ultra-wideband (UWB) ranging technologies have been algorithm by Moore et al. [21] achieves the network localiz-
0 proposed [10], [22], [11]. Measurement technologies include abilitybygluinglocallyobtainedquadrilaterals,thuseffectively
2 (a)RSS-based(received-signal-strength),(b)TOA-based(time- reducing the likelihood of flip ambiguities. Kannan et al. [12],
1 of-arrival), and (c) AoA-based (angle-of-arrival). [13]formulateflipambiguityproblemtofacilitaterobustsensor
:
v RSS-based (received-signal-strength) technologies are most network localization.
i popular mainly due to the ubiquity of WiFi and cellular Lederer et al. [15], [16] and Priyantha et al. [23] revolve
X
networks. The basic idea is to translate RSS into distance around anchor-free localization problem in which none of the
r
a estimates. The performance of RSS-based measurements are nodes know their positions. The goal is to construct a global
environmentally dependent, due to shadowing and multipath networklayoutbyusingnetworkconnectivity.Theydevelopan
effects [22], [25], [3]. TOA-based technologies such as UWB algorithm for constructing a globally rigid Delaunay complex
andGPSmeasuresthepropagation-inducedtimedelaybetween for localization of a large sensor network with complex shape.
a transmitter and a receiver. The hallmark of TOA-based Bruck, Gao and Jiang [6] establish the condition of network
technologies is the receiver’s ability to accurately deduce the localization using the local angle information. They show that
arrival time of the line-of-sight (LOS) signals. TOA-based embedding a unit disk graph is NP-hard even when the angles
approachesexcel RSS-basedones inboth theranging accuracy between adjacent edges are given.
and reliability. In particular, UWB-based ranging technologies Patwari et al. [22] used the Cramer-Rao bound (CRB) to
appear to be promising for indoor positioning as UWB signal establishperformanceboundsforlocalizingstationarynodesin
can penetrate most building materials [10], [22], [11]. AoA- sensor networks under different path-loss exponents. Dulman
basedtechnologiesmeasureanglesofthetargetnodeperceived et al. [8] propose an iteration algorithm for the placement
by anchors by means of an antenna array using either RRS or of three anchors for a given set of stationary nodes: in each
TOAmeasurement[22],[11],[2].ThusAoA-basedapproaches iteration, the new position of one chosen anchor is computed
are viewed as a variant of RRS or TOA technologies. using the noise-resilience metric. Bishop et al. [2], [4], [5], [3]
study the geometric impact of anchor placement with respect the direction of the negative gradient −∇f(x(0),y(0)).
tpolaocneemsetnattimoneatrhyodnsod[e7.].BBulausseudeotnala.cptruoaplolsoecdalaizdaatpiotinveearrnocrhoart (cid:18)x(i+1)(cid:19)=(cid:18)x(i)(cid:19)−η(cid:32)∂f∂(xx,y)(cid:33), (2)
different places in the region, their algorithms can empirically y(i+1) y(i) ∂f(x,y)
∂y
determine good places to deploy additional anchors.
where η refers to the iteration step size, i denotes the ith
Our work primarily focuses on the geometric impact quan-
iterations. One starts with an initial value (x(0),y(0)), then
tification of an anchor placement over a traversal area. As a
applies (2) iteratively to reach a local minimum (xˆ,yˆ), i.e.,
byproduct, it also forms the basis for optimal anchor selection
f(xˆ,yˆ)≤f(x(i),y(i)), 1≤i≤n.
for mitigating the impact of measurement noise. To our best
TherearethreeissuesassociatedwithGDM:1)initialvalue;
knowledge, only a few papers [8], [7], [2], [22], [4], [5], [3]
2) iteration step size and 3) convergence rate. The convergence
mentioned about the geometric effect of anchor placement but
ratecouldbesensitivetotheinitialvalueaswellastheiteration
underadifferentcontextfromthispaper.Theideaofquantify-
step size. While in general there is no theoretical guidance
ing the geometric impact of anchor placement on localization
for selecting the initial value in practice, the value obtained
accuracy over a traversal area has not been considered before.
by a linearized method is often chosen heuristically as the
The rest of this paper is structured as follows. Section 2
initial value. This nonlinear optimization problem in (1) can
shows examples to illustrate the anchor placement impact
be linearized by the least-square method [14], [19] as follows:
on localization accuracy. Section 3 proposes a method for
quantifying the anchor placement impact. Section 4 presents a (ATA)(cid:18)x(0)(cid:19)=ATM, where (3)
two-phaselocalization methodSection 5conducts asimulation y(0)
study to validate the theoretical results. Section 6 presents
the field study using the UWB ranging technology. Section 7 (x −x ) (y −y ) d(cid:98)2−d(cid:98)2+r2−r2 
2 1 2 1 1 2 2 1
concludes this paper. A=2(x3−x1) (y3−y1),M =d(cid:98)21−d(cid:98)23+r32−r12 ,
 ··· ···   ··· 
2. EFFECTOFANCHORPLACEMENT (xm−x1) (ym−y1) d(cid:98)21−d(cid:98)2m+rm2 −r12
(4)
Throughout this paper we use the term anchor to denote
(cid:112)
a node with known position. The positions of anchors are AT isthetransposeofA,andri = x2i +yi2,1≤i≤m.The
stationary and available to each mobile node (MN hereafter) least-square method (LSM hereafter) uses noisy measurements
in a traversal area [22]. The goal is to establish the position of to estimate the position of MN (x(0),y(0))T.
aMNthroughrangingmeasurementstoavailableanchors.The
problem is formulated as follows: Let p =(x ,y ), (1 ≤ i ≤ 100
i i i
m)denotetheknownpositionoftheithanchor,andptheactual
positionoftheMNofinterest.Thedistancebetweenpandp is 80
i
(cid:112)
thus expressed as d = d(p,p )= (x−x )2+(y−y )2. In
i i i i
60
practical terms, the obtained distance measurement is affected
by measurement noises. Thus the obtained distance is as
40
d(cid:98)i =di+(cid:15)iwheredithetruedistance,and(cid:15)iiswidelyassumed
to be a Gaussian noise with zero mean and variance σ2 [30],
i 20
[29], [4], [14], [8], [22], [11]. Define a function
m 0
f(x,y)=(cid:88)(cid:16)(x−xi)2+(y−yi)2−d(cid:98)i2(cid:17)2, 0 20 40 60 80 100
Fig.1. HilbertTrajectory
i=1
To shed light on the anchor placement effect on localization
where m is the number of accessible anchors, (x ,y ) is the
i i accuracy, consider a Hilbert traversal trajectory with different
positionoftheith anchor,andd(cid:98)i noisyrangingbetweentheith anchorplacement(AP)setupsinTable1.TheHilberttrajectory
anchor and the MN. The localization problem [22], [8], [14] is
is formed by the piecewise connection of 8190 points in an
formulated as
100×100 region (see Fig(1)). A Hilbert trajectory (HT here-
afer) has a space-filling property. As a result, the localization
min f(x,y),(cid:60) is the real number set. (1)
performance obtained on a HT is a good approximation to that
(x,y)∈(cid:60)2
of the underlying area.
(1)presentsanonlinearoptimizationproblemthatcanbesolved We conduct an experiment as follows: start from the upper-
byanumberofalgorithms.Thispaperusesthegradientdescent left corner, the MN moves along the HT. At each point p (the
method (GDM hereafter) for its simplicity [1]. GDM is based position of MN), noisy distance measurements d(cid:98)i = di +(cid:15)i
on an intuitive idea that if f(x(0),y(0)) is differentiable in the from p to the anchors are generated. Both LSM and GDM are
vicinty of (x(0),y(0)), then f((x(0),y(0)) decreases fastest in then used to derive the estimated position. To avoid artifacts,
for a given AP and noise level, the error statistics in Table 1 where (sin(α),cos(α)) and (sin(β),cos(β)) denote the direc-
are obtained by traversing the HT 10 times. Each HT traversal tion cosines from p to anchors at p and p . The anchor-pair
i j
involves the establishment of 8190 positions using both LSM GDOP function g (p ,p )(p) is
2 i j
andGDM(GDMusestheestimatedpositionderivedfromLSM
(cid:113)
as its iteration initial value), with a step size of 0.00001. The g (p ,p )(p)= tr((hTh)−1), (7)
2 i j
terminationconditionofGDMissetas||p(i+1)−p(i)|| <0.001
2
where T/tr denotes the transpose/trace of a matrix, and A−1
and the maximum number of iterations is set as 100.
refers to the inversion of A. A simple manipulation obtains
TABLE1 det(hTh)=sin2(β−α) and
ANCHORPLACEMENT&LOCALIZATIONACCURACY
(cid:115)
Least-SquareMethod (cid:113) 2
positionsofanchors σ ave std time g2(pi,pj)(p)= tr((hTh)−1)= sin2(β−α). (8)
0.3 0.42 0.26 2.52
(0,100),(0,0),(100,0)
1.0 1.40 0.85 2.54
(0,100),(7,50),(3,40) 0.3 4.01 3.65 2.51 Substituting sin2(β −α)= 1−(d2i +d2j −||pi−pj||2)2 into
1.0 13.45 12.15 2.57
2d d
GradientDescentMethod (8) yields (5). i j (cid:3)
positionsofanchors σ ave std time
Theorem 1 asserts that β −α degree separation represents
0.3 0.37 0.21 63
(0,100),(0,0),(100,0)
1.0 1.22 0.69 81 the effect of anchor pair at p ,p on the localization accu-
i j
(0,100),(7,50),(3,40) 01..30 02..7964 07..9421 227953 racy at p. It im√plies that β − α = π/2 degree separation
(g (p ,p )(p)= 2)arebestinlocalizationaccuracy,whereas
GaussiannoiseN(0,σ2) 2 i j
β−α=0degreeseparation(colinear)(g (p ,p )(p)=∞)are
2 i j
worst. The smaller the g (p ,p )(p), the better the accuracy of
The ave and std fields in Table 1 denote average local- 2 i j
localization. This is a well-known result also obtained in [22],
ization error and standard deviation, and the time field the
[4], [5], [2], [8].
elapsed time per HT traversal (seconds). Table 1 shows that
We now extend the anchor-pair GDOP function into the
AP (0,100),(0,0),(100,0) yields a much better localization
multi-anchor GDOP function. Let p ,··· ,p be a set of
accuracy than AP (0,100),(7,50),(3,40) for both LSM and 1 m
positions of anchors accessible to MN at p. The multi-anchor
GDM.Thisindicates,afortiori,thatananchorplacement(AP)
GDOPfunctiong (p ,··· ,p )(p)islinkedtotheanchor-pair
has a significant bearing on localization accuracy over an area. m 1 m
GDOP function as follows:
This observation raises a question: can we quantify the impact
ofananchorplacement(AP)withrespecttoanarea?Theaim g (p ,··· ,p )(p)= min g (p ,p )(p) (9)
m 1 m 2 i j
of this paper is to answer this question. 1≤i,j≤m
i(cid:54)=j
3. ANCHORPLACEMENTEFFECTQUANTIFICATION We call an anchor pair (pi,pj) the optimally selected
anchor pair (OSAP) if g (p ,p )(p) has a minimum value
The section focuses on the effect quantification of anchor 2 i j
among all anchor pairs from from p ,··· ,p . (9) shows that
placement. Our main device is based on the notion of the 1 m
g (p ,··· ,p )(p) is determined by the OSAP, which in turn
geometric dilution of precision (GDOP) [18], [17], [22], [8]. m 1 m
varies with p and accessible anchors.
We begin with the anchor-pair GDOP function as follows:
Figs(2)-(3) visualize g (···)(p) function under the two
Theorem 1: Let p and p be the positions of anchor pair 3
i j anchor placements over the 100 × 100 region. A three-
i and j, and d and d be the actual distances from MN at
i j dimensional graph in Figs(2)-(3), which is called the least
p = (x,y) to the anchors i,j, then the geometric dilution of
vulnerabilitytomography(LVT),geometrizestheanchorplace-
precisionatpwithrespecttop ,p ,denotedbytheanchor-pair
i j ment effect: the LVT elevation has an implication: when
GDP function g (p ,p )(p), is
2 i j in a trough area, the noise has less impact on localization
(cid:118)(cid:117) 2 accuracy than when in a peak area. Fig(2) shows that the
g2(pi,pj)(p)=(cid:117)(cid:116)1−(cid:16)d2i+d2j−||pi−pj||2(cid:17)2, (5) LaVeTlevoaftioAnPv(a1ri0a0ti,o0n),(f0ro,m0),√(02,10to0)2.haIsntecrornatirnaswt,atvhees LwVitTh
2didj
of AP (0,100),(7,50),(3,40) shown in Fig(3) is relatively
where ||p −p || is the distance between p =(x ,y ) and p =
i j i i i j flat for the most part of the region but has a dramatic
(x ,y ).
j j elevation variation from 2 to 16 in the vicinity of (0,0)
Proof: Let h(pi,pj)(p) be a matrix defined as and (0,100). Overall, the terrain waves in Fig(3) has a
(cid:18) (cid:19) much higher elevation than that in Fig(2). This offers an
sin(α) cos(α)
h(pi,pj)(p)= sin(β) cos(β) (6) explanation: why AP (0,100),(0,0),(100,0) outperforms AP
(0,100),(7,50),(3,40) as shown in Table 1.
√ (x−xi) , √ (y−yi) 
The LVTs in Figs(2)-(3) induce us to extend the GDOP
=√(x−x(xi)−2x+j()y−yi)2, √(x−x(yi)−2y+j()y−yi)2, function from a point into an area. Let gm(p1,··· ,pm)(Ω)
(x−xj)2+(y−yj)2 (x−xj)2+(y−yj)2 denotetheaverageelevationofgm(p1,··· ,pm)(p)overΩ.The
that the localization accuracy over the AP is better than that
1
2
1.9 under AP2 as g3(p1,p2,p3)(Ω) < g3(p1,p4,p5)(Ω), which
1.8
1.7 agreewiththeresultsobtainedinnoisyenvironmentsinTable1.
11110246 100 11..56 Comparing g3(p1,p2,p3)(Ω) and g3(p1,p2,p3)(Γ) in Table 2
468 80 1.4 showsthatthedifferencebetweenthemisverysmall.However,
2
0 60
40 Y Axis computation of g3(p1,p2,p3)(Ω) takes about 23 seconds, as
0 20 4X0 Axis 60 80 100 0 20 opposed to 0.2 seconds in computing g3(p1,p2,p3)(Γ).
Fig.2. LVT of anchor placement (0,100),(0,0),(100,0) 4. TWO-PHASELOCALIZATIONALGORITHM
ItisclearfromFig(2)thatg (p ,··· ,p )(p)isdetermined
m 1 m
16
14 by an OSAP and the OSAP varies by area. Fig(4) shows the
12
810 OSAP areas under AP (0,100),(0,0),(100,0). Observe the
16 6
111024 100 24 noise-free scenario in the left-hand side graph in Fig(4) where
468 80 0 eachcoloredarearepresentsananchorpairchosenoveritstwo
2
0 60
40 Y Axis alternativesbasedon(9).Thewhitecoloredareasrepresentthe
0 20 4X0 Axis 60 80 100 0 20 rTehgeiobnlsueincowlohriecdh athreeasanrcehporerspenaitrthaet (re0g,i1o0n0s),w(h0e,r0e)tihsecahnocsheonr.
Fig.3. LVT of anchor placement (0,100),(7,50),(3,40) pair at (0,0),(100,0) has its geometric advantage over its
alternative anchor pairs. The red colored areas represent the
regions where the anchor pair at (100,0),(0,100) is expected
relation between g (p ,··· ,p )(Ω) and g (p ,··· ,p )(p)
m 1 m m 1 m to produce the most accurate localization. The right-hand side
(g (p ,··· ,p )(x,y)) becomes
m 1 m graph in Fig(4) plots the OSAP areas under the noise level
1 (cid:90) of σ = 1.0. The presence of noise makes the borderlines on
g (p ,··· ,p )(Ω)= g (p ,··· ,p )(x,y)dxdy
m 1 m |Ω| m 1 m different colored areas rough and unsmooth. This is because
Ω
(10) that borderlines correspond to the isolines by two anchor pairs
and hence are highly sensitive to noise.
Similarly, the notion of g (p ,··· ,p )(p) can be extended
m 1 m
from a point p to a trajectory Γ as
100 100
1 (cid:90)
gm(p1,··· ,pm)(Γ)= |Γ| gm(p1,··· ,pm)(x,y)dl (11) 80 80
Γ
The discrete form of (11) becomes 60 60
(cid:80)
g(p ,··· ,p )(x(t ),y(t )) 40 40
1 m k k
1≤k≤n
g (p ,··· ,p )(Γ)= .
m 1 m n 20 20
(12)
0 0
0 20 40 60 80 100 0 20 40 60 80 100
(10) provides a means for quantifying the impact of an AP
over an area. In practice, due to the arbitrariness of the area Fig.4. a) noise free b) noise level σ(1.0): white area by anchor pair
at(0,100),(0,0),blackareabyanchorpairat(0,0),(100,0),redarea
boundary and of an anchor placement, it would be impossible by anchor pair at (0,100),(100,0)
to derive a closed-form expression. In this paper, we use
Trapezium rule method to compute (10). It is done by first ThisobservationofFig(4)leadstoanewtwo-phaselocaliza-
splitting the area into 10,000 non-overlapping sub-areas, then tionalgorithm(TPLM)thatdifferssignificantlyfromLSMand
applying the Trapezium rule on each of these sub-areas. GDM. TMPL differs from LSM and GDM. It proceeds in two
phases: In the first phase called optimal anchor pair selection,
TABLE2 an optimal anchor pair at (p ,p ) is identified based on (9),
i j
ANCHORPLACEMENTIMPACTQUANTIFICATION then the position(s) can be estimated by solving
AnchorPlacement
p1=(0,100),p2=(0,0),p3=(100,0),p4=(7,50),p5=(3,40) ||p(cid:98)−pi||=d(cid:98)i, ||pˆ−pj||=d(cid:98)j (13)
AnchorPlacementImpactoverΩ
g3(p1,p2,p3)(Ω) g3(p1,p4,p5)(Ω) Through a lengthy manuplation, two possible solutions to (13)
1.501 2.615
AnchorPlacementImpactoverΓ are obtained as follows
g3((p1,p2,p3)(Γ) g3(p1,p4,p5)(Γ) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
x cos(θ) sin(θ) u x
1.499 2.622 (cid:98) = + i
Ω:100×100area,Γ:HT y(cid:98) sin(θ) cos(θ) v yi
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
x cos(θ) sin(θ) u x
Table 2 provides a numerical calculation of AP impact over (cid:98) = + i (14)
y sin(θ) cos(θ) −v y
Ω and over Γ. The AP impact calculated in Table 2 implies (cid:98) i
where θ =arctan(xyjj−−yxii), dij =||pi−pj||, and 5. SIMULATIONSTUDY
(cid:118) The aim of this section is threefold: first, to conduct the
u= (d2ij +2d(cid:98)di2−d(cid:98)j2), v =(cid:117)(cid:117)(cid:117)(cid:116)d2i −d2ij +2d(cid:98)di2−d(cid:98)j22 pinernfoorimseanecnevircoonmmpeanrtiss;onsecboentwd,eeton sTtuPdLyMt,heLSimMp,acatndofGnDoiMse
ij ij models; and third, to investigate the random anchor placement
(15) impact.
In the second phase called disambiguation, the reference point A. Visualizing Noise-induced Distortion
obtainedfromLSMisusedtosingleoutonefromtwopossible
To visualize the difference among LSM, GDM, and TPLM,
positions (14). It is done by choosing one position that has a
we plot the restored HT by LSM, GDM, and TPLM under
shorter distance to the reference point.
Gaussian noise model in Figs(5)-(7) using AP in Table 4.
4
Notice that anchor positions are plotted as solid black circles.
Algorithm 1 Two-phase Localization Method
Input: p =(x ,y ),1≤i≤m: ith anchor’s position TABLE4
i i i
Input: d¯: distance measurement from jth anchor to MN p ANCHORPLACEMENTSETUP
i
Input: p˘=(x˘,y˘): a reference point obtained via LSM AnchorPosition
1: gmin ←∞,p+ ←(0,0),p++ ←(0,0) pp41==((30,,4100)0,)p,5p2==(1(070,,500)),,pp63==((10,,908))
2: for i=1 to m−1 do AnchorPlacementSetup
3: for j =i+1 to m do AP1 (0,100),(0,0),(100,0)
4: if g2(pi,pj)(p)<gmin then AAPP23 (0,1(000,)1,0(00),,0()7,,(5100)0,,(03),,4(01),98)
5: gmin ←g(pi,pj)(p),(p+,p++)←(pi,pj) AP4 (0,100),(7,50),(3,40),(1,98)
6: end if
7: end for
8: end for 100 100
9: d=|p+−p++|,d+ =|p−p+|,d++ =|p−p++|
10: θ =arctan y++−y+,u= d2+−d2+++d2,v =(cid:113)d2 −u2 80 80
x++−x+ 2d +
11: p∗ =(csoins((θθ))−cosisn(θ(θ)))(cid:0)uv(cid:1)+(cid:0)xy++(cid:1) 60 60
12: p∗∗ =(csoins((θθ))−cosisn(θ(θ)))(cid:0)−uv(cid:1)+(cid:0)xy++(cid:1) 40 40
13: if |p∗−p˘|<|p∗∗−p˘| then
14: return p∗ 20 20
15: end if
0 0
16: return p∗∗ 0 20 40 60 80 100 0 20 40 60 80 100
(a) (b)
Fig.5. restored HT by LSM: a) noise (0.02); b) noise (0.2)
The code between lines 2-8 examines each anchor pair in
turn and chooses an optimal anchor pair, which corresponds
to (9). The complexity is (cid:0)m(cid:1). Lines 2-12 constitute the basic 100 100
2
blockoftheoptimalanchorselectionphase.Thecodebetween 80 80
lines 13-16 is used to disambiguate the two possible solutions
using the reference position p˘obtained by LSM. 60 60
TABLE3 40 40
TWO-PHASELOCALIZATIONMETHOD
20 20
positionsofanchors σ ave std time
0.3 0.40 0.22 3.17
(0,100),(0,0),(100,0) 0 0
1.0 1.32 0.73 3.17
0 20 40 60 80 100 0 20 40 60 80 100
0.3 0.70 0.88 3.44 (a) (b)
(0,100),(7,50),(3,40) 1.0 2.76 5.67 3.15 Fig.6. restored HT by GDM: a) noise (0.02); b) noise (0.2)
GaussiannoiseN(0,σ2)
Visual inspection reveals the apparent perceived difference
We present the results of TPLM in Table 3 using the same among the restored HTs by LSM, GDM and TPLM: when
anchorplacementsetupsinTable1.Inallcases,TPLMgivesa the noise level σ is 0.2, the restored HT by LSM becomes
significant error reduction over LSM with a minor degradation completely unrecognizable in Fig(5)(b). While in Fig(6)(b) the
as TPLM uses the reference point obtained from LSM. It is upperrightportionoftherestoredHTbyGDMtosomedegree
fairly clear that TPLM is an order of magnitude faster than preserves the hallmark of the HT, but the most part of the
GDM, and performs slightly better than GDM in terms of recovered HT is severely distorted and barely recognizable.
localization accuracy. The restored curve by TPLM contrasts sharply with those
1 8000 1 8000 meter) 45 AAAAPPPP1234
4600 4600 localization Error ( 123
0
20 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
by 0 FL 0igS.M7. 20arnesdt 4o0Gre(adD) 6MH0Tibn 8y0iTtsP 1L0p0Mre:s 0ear) 0vnaotiiso e2n0(0o.f0 420fi)n(;be)b )6d0neotiasi 8el0s(0o.f2 10)0the localization Error (meter) 12345 AAAAPPPP1234
0
HT for the most part, while having minor distortion in the 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
uniform noise: noise level
lower- and upper-left corner areas in Fig(7)(b). Such spatially
Fig.9. Accuracy: (top) GDM; (bottom) TPLM (uniform noise)
unevenlocalizationperformancecanbeexplainedbyexamining
Fig(3),whichshowsthatthelower-andupper-leftareasexactly
correspond to the LVT peak areas. The perceived differences
noisemodels,TPLMoutperformsGDMbyhugemarginsunder
between GDM and TPLM in Figs(5)-(7) can be quantified as
AP and AP . Both GDM and TPLM perform indistinguish-
2 4
follows: GDM has the average error of 0.69 per HT traversal,
ablyunderAP andAP .Whiletheimpactdifferencebetween
1 3
while TPLM produces the average error of 0.465. In addition,
AP and AP is barely noticed as their performance curves
1 3
GDM takes about 218.48 seconds per HT traversal, in contrast
areoverlapped,theperformancecurvesbetweenAP andAP
1 2
to 2.94 seconds taken by TPLM.
and between AP and AP are clearly separated, thus the
3 4
impact difference AP and AP and between AP and AP
B. Gaussian Noise vs. Non-Gaussian Noise 1 2 3 4
are evident.
Inthissubsection,wewillcomparetheperformanceofGDM
andTPLMunderGaussiananduniformnoisemodels,thereare 500
s[Gd7oiDf]mf,eMe[r2esa0ncn]te.dnnIaoTnriPisoteLhsMemweohixsdeperceelosrmr.imaFpneoagnrrietnGadglaunsuntosudisdseieayrn,matnhsaoayemisfpeoeelrmnlfoooowirdsmeeulanleniNfvcoee(rl0mbw,eσimttw2ho)etd,eheσnel Elapsed Time (seconds) 123400000000 AAAAPPPP1234
referstothenoiselevelinFig(8),andforauniformnoisemodel 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
U(−a,a), the noise level in Fig(9) is expressed as a2/3. The
5
dovatear 1p0resHeTntetrdavreerflseaclstsutnhdeeravtheeraagnecherorrorploafceGmDenMtsainndTaTbPleLM2. me (seconds) 34 AAAAPPPP1234
meter) 567 AAAAPPPP1234 Elapsed Ti 012 0.1 0.2 0.3 0.4 gauss i0a.n5 noise: no 0is.6e level 0.7 0.8 0.9 1
localization Error ( 1234 GFDigT.Mh1e0t.rheasEnulaltthpsesesudhnotiwifmoetrh:ma(ttootnph)ee.GGBDayMucs;os(mibaonptatnoroimsios)enT,hPtaLhseMmp(eonrrofeormrimmalpannaocciteseoo)nf
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TPLM is insensitive to noise models. Fig(10) plots the elapsed
7 timebyGDMandTPLMperHTtraversal,showingthatTPLM
meter) 56 AAAAPPPP1234 is orders of magnitude faster than GDM.
localization Error ( 1234 C.WReansdtuodmyAthnechiomrpPalcatceomfernatndom anchor placement on the
performance of LSM, GDM and TLPM. To achieve this, the
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 number of anchors are randomly placed in 100 × 100 and
gaussian noise: noise level
50×100 regions. For each randomly generated anchor place-
Fig.8. Accuracy: (top) GDM; (bottom) TPLM (Gaussian noise)
ment (RGAP) in a region, the g (p ,··· ,p )(Γ) function is
m 1 m
The graphs in Figs(8)-(9) represent the localization error calculated, and the error statistics of LSM, GDM, and TPLM
curves of GDM and of TPLM, respectively. Under both the under Gaussian noise level of 0.3 are gathered and compared.
Fig(11)(a)-(b) show the actual g (p ,··· ,p )(Γ) (average 18 TPLM
m 1 m 16 LSM
LVT elevation over Γ)) distribution by 100 RGAPs. In the n 14 GDM
o
ttorapvegrrsaaplha,re3aawnchhiloersinarteherabnodtotommlygrpalpache3da(rn.uch.po.r)sinarethre.ue.pn.tiirne or deviati 11 802
the upper half of the traversal area. Fig(11)(a) clearly exhibit err 6
4
positive skewness. This implies that a RGAP over the entire 2
traversal area in general yields good localization accuracy. 0
1.5 2 2.5 3 (a) 3.5 4 4.5 5
Recall that the smaller g (p ,··· ,p )(Γ) is, the better local-
m 1 m 20
TPLM
iizsamtioonreisd.isTpheresegdmt(hpa1n,·t·h·at,pinm)t(hΓe)Fdigis(1tr1ib)(uat)io,nmeinanFinigg(1th1a)(tba) error 15 GLDSMM
RRGGAAPP oovveerr tthhee ehnatlifretratrvaevresraslaalraeraeain. general underperforms a average 10
5
12
0
10 1.5 2 2.5 3 (b) 3.5 4 4.5 5
8
6 Fig. 12. (a) error deviation; (b) average error (100 RGAPs with 3
4 anchors over the 100×100 region, σ=0.3)
2
0
1 2 3 4 5(a) 6 7 8 9
performance of LSM, GDM, and TPLM deteriorates as the
12
value of g (p ,..,p )(Γ) increases. This solidifies the fact
10 m 1 m
8 thatgm(p1,..pm)(Γ)isaneffectivediscriminatorfortheanchor
6 placement impact on localization performance.
4
2 6. FIELDEXPERIMENTALSTUDY
0
1 2 3 4 5 (b) 6 7 8 9 ThissectionfocusesonthefieldtestofaDARPA-sponsored
Fig.11. (a) g3(...)(Γ) distribution by 100 RGAPs with 3 anchors in research project, using the UWB-based ranging technology
100×100; (b) g3(...)(Γ) distribution by 100 RGAPs with 3 anchors from Multispectrum Solutions (MSSI) [10]. and Trimble dif-
in 50×100
ferential GPS (DGPS). Our field testbed area was a 100×100
square meters. It consisted of an outdoor space largely oc-
TABLE5 cupied by surface parking lots and an indoor space inside a
RANDOMANCHORPLACEMENT
warehouse. This testbed area was further divided into 10,000
RandomAnchorPlacementinTheTraversalArea
non-overlapping 1 × 1 grids. Each grid represents the finest
TPLM LSM GDM
Anchors gm(p1,..pm)(Γ) ave ave ave positioning resolution to evaluate RF signal variation.
3 2.05 0.57 1.52 0.87
4 1.69 0.44 0.79 0.49 TABLE6
5 1.53 0.40 0.59 0.38 FIELDTESTBEDAREAANDANCHORPLACEMENT
6 1.48 0.39 0.51 0.33 x(cid:48) y(cid:48) x y α
RandomAnchorPlacementintheHalfTraversalArea 0 0 f f
−74.476069 40.537808 84719 111045 0.381583
TPLM LSM GDM
Anchors gm(p1,..pm)(Γ) ave ave ave Anchorplacement
GPSposition transformedposition
3 2.72 0.97 2.82 1.54
4 1.93 0.52 1.37 0.69 −74.475585 40.538468 p4=(65.345179,52.75145)
5 1.77 0.46 0.81 0.46 −74.475287 40.53856 p5=(92.580022,52.83239)
6 1.69 0.44 0.72 0.40 −74.475186 40.538294 p6=(89.52274,22.232383)
Inthefieldtest,DGPSdevicesweremainlyusedforoutdoor
Fig(12)presentstheperformancecurvesofLSM,GDM,and positioning while MSSI UWB-based devices were used for
TPLM under RGAPs, where the x axis refers to the value of indoor positioning. The experimental system was composed of
g (p ,..p )(Γ) and the y axis the error deviation and average four MNs equipped with both the Trimble DGPS and MSSI
m 1 m
error. As is clear from Fig(12), TPLM outperforms both LSM UWB-based ranging devices. Three UWB devices were used
andGDMinlocalizationaccuracyandreliability:boththeerror as anchors being placed at known fixed positions. Using the
deviation and average error curves of TPLM are consistently known positions of the UWB anchors and real-time distance
lower than those of LSM and GDM. measurements,eachMNcouldestablishthecurrentpositionas
Table 5 tabulates the results of RGAPs with different num- well as that of other MNs locally, at a rate of approximately 2
ber of anchors, where g (p ,..,p )(Γ) denotes the average samplespersecond.EachMNcouldalsoestablishtheposition
m 1 m
g (p ,..p )(Γ) over 100 RGAPs with a fixed number of via the DGPS unit at a rate of 1 sample per second when
m 1 m
anchors. It shows that the localization accuracy of TPLM over residing in the outdoor area. The field test area was divided
GDM is diminished as the number of anchors increases. It into four slightly overlapping traversal quadrants denoted as
becomes evident that under the noise level of 0.3, the actual (Γ ,Γ ,Γ ,Γ ). A laptop (MN) equipped with DGPS and
0 1 2 3
100
UWB ranging devices was placed in a modified stroller as 200
80
shown in Fig(13), and one tester pushed the stroller, traveling
60
along a predefined trajectory inside a quadrant (see Fig(14)).
Each run lasted about 30 minutes of walk. 40 150
20
0
GPS Antenna 0 20 40 60 100
(a) Trajectory 100
UWB radio 120
100
80 50
Rain Cover Laptop running 60
system s/w
40
Basket w/GPS 0
unit & batteries 20
Speedo warehouse
0
0 20 40 (b) 60 warehouse 100 0 20 40 (c) 60 80 100
Fig.16. RestoredTrajectoriesby(a)TPLM,(b)GDM,(c)LSM(black
circles: positions of UWB anchors)
Fig.13. StrollerwithalaptopattachedwithDGPSandUWBranging
devices
Inthefieldtestbed,thetrajectoryofeachMNwascontrolled
100 by an individual tester during a 30-minute walk. For purpose
Γ
1 Γ of the primary experiment being conducted, the strollers were
2
fitted with bicycle speedometers to allow the tester to control
80
his speed, in order to produce sufficient reproducibility for the
primaryexperiment,butnotforourtestingoflocalization.Due
60 to inherent variability in each individual movement, an objec-
tive assessment of localization accuracy without a ground truth
reference is almost impossible. For a performance comparison,
40
Γ we used GPS trajectories as a reference for visual inspection
3
of the restored trajectories by LSM, GDM, and TPLM.
20 Three curves in Fig(15)(a)-(d) represent the field distance
warehouse Γ measurements between a tester and the three UWB anchor de-
0
vicesduringa30-minutewalkintrajectories(Γ ,Γ ,Γ ,Γ ).it
0 0 1 2 3
isfairlyobviousthatmeasurementnoiseindifferenttrajectories
0 20 40 60 80 100
vary widely: The distance measurement curves in trajectories
Fig.14. Fourtrajectories(Γ0,Γ1,Γ2,Γ3),blackcircles:thepositions (Γ0,Γ3) are discontinuous and jumpy in Figs(15)(a)&(d), in
of anchors, yellow rectangle: warehouse
contrast to the relatively smooth distance curves in trajectories
Withrepeatedtrialruns,wefoundthattheUWBsignalcould (Γ ,Γ ) in Fig(15)(b)&(c). Such a discontinuity in measure-
1 2
barelypenetrateonecementwallofthewarehouse.Toestablish ments occurred when testers were traveling in an area where
positions inside the warehouse, we placed one anchor close to UWB devices on stroller had no direct line of sight to UWB
the main entrance of the warehouse while placing two other anchors, thus introducing additional noise in Γ ,Γ .
0 3
anchors around the center of the field testbed area. Fig(14) Figs(16)(a)-(c) show the restored trajectories by TPLM,
showed the four trajectories (Γ0,Γ1,Γ2,Γ3) in the field test, GDM, and LSM using the UWB ranging technology in the
which are translated from the GPS trajectories. A part of Γ0 fieldtest.AvisualinspectionsuggeststhatLSMwasextremely
trajectory was inside the warehouse, thus the part of GPS of prone to noise as its restored trajectories were appreciably
Γ0 was not available. Table7 provides the calculated results distorted beyond recognition in some parts. As indicated in
of g3(p4,p5,p6)(Γ) using the GPS trajectory data, indicating Fig(16)(b), GDM gave an obvious error reduction over LSM
that Γ0 would yield the most accurate localization while Γ1 but at the expense of computational cost. The most visually
produced the least accurate localization. perceived difference between LSM and GDM can be seen in
the circled areas in Figs(16)(b)-(c). By contrast, the difference
TABLE7
betweenGDMandTPLMcanbevisualizedinthecircledareas
g3(p4,p5,p6)(Γ)OFTRAJECTORIESINFIELDTESTBED
Figs(16)(a)-(b) where TPLM produced a detail-preserved but
g3(p4,p5,p6)(Γ0) g3(p4,p5,p6)(Γ1)
slightly distorted contour of the trajectory. A further offline
1.667025 1.967605
g3(p4,p5,p6)(Γ2) g3(p4,p5,p6)(Γ3) analysis showed that on average GDM takes 18.87ms per
1.787370 1.835048 position establishment, while TPLM/LSM take 0.4/0.35ms.
100 DDDiiissstttaaannnccceee bbbttwwwtnnn MMMNNN000 AAAnnnccchhhooorrr 654 120 DDDiiissstttaaannnccceee bbbtwtwwtnnn MMMNNN111 AAAnnnccchhhooorrr 465 100 DDDiiissstttaaannnccceee bbbtwtwwtnnn MMMNNN222 AAAnnnccchhhooorrr 456 112400 DDDiiissstttaaannnccceee bbbwttwwtnnn MMMNNN333 AAAnnnccchhhooorrr 564
80 100 80
100
Distance [meter] 4600 Distance [meter] 6800 Distance [meter] 4600 Distance [meter] 6800
40
20 40 20
20
0 0 200 400 600 800 1000 1200 1400 1600 1800 20 0 200 400 600 800 1000 1200 1400 1600 1800 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 0 200 400 600 800 1000 1200 1400 1600
Time [sec] Time [sec] Time [sec] Time [sec]
(a) (b) (c) (d)
Fig.15. DistancemeasurementstothreeUWBanchors:(a)Γ0,(b)Γ1,(c)Γ2,(d)Γ3
7. CONCLUSION [10] R.J.Fontana. Recentsystemapplicationsofshort-pulseultra-wideband
technology. IEEE Trans. on Microwave Theory and Techniques,
This paper studies the geometric effect of anchor placement 52(9):2087–2104,September2004.
onlocalizationperformance.Theproposedapproachallowsthe [11] S. Gezici, Z. Tian, G.B. Giannakis, H. Kobayashi, A.F. Molisch, H.V.
Poor, and Z. Sahinoglu. Localization via ultra-wideband radios. IEEE
construction of the least vulnerability tomography (LVT) for
SignalProcessingMag,22(4),2005.
comparingthegeometricimpactofdifferentanchorplacements. [12] A. Kannan, B. Fidan, and G. Mao. Derivation of flip ambiguity
As a byproduct, we propose a two-phase localization method. probabilities to facilitate robust sensor network localization. In IEEE
WirelessCommunicationsandNetworkingConference,2009.
To validate theoretical results, we conduct comprehensive
[13] A.Kannan,B.Fidan,andG.Mao.Analysisofflipambiguitiesforrobust
simulation experiments based on randomly generated anchor sensornetworklocalization.IEEETrans.onVehicularTechnology,59(4),
placementsanddifferentnoisemodels.Theexperimentalresults 2010.
[14] K.LangendoenandN.Reijers. DistributedLocalizationAlgorithms. In
agree with our anchor placement impact analysis. In addition,
R. Zurawski, editor, Embedded Systems Handbook. CRC press, August
we show that TPLM outperforms LSM by a huge margin in 2005.
accuracy. TPLM performs much faster than GDM and slightly [15] S.Lederer,Y.Wang,andJ.Gao.Connectivity-basedlocalizationoflarge
scalesensornetworkswithcomplexshape. InINFOCOM,May2008.
betterthanGDMinlocalizationaccuracy.Thefieldstudyshows
[16] S. Lederer, Y. Wang, and J. Gao. Connectivity-based localization of
that TPLM is more robust against noise than LSM and GDM. largescalesensornetworkswithcomplexshape. ACMTrans.onSensor
Networks,5(4),November2009.
In this paper, we adopt a widely used assumption that
[17] H.B. Lee. Accuracy limitation of hyperbolic multilateration systems.
measurementnoiseisindependentofdistance.Thisassumption IEEE Trans. on Aerospace and Electronic Systems, AES-11, January
issomewhatunrealisticinsomepracticalsettings.Inourfuture 1975.
[18] H.B. Lee. A novel procedure for assessing the accuracy of hyperbolic
research, we will perform experiments using UWB ranging
multilateration systems. IEEE Trans. on Aerospace and Electronic
technology to study the noise-distance relation in indoor and Systems,AES-11,January1975.
outdoor environments. The anchor placement impact analysis [19] Z.Li,W.Trappe,Y.Zhang,andB.Nath. Robuststatisticalmethodsfor
securingwirelesslocalizationinsensornetworks. InIPSN,2005.
will be refined to incorporate more realistic noise-distance
[20] L. Liu and M.G. Amin. Performance analysis of gps receivers in non-
models. gaussiannoiseincorporatingprecorrelctionfilterandsamplingrate.IEEE.
Trans.OnSignalProcessing,56(3),March2008.
REFERENCES [21] D. Moore, J. Leonard, D. Rus, and S. Teller. Robust Distributed
Network Localization with Noisy Range Measurements. In Proceeding
ofSenSys’04,November2004.
[1] G.B.ArfkenandH.J.Weber.MathematicalMethodsforPhysicists,Sixth
[22] N. Patwari, J.N. Ash, S. Kyperountas, A.O. Hero III, R.L. Moses, and
Edition:AComprehensiveGuide. AcademicPress,2005.
N.S. Correal. Locating the Nodes: Cooperative localization in wireless
[2] A.N.Bishop,B.Fidan,B.D.OAnderson,K.Dogancay,andN.Pathirana.
sensornetworks. IEEESignalProcess.Mag,22(4),2005.
Optimality analysis of sensor-target geometries in passive location: Part
[23] N.B.Priyantha,H.Balakrishnan,E.Demaine,andS.Teller. Anchor-free
1-bearing-onlylocalization. InISSNIP,September2007.
DistributedLocalizationinSensorNetworks. InTechReport#892,MIT
[3] A.N. Bishop, B. Fidan, B.D.O. Anderson, K. Dogancay, and P.N.
LaboratoryforCompterScience,April2003.
Pathirana. Optimality analysis of sensor-target localization geometries.
[24] A.M. So and Y. Ye. Theory of semidefinite programming for sensor
Automatic,46(3),2010.
networklocalization. InSODA,2005.
[4] A.N.BishopandPJensfelt. Optimalityanalysisofsensor-targetgeome-
[25] R.Stoleru,T.He,andJ.A.Stankovic.WalkingGPS:APracticalSolution
tries in passive location: Part 2- time-of-arrival based localization. In
forLocalizationinManuallyDeployedWirelessSensorNetworks.InPro-
ISSNIP,September2007.
ceedingsofIEEEWorkshoponEmbeddedNetworkedSensors,November
[5] A.N. Bishop and P Jensfelt. An optimality analysis of sensor-target
2004.
geometriesforsignalstrengthbasedlocalization. InISSNIP,September
[26] P.Tseng. Second-orderconeprogrammingrelaxationofsensornetwork
2009. localization. SIAMJ.Optim,18,2007.
[6] J. Bruck, J. Gao, and A. Jiang. Localization and routing in sensor [27] K.WhitehouseandD.Culler.Arobustnessanalysisofmulti-hopranging-
network by local angle information. ACM Trans. on Sensor Networks, basedlocalozationapproximations. InIPSN,2006.
5(1),February2009. [28] K.Whitehouse,C.Karlof,A.Woo,F.Jiang,andD.Culler. Theeffects
[7] N. Bulusu, J. Heidemann, and D. Estrin. Adaptive beacon placement. ofrangingnoiseonmultihoplocalozation:anempiricalstudy. InIPSN,
InThe21thInternationalConferenceonDistributedComputingSystems, 2005.
April2001. [29] M.YoussefandA.Agrawala. TowardsanOptimalStrategyforWLAN
[8] S. O. Dulman, A. Baggio, P.J.M. Havinga, and K.G. Langendoen. A LocationDeterminationSystems.InternationalJournalofModellingand
geometricalperspectiveonlocalization. Inthe14thAnnualInternational Simulation,27,2007.
ConferenceonMobileComputingandNetworking,2008. [30] G. Zhao, T. He, and J. A. Stankovic. Impact of Radio Asymmetry
[9] T.Eren,D.K.Goldenberg,W.Whiteley,Y.R.Yang,A.S.Morse,D.O. on Wireless Sensor Networks. In Proceedings of ACM Conference on
Anderson,andP.N.Belhumeur.Rigidity,computation,andrandomization EmbeddedNetworkedSensorSystems,November2004.
innetworklocalization. InINFOCOM,pages2673–2684,March2004.