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Liu Y, Yang Z, Wang X et al. Location, localization, and
localizability. JOURNAL OF COMPUTER SCIENCE AND
TECHNOLOGY 25(2): 274–297 Mar. 2010
Location, Localization, and Localizability
Yunhao Liu (刘云浩), Member, ACM, Senior Member, IEEE, Zheng Yang
(杨 铮), Student Member, ACM, IEEEXiaoping Wang (王小平), Student
Member, IEEE, and Lirong Jian (简丽荣), Student Member, IEEE
Department of Computer Science and Engineering, Hong Kong
University of Science and Technology, Hong Kong, China
E-mail: {liu, yangzh, xiaopingwang, jlrphx}@cse.ust.hkReceived
October 28, 2009; revised January 6, 2010.
Abstract Location-aware technology spawns numerous unforeseen
pervasive applications in a wide range of living, pro-duction,
commence, and public services. This article provides an overview of
the location, localization, and localizabilityissues of wireless
ad-hoc and sensor networks. Making data geographically meaningful,
location information is essential formany applications, and it
deeply aids a number of network functions, such as network routing,
topology control, coverage,boundary detection, clustering, etc. We
investigate a large body of existing localization approaches with
focuses on errorcontrol and network localizability, the two rising
aspects that attract significant research interests in recent
years. Errorcontrol aims to alleviate the negative impact of noisy
ranging measurement and the error accumulation effect during
coope-rative localization process. Network localizability provides
theoretical analysis on the performance of localization
approaches,providing guidance on network configuration and
adjustment. We emphasize the basic principles of localization to
under-stand the state-of-the-art and to address directions of
future research in the new and largely open areas of
location-awaretechnologies.
Keywords location-based services (LBS), localization, error
control, localizability, wireless ad-hoc and sensor networks
1 Location
The proliferation of wireless and mobile devices hasfostered the
demand for context-aware applications, inwhich location is viewed
as one of the most signifi-cant contexts. For example, pervasive
medical careis designed to accurately record and manage
patientmovements[1-2]; smart space enables the interaction be-tween
physical space and human activities[3-4]; mod-ern logistics has
major concerns on goods transporta-tion, inventory, and
warehousing[5-6]; environmentalmonitoring networks sense air,
water, and soil qualityand detect the source of pollutants in real
time[7-11];and mobile peer-to-peer computing encourages
contentsharing and contributing among mobile hosts in
thevicinity[12-13]. In brief, location-based service (LBS)is a key
enabling technology of these applications andwidely exists in
nowadays wireless communication net-works from the short-range
Bluetooth to the long-rangetelecommunication networks, as
illustrated in Fig.1.
Recent technological advances have enabled the de-velopment of
low-cost, low-power, and multifunctionalsensor devices. These nodes
are autonomous deviceswith integrated sensing, processing, and
communi-cation capabilities. With the rapid development ofwireless
sensor networks (WSNs), location information
Fig.1. Location-based services for a wide range of wireless
net-
works.
becomes critically essential and indispensable. Theoverwhelming
reason is that WSNs are fundamentallyintended to provide
information on spatial-temporalcharacteristics of the physical
world; hence, it is impor-tant to associate sensed data with
locations, makingdata geographically meaningful. For example, a
num-ber of applications, such as object tracking and environ-ment
monitoring, inherently rely on location informa-tion. A detailed
survey on location-based applicationscan be found in [14-15].
Location information also supports many fundamen-tal network
services, including network routing, topol-ogy control, coverage,
boundary detection, clustering,etc. We give a brief overview as
follows.
Survey©2010 Springer Science +Business Media, LLC & Science
Press, China
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Yunhao Liu et al.: Location, Localization, and Localizability
275
• RoutingRouting is a process of selecting paths in a net-
work along which to send data traffic. Most routingprotocols for
multi-hop wireless networks utilize physi-cal locations to
construct forwarding tables and delivermessages to the node closer
to the destination in eachhop[16]. Specifically, when a node
receives a message,local forwarding decisions are made according to
thepositions of the destination and its neighboring nodes.Such
geographic routing schemes require localized infor-mation, making
the routing process stateless, scalable,and low-overhead in terms
of route discovery.• Topology ControlTopology control is one of the
most important tech-
niques used in wireless ad-hoc and sensor networks forsaving
energy and eliminating radio interference[17-18].By adjusting
network parameters (e.g., the transmit-ting range), energy
consumption and interference canbe effectively reduced; meanwhile
some global net-work properties (e.g., connectivity) can still be
wellretained. Importantly, using location information asa priori
knowledge, geometry techniques (e.g., spannersubgraphs and
Euclidean minimum spanning trees) canbe immediately applied to
topology control[17].• CoverageCoverage reflects how well a sensor
network observes
the physical space; thus, it can be viewed as the quali-ty of
service (QoS) of the sensing function. Previ-ous designs fall into
two categories. The probabilisticapproaches[19-21] analyze the node
density for ensuringappropriate coverage statistically, but
essentially haveno guarantee on the result. In contrast, the
geometricapproaches[22] are able to obtain accurate and
reliableresults, in which locations are essential.• Boundary
DetectionBoundary detection is to figure out the overall
boundary of an area monitored by a WSN. There aretwo kinds of
boundaries: the outer boundary showingthe under-sensed area, and
the inner boundary indica-ting holes in a network deployment. The
knowledge ofboundary facilitates the design of routing, load
balanc-ing, and network management[23]. As direct evidence,location
information helps to identify border nodes andfurther depict the
network boundary.• ClusteringTo facilitate network management,
researchers of-
ten propose to group sensor nodes into clusters andorganize
nodes hierarchically[24]. In general, ordinarynodes only talk to
the nodes within the same cluster,and the inter-cluster
communications rely on a specialnode in each cluster, which is
often called cluster head.Cluster heads form a backbone of a
network, basedon which the network-wide connectivity is
maintained.
Clustering brings numerous advantages on network op-erations,
such as improving network scalability, local-izing the information
exchange, stabilizing the networktopology, and increasing network
life time. Among allpossible solutions, location-based clustering
approachesare greatly efficient by generating non-overlapped
clus-ters. In addition, location information can also be usedto
rebuild clusters locally when new nodes join the net-work or some
nodes suffer from hardware failure[24].
2 Localization
Network localization has attracted a lot of researchefforts in
recent years. One method to determine thelocation of a device is
through manual configuration,which is often infeasible for
large-scale deployments ormobile systems. As a popular system,
Global Position-ing System (GPS) is not suitable for indoor or
under-ground environments and suffers from high hardwarecost. Local
Positioning Systems (LPS) rely on high-density base stations being
deployed, an expensive bur-den for most resource-constrained
wireless ad hoc net-works.
The limitations of existing positioning systems mo-tivate a
novel scheme of network localization, in whichsome special nodes
(a.k.a. anchors or beacons) knowtheir global locations and the rest
determine their loca-tions by measuring the geographic information
of theirlocal neighboring nodes. Such a localization scheme
forwireless multi-hop networks is alternatively describedas
“cooperative”, “ad-hoc”, “in-network localization”,or “self
localization”, since network nodes cooperativelydetermine their
locations by information sharing.
In this section, we first review the state-of-the-art
lo-calization approaches from two aspects: physical mea-surements
and network-wide localization algorithms.We then discuss a number
of techniques for controllinglocalization errors caused by noisy
physical measure-ments and algorithmic defects.
Almost all existing localization algorithms consist oftwo
stages: 1) measuring geographic information fromthe ground truth of
network deployment; 2) computingnode locations according to the
measured data. Geo-graphic information includes a variety of
geometric re-lationships from coarse-grained neighbor-awareness
tofine-grained inter-node rangings (e.g., distance or an-gle).
Based on physical measurements, localization al-gorithms solve the
problem that how the location infor-mation from beacon nodes
spreads network-wide. Gen-erally, the design of localization
algorithms largely de-pends on a wide range of factors, including
resourceavailability, accuracy requirements, and deployment
re-strictions; and no particular algorithm is an absolutefavorite
across the spectrum.
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276 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
2.1 Physical Measurements and Single-HopPositioning
Clearly, it is difficult, if not infeasible, to do localiza-tion
without knowledge of the physical world. Accord-ing to the
capabilities of diverse hardwares, we classifythe measuring
techniques into six categories (from fine-grained to
coarse-grained): location, distance, angle,area, hop count, and
neighborhood, as shown in Fig.2.
Fig.2. Physical measurements.
Among them, the most powerful physical measure-ment is directly
obtaining the position without any fur-ther computation. GPS is
such a kind of infrastructure.The other five measurements are used
in the scenariosof positioning an unknown node by given some
refer-ence nodes. The terms of “reference” and “unknown”nodes refer
to the nodes being aware and being NOTaware of their locations,
respectively. Distance and an-gle measurements are obtained by
ranging techniques.Hop count and neighborhood are basically based
on ra-dio connectivity. In addition, area measurement relieson
either ranging or connectivity, depending on how thearea constrains
are formed.
2.1.1 Distance Measurements
The distances from an unknown node to several ref-erences
constrain the presence of this node, which is thebasic idea of the
so called multilateration. Fig.3 showsan example of trilateration,
a special form of multilat-eration which utilizes exact three
references. A to-be-located node (node 0) measures the distances
from itselfto three references (nodes 1, 2, 3). Obviously, node
0should locate at the intersection of three circles centeredat each
reference position. The result of trilateration isunique as long as
three references are non-linear.
Suppose the location of the unknown node is (x0, y0)and it is
able to obtain the distance estimates d′i to thei-th reference node
locating at (xi, yi), 1 6 i 6 n. Letdi be the actual Euclidean
distance to the i-th referencenode, i.e.,
di =√
(xi − x0)2 + (yi − y0)2.
The difference between the measured and the actualdistances can
be represented by ρi = d′i − di. Owingto ranging noises in d′i, ρi
is often non-zero in practice.
The least squares method is used to assign a value to(x0, y0)
that minimizes
∑ni=1 ρ
2i . This problem can be
solved by a numerical solution to an over-determinedlinear
system[25].
Fig.3. Trilateration. (a) Measuring distance to 3 reference
nodes.
(b) Ranging circles.
The over-determined linear system can be obtainedas follows.
Rearranging and squaring terms of the ac-tual distances, we have n
such equations:
x2i + y2i − d2i = 2x0xi + 2y0yi − (x20 + y20).
By subtracting out the n-th equation from the rest,we have n− 1
equations of the following form:x2i +y
2i −x2n−y2n−d2i +d2n = 2(xi−xn)x0 +2(yi−yn)y0
which yields the linear relationship
Ax = B
where A is an (n−1)×2 matrix, such that the i-th rowof A is
[2(xi − xn) 2(yi − yn)], x is the column vectorrepresenting the
coordinates of the unknown location[x0 y0]T, and B is the (n − 1)
element column vectorwhose i-th term is (x2i + y
2i − x2n− y2n− d2i + d2n). Prac-
tically, we cannot determine B, since the real distancesare not
known to us, so computation is performed onB′, which is the same as
B with d′i substituting fordi. Now the least square solution is an
estimate forx′ that minimizes ‖Ax′ −B′‖2, which is provided byx′ =
(ATA)−1ATB′.
So far, for distance-based positioning, the only thingomitted is
how to measure distances in the physicalworld. Many ranging
techniques are proposed and de-veloped; among them, the radio
signal strength basedand time based ranging are two of the most
widely usedones in existing designs.
(a) Radio Signal Strength Based Distance Measure-ment
Radio Signal Strength (RSS) based ranging tech-niques are based
on the fact that the strength of radiosignal diminishes during
propagation. As a result, theunderstanding of radio attenuation
helps to map thesignal strength to the physical distance. In
theory, ra-dio signal strengths diminish with distance according
to
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Yunhao Liu et al.: Location, Localization, and Localizability
277
a power law. A generally employed model for wirelessradio
propagation is as follows[26]:
P (d) = P (d0)− η10 log( d
d0
)+ Xσ
where P (d) is the received power at distance d, P (d0)the
received power at some reference distance d0, ηthe path-loss
exponent, and Xσ a log-normal randomvariable with variance σ2 that
accounts for fading ef-fects. Hence, if the path-loss exponent for
a given envi-ronment is known, the received signal strength can
betranslated to the signal propagation distance.
In practice, however, RSS-based ranging measure-ments contain
noises on the order of several meters.The ranging noise occurs
because radio propagationtends to be highly dynamic in complicated
environ-ments.
On the whole, RSS based ranging is a relatively“cheap” solution
without any extra devices, as all net-work nodes are supposed to
have radios. It is believedthat more careful physical analysis of
radio propaga-tion may allow better use of RSS data.
Nevertheless,the breakthrough technology is not there today.
(b) Time Difference of Arrival (TDoA)A more promising technique
is the combined use of
ultrasound/acoustic and radio signals to estimate dis-tances by
determining the Time Difference of Arrival(TDoA) of these
signals[25,27-28]. In such a scheme, eachnode is equipped with a
speaker and a microphone, asillustrated in Fig.4. Some systems use
ultrasound whileothers use audible frequencies. The general
rangingtechnique, however, is independent of particular
hard-ware.
Fig.4. TDoA hardware model.
The idea of TDoA ranging is conceptually simple,as illustrated
in Fig.5. The transmitter first sends aradio signal. It waits for
some fixed internal of time,tdelay (which might be zero), and then
produces a fixedpattern of “chirps” on its speaker. When
receivershear the radio signal, they record the current
time,tradio, and then turn on their microphones. When their
microphones detect the chirp patter, they again recordthe
current time, tsound. Once they have tradio, tsound,and tdelay, the
receivers can compute the distance d tothe transmitter by
d =vradio · vsoundvradio − vsound · (tsound − tradio −
tdelay),
where vradio and vsound denote the speeds of radio andsound
waves respectively. Since radio waves travel sub-stantially faster
than sound in air, the distance can besimply estimated as d =
vsound · (tsound− tradio− tdelay).If the design is transmitting
radio and acoustic signalssimultaneously, i.e., tdelay = 0, the
estimation can befurther simplified as vsound · (tradio −
tsound).
Fig.5. TDoA computation model.
TDoA methods are impressively accurate under line-of-sight
conditions. For instance, it is claimed in [25]that distance can be
estimated with error no more thana few centimeters for node
separations under 3 meters.The cricket ultrasound system[27] can
obtain close tocentimeter accuracy without calibration over ranges
ofup to 10 meters in indoor environments.
Being accurate, TDoA systems suffer from highcost and are
constrained by the line-of-sight condition,which can be difficult
to meet in some environments.In addition, TDoA systems perform
better when theyare calibrated properly, since speakers and
microphonesnever have identical transmission and reception
charac-teristics. Furthermore, the speed of sound in air varieswith
air temperature and humidity, which introduce in-accuracy into
distance estimation. Acoustic signals alsoshow multi-path
propagation effects that may affect theaccuracy of signal
detection. These can be mitigated toa large extent using simple
spread-spectrum techniques,like those described in [29]. The basic
idea is to senda pseudo-random noise sequence as the acoustic
signaland use a matched filter for detection, instead of usinga
simple chirp and threshold detection.
Recently, researchers observe that two intrinsic un-certainties
in TDoA measuring process can contributeto the ranging inaccuracy:
the possible misalignmentbetween the sender timestamp and the
actual signalemission, and the possible delay of a sound signal
ar-rival being recognized at the receiver[30]. Indeed, manyfactors
can cause these uncertainties in a real system,
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278 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
such as the lack of real-time control, software delay,
in-terrupt handling delay, system loads, etc. These twodelays, if
not carefully controlled, can easily add upto several milliseconds
on average, which translates toseveral feet of ranging error.
BeepBeep[30], a recentlydesigned high-accuracy acoustic-based
ranging system,achieves the localization accuracy as good as one or
twocentimeters within a range of more than ten meters,which is so
far the best ranging result for off-the-shelfcell phones.
In conclusion, many localization algorithms useTDoA simply
because it is dramatically more accuratethan radio-only methods.
The tradeoff is that nodesmust be equipped with acoustic
transceivers in addi-tion to radio transceivers, significantly
increasing boththe complexity and the cost of the system.
2.1.2 Angle Measurement
Another approach for localization is the use ofangular estimates
instead of distance estimates. Intrigonometry and geometry,
triangulation is the processof determining the location of a point
by measuring an-gles to it from two known reference points at
either endof a fixed baseline, using the law of sines.
Triangulationwas once used to find the coordinates and sometimesthe
distance from a ship to the shore.
The Angle of Arrival (AoA) data is typically gath-ered using
radio or microphone arrays, which allow areceiver to determine the
direction of a transmitter.Suppose several (3∼4) spatially
separated microphoneshear a single transmitted signal. By analyzing
thephase or time difference between the signal arrivals atdifferent
microphones, it is possible to discover the AoAof the signal.
Those methods can obtain accuracy within a fewdegrees[31]. A
straightforward localization technique,involving three rotating
reference beacons at the boun-dary of a sensor network providing
localization for allinterior nodes, is described in [32]. A more
detailed de-scription of AoA-based triangulation techniques is
pro-vided in [33].
Unfortunately, AoA hardware tends to be bulkierand more
expensive than TDoA ranging hardware,since each node must have one
speaker and several mi-crophones. Furthermore, the need of spatial
separationbetween microphones is difficult to be accommodatedin
small size sensor nodes.
2.1.3 Area Measurement
If the radio or other signal coverage region can bedescribed by
a geometric shape, locations can be es-timated by determining which
geometric areas that anode is in. The basic idea of area estimation
is to
compute the intersection of all overlapping coverage re-gions
and choose the centroid as the location estimate.Along with the
increasing number of constraining areas,higher localization
accuracy can be achieved.
According to how area is estimated, we classify theexisting
approaches into two categories: single referencearea estimation and
multi-reference area estimation.
(a) Single Reference Area EstimationIn this case, constraining
areas are obtained accord-
ing to a single reference. For instance, the region of ra-dio
coverage may be upper-bounded by a circle of radiusRmax. In other
words, if node B hears node A, it knowsthat it must be no more than
a distance Rmax from A.If an unknown node hears from several
reference nodes,it can determine that it must lie in the geometric
regiondescribed by the intersection of circles of radius
Rmaxcentered at these nodes, as illustrated in Fig.6(a). Thiscan be
extended to other scenarios. For instance, whenboth lower bound
Rmin and upper bound Rmax can bedetermined, the shape of a single
node’s coverage isan annulus, as illustrated in Fig.6(c); when an
angu-lar sector (θmin, θmax) and a maximum range Rmax canbe
determined for some radio antennas, the shape for asingle node’s
coverage would be a cone with given angleand radius, illustrated in
Fig.6(d).
Fig.6. Area measurements.
Localization techniques using geometric regions arefirst
described by [34]. One of the nice features of thesetechniques is
that not only can the unknown nodes usethe centroid of the
overlapping region as a specific lo-cation estimate if necessary,
but also they can deter-mine a bound on the location error using
the size ofthis region. When the upper bounds on these regionsare
tight, the accuracy of this geometric approach canbe further
enhanced by incorporating “negative infor-mation” about which
reference nodes are not withinrange[35]. Although arbitrary shapes
can be potentially
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Yunhao Liu et al.: Location, Localization, and Localizability
279
computed in this manner, a computational simplifica-tion to
determine this bounded region is to use rect-angular bounding boxes
as location estimates. Thebounding-box algorithm is a
computationally efficientmethod to localize nodes given their
ranges to severalreferences. Essentially, it is assumed that each
node lieswithin the intersection of its reference bounding
boxes.
(b) Multi-Reference Area EstimationAnother approach of area
estimation is the appro-
ximate point in triangle (APIT)[36]. Its novelty lies inhow the
regions are defined. Actually, bounding tri-angles are obtained
according to any group of threereference nodes, rather than the
coverage of a singlenode.
APIT consists of two key processes: triangle inter-section and
point in triangle (PIT) test. Nodes areassumed to hear a fairly
large number of beacons. Anode forms some number of “reference
triangles”: Thetriangle formed by three arbitrary references. The
nodethen decides whether it is inside or outside a given tri-angle
by PIT test. Once this process is complete, thenode simply finds
the intersection of the reference tri-angles that contains it and
chooses the centroid as itsposition estimate, as illustrated in
Fig.6(b). In this pro-cess, APIT does not assume that nodes can
really rangeto these beacons.
The PIT test is based on geometry. For a given tri-angle with
points A, B, and C, a point M is outsidetriangle ABC, if there
exists a direction such that apoint adjacent to M is further/closer
to points A, B,and C simultaneously. Otherwise, M is inside
triangleABC. Unfortunately, given that typically nodes cannotmove,
an approximate PIT test is proposed based ontwo assumptions. The
first one is that the range mea-surements are monotonic and
calibrated to be compara-ble but are not required to produce
distance estimates.The second one assumes sufficient node density
for ap-proximating node movement. If no neighbor of M isfurther
from/closer to all three anchors A, B, and C si-multaneously, M
assumes that it is inside triangle ABC.Otherwise, M assumes it
resides outside this triangle.In practice, however, this
approximation does not re-alize the PIT test well. Nevertheless,
APIT providesa novel point of view to conduct localization based
onarea estimation.
2.1.4 Hop Count Measurements
Based on the observation that if two nodes can com-municate by
radio, their distance from each other is lessthan R (the maximum
range of their radios) with highprobability, many delicate
approaches are designed foraccurate localization. In particular,
researchers havefound “hop count” to be a useful way to compute
inter-node distances. The local connectivity informa-tion
provided by the radio defines an unweighted graph,where the
vertices are wireless nodes and edges repre-sent direct radio links
between nodes. The hop counthij between nodes si and sj is then
defined as the lengthof the shortest path from si to sj .
Obviously, the phys-ical distance between si and sj , namely, dij ,
is less thanR× hij , the value which can be used as an estimate
ofdij if nodes are densely deployed.
It turns out that a better estimate can be made ifwe know
nlocal, the expected number of neighbors pernode. As shown by
Kleinrock and Silvester[37], it ispossible to compute a better
estimate for the distancecovered by one radio hop:
dhop = R(1+e−nlocal−
∫ 1−1
e−(nlocal/π)arccos t−t√
1−t2dt).
Then dij ≈ hij × dhop. Experimental studies[38]show that the
equation above can be quite accuratewhen nlocal grows above 5.
However, when nlocal > 15,dhop approaches R, so the equation of
dhop becomesless useful. Nagpal et al.[38] demonstrate by
algorithmthat even better hop-count distance estimates can
becomputed by averaging distances with neighbors. Thisbenefit does
not appear until nlocal > 15; while, it canreduce hop-count
error down to as little as 0.2R.
Another method to estimate per-hop distance is toemploy a number
of reference nodes, as illustrated inFig.7(a). Since the locations
of reference nodes areknown, the pairwise distances among them can
be com-puted. Hence, if the hop count hij between two refer-ences
(si and sj) and the distance dij are available, theper-hop distance
can be estimated by dhop = dij/hij .
Due to the hardware limitations and energy con-straints of
wireless devices, hop count based localiza-tion approaches are
cost-effective alternatives to rang-ing based approaches. Since
there is no way to measurephysical distances between nodes,
existing hop-countbased approaches largely depend on a high density
ofseeds.
Most existing approaches, however, would failin anisotropic
network topologies, where holes existamong wireless devices, as
shown in Fig.7(b). Inanisotropic networks[39], the Euclidean
distance be-tween a pair of nodes may not correlate closely withthe
hop count between them because the correspondingshortest path may
have to curve around intermediateholes, leading to poor distance
estimation. Unfortu-nately, anisotropic networks are more likely to
exist inpractice for several reasons. First, in many real
appli-cations, sensor nodes/seeds can rarely be uniformly de-ployed
over the field due to the geographical obstacles.Second, even if we
assume that the initial sensor
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280 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
Fig.7. Hop count measurement. (a) Per-hop distance measure-
ment. (b) Distance mismatch.
network is isotropic, unbalanced power consumptionamong nodes
will easily create holes in the network.Recently, a distributed
method[40] has been proposedto detect hole boundary by using only
the connecti-vity information. Based on that work, REP[41] is
pro-posed to deal with the “distance mismatch” problem
inanisotropic networks.
2.1.5 Neighborhood Measurement
The radio connectivity measurement can be con-sidered economic
since no extra hardware is needed.Perhaps the most basic
positioning technique is thatof one neighbor proximity, involving a
simple decisionof whether two nodes are within the reception
rangeof each other. A set of reference nodes is placed inthe
network with some non-overlapping (or nearly non-overlapping)
sub-regions. Reference nodes periodicallyemit beacons including
their location IDs. Unknownnodes use the received locations as
their own location,achieving a course-grained localization. The
major ad-vantage of such a neighbor proximity approach is
thesimplicity of computation.
The neighborhood information can be more useful
when the density of reference nodes is sufficiently highso that
there are often multiple reference nodes withinthe range of an
unknown node[42]. Let there be k refer-ence nodes within the
proximity of the unknown node.As shown in Fig.8, we use the
centroid of the polygonconstructed by the k reference nodes as the
estimatedposition of the unknown node. This is actually a
k-nearest-neighbor approximation in which all referencenodes have
equal weights.
Fig.8. k-neighbor proximity.
This centroid technique has been investigated usinga model with
each node having a simple circular rangeR in an infinite square
mesh of reference nodes spaceda distance d apart[43]. It is shown
through simulationthat, as the overlap ratio R/d is increased from
1 to 4,the average error in localization decreases from 0.5d
to0.25d.
The k-neighbor proximity approach inherits themerit of
computational simplicity from the single neigh-bor proximity
approach; while at the same time, it pro-vides more accurate
localization results statistically.
2.1.6 Comparative Study and Directions of FutureResearch
A comparative study is presented in this subsectionfor existing
physical measurement approaches. Table 1provides an overview of
these approaches in terms of ac-curacy, hardware cost, and
environment requirements.All approaches have their own merits and
drawbacks,making them suitable for different scenarios.
Recent technical advances foster two novel ranging
Table 1. Comparative Study of Physical Measurements
Physical Measurements Accuracy Hardware Computa-
Cost tion Cost
Distance RSS Median Low Low
TDoA High High Low
Angle AoA High High Low
Area Single reference Median* Median* Median
Multi-reference Median* Median* High
Hop Count Per-hop distance Median Low Median
Neighborhood Single neighbor Low Low Low
Multi-neighbor Low Low Low
∗: depends on the diverse geometric constrains
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Yunhao Liu et al.: Location, Localization, and Localizability
281
approaches. Ultra-WideBand (UWB) is a radio tech-nology that can
be used at very low energy levels forshort-range high-bandwidth
communications by usinga large portion of the radio spectrum[44].
It has rela-tive bandwidth larger than 20% or absolute bandwidthof
more than 500 MHz. Such wide bandwidth offersa wealth of advantages
for both communications andranging applications. In particular, a
large absolutebandwidth offers high resolution with improved
rang-ing accuracy of centimeter-level.
UWB has a combination of attractive properties forin-building
location systems. First, it is a non-line-of-sight technology with
a range of a few tens of meters,which makes it practical to cover
large indoor areas; sec-ond, it is easy to filter the signal to
minimize the multi-path distortions that are the main cause of
inaccuracyin RF based location systems. With conventional
RF,reflections in in-building environments distort the di-rect path
signal, making accurate pulse timing difficult;while with UWB, the
direct path signal can be distin-guished from the reflections.
These properties providea good cost-to-performance ratio of all
available indoorlocation technologies.
The second promising technique is Chirp SpreadSpectrum (CSS)
designed by Nanotron Technologies[45]
and adopted by IEEE 802.15.4a. CSS is a cus-tomized application
of Multi-Dimensional Multiple Ac-cess (MDMA) for the requirements
of battery-poweredapplications, where the reliability of the
transmissionand low power consumption are of special importance.CSS
operates in the 2.45 GHz ISM band and achieves amaximum data rate
of 2 Mbps. Each symbol is trans-mitted with a chirp pulse that has
a bandwidth of80MHz and a fixed duration of 1µs.
Nanotron Technologies have developed a ToAmethod that employs a
ranging signal sent by a readerand an acknowledgement sent back
from the tag to can-cel out the requirements for clock
synchronization. Thissolution provides protection against
multi-path propa-gation and noise by its CSS modulation. To
eliminatethe effect of clock drift and offset, ranging
measure-ments are taken by both the tag and the reader to pro-vide
two measurements that can then be averaged. Thisranging result is
reasonably accurate with no more than1 meter error, even in the
most challenging environ-ments. The method is called Symmetric
Double SidedTwo Way Ranging, or SDS-TWR.
2.2 Network-Wide Localization
2.2.1 Computation Organization
This subsection defines taxonomy for localization al-gorithms
based on their computational organization.
Centralized algorithms are designed to run on a cen-tral machine
with powerful computational capabilities.Network nodes collect
environmental data and sendback to a base station for analysis,
after which the com-puted positions are delivered back into the
network.Centralized algorithms resolve the computational
limi-tations of nodes. This benefit, however, comes fromaccepting
the communication cost of transmitting databack to a base station.
Unfortunately, communicationgenerally consumes more energy than
computation inexisting network hardware platforms.
In contrast, distributed algorithms are designed torun in
network, using massive parallelism and inter-node communication to
compensate for the lack of cen-tralized computing power, while at
the same time to re-duce the expensive node-to-sink communications.
Dis-tributed algorithms often use a subset of the data tolocate
each node independently, yielding an approxima-tion of a
corresponding centralized algorithm where allthe data are
considered and used to compute the posi-tions of all nodes
simultaneously. There are two impor-tant categories of distributed
localization approaches.The first group, beacon-based distributed
algorithms,typically starts a localization process with beacons
andthe nodes in vicinity of beacons. In general, nodesobtain
distance measurements to a few beacons andthen determine their
locations. In some algorithms,the newly localized nodes can become
beacons to helplocating other nodes. In such iterative localization
ap-proaches, location information diffuses from beacons tothe
border of a network, which can be viewed as a top-down manner. The
second group of approaches per-forms in a bottom-up manner, in
which localization isoriginated in a local group of nodes in
relative coordi-nates. After gradually merging such local maps,
entirenetwork localization is achieved in global coordinates.
2.2.2 Centralized Localization Approaches
(a) Multi-Dimensional Scaling (MDS)Multi-Dimensional scaling
(MDS)[46] was originally
developed for use in mathematical psychology. The in-tuition
behind MDS is straightforward. Suppose thereare n points, suspended
in a volume. We do not knowthe positions of the points, but we know
the distancesbetween each pair of points. MDS is an O(n3)
algo-rithm that uses the law of cosines and linear algebra
toreconstruct the relative positions of the points based onthe
pairwise distances. The algorithm has three stages:
1) Generate an n × n matrix M , whose (i, j) entrycontains the
estimated distance between nodes i and j(simply run Floyd’s
all-pairs shortest-path algorithm).
2) Apply classical metric-MDS on M to determinea map that gives
the locations of all nodes in relative
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282 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
coordinates.3) Transform the solution into global
coordinates
based on some number of fixed anchor nodes.MDS performs well on
RSS data, getting perfor-
mance on the order of half the radio range when theneighborhood
size nlocal is higher than 12[47]. The mainproblem with MDS,
however, is its poor asymptotic per-formance, which is O(n3) on
account of stages 1 and 2.
(b) SemiDefinite Programming (SDP )The semidefinite programming
(SDP) approach was
pioneered by Doherty et al.[34] In their algorithm, ge-ometric
constraints between nodes are represented aslinear matrix
inequalities (LMIs). Once all the con-straints in the network are
expressed in this form, theLMIs can be combined to form a single
semidefiniteprogram, which is solved to produce a bounding
regionfor each node. The advantage of SDP is its eleganceon concise
problem formulation, clear model represen-tation, and elegant
mathematic solution.
Solving the linear or semidefinite program has to bedone
centrally. The relevant operation is O(k2) for an-gle of arrival
data, and O(k3) when radial (e.g., hopcount) data is included,
where k is the number of con-vex constraints needed to describe the
network. Thus,the computation complexity of SDP is likely to
precludeitself in practice.
Unfortunately, not all geometric constraints can beexpressed as
LMIs. In general, only constraints thatform convex regions are
amenable to representation asan LMI. Thus, AoA data can be
represented as a trian-gle and hop count data can be represented as
a circle,but precise range data cannot be conveniently
repre-sented, as rings cannot be expressed as convex con-strains.
This inability to accommodate precise rangedata might prove to be a
significant drawback.
2.2.3 Distributed Localization Approaches
(a) Beacon Based LocalizationBeacon based localization
approaches utilize esti-
mates of distances to reference nodes that may be se-veral hops
away[48-49]. These distances are propagatedfrom reference nodes to
unknown nodes using a basicdistance-vector technique. Such a
mechanism can beseen as a top-down manner due to the progressive
pro-pagation of location information from beacons to anentire
network. There are three types as follows.
1) DV-hop: In this approach, each unknown nodedetermines its
distance from various reference nodes bymultiplying the least
number of hops to the referencenodes with an estimated average
distance per hop thatdepends upon the network density.
2) DV distance: If inter-node distance estimatesare directly
available for each link in the graph, the
distance-vector algorithm is used to determine the dis-tance
corresponding to the shortest distance path be-tween the unknown
nodes and reference nodes.
3) Iterative localization: One variant of above ap-proaches is
indirect use of beacon nodes. Initially anunknown node, if
possible, is located based on its neigh-bors by multilateration or
other positioning techniques.After being aware of its location, it
becomes a refer-ence node to localize other unknown nodes in the
sub-sequent localization process. This step continues iter-atively,
gradually turning the unknown nodes to theknown. The process of
iterative localization is illus-trated in Fig.9.
Fig.9. Iterative localization.
Iterative trilateration only involves local
information(information within neighborhood) and accordingly
re-duces communication cost. Nevertheless, the use of lo-calized
unknown nodes as reference nodes inherentlyintroduces substantial
cumulative error. Some workscharacterize the error propagation in
multihop locali-zation approaches and make efforts to control
erroraccumulation[50-51].
Experimental studies show that multilateration-based algorithms
require an average node degree be-yond 10 to properly localize most
of the nodes in a ran-domly deployed network[52]. When the average
degreeis below 8, iterative multilateration will fail for most
ofthe nodes, since it makes the nodes form a chain of de-pendence
and a single-point failure on one node wouldlead to further
failures on a set of subsequent nodes.To make localization
applicable for sparse networks,Sweeps[53] partially relaxes the
requirement of node de-pendence. In contrast to the traditional
unique positioncomputation, Sweeps introduces a novel concept of
fi-nite localization which locates a target node to a setof
possible positions, called candidate positions. Finitelocalization
guarantees that the ground truth positionof a node is one of its
candidate positions. Further,Sweeps adopts a new positioning
scheme, called bilate-ration, to compute the candidate positions of
a node byutilizing only two ranging measurements. As shown
inFig.10, bilateration produces two candidate positionsfor a node
and one of them is the ground truth posi-tion. Similar to
multilateration, the finitely localizednode, called swept node, can
act as a reference node tolocalize other nodes. The only difference
is that all can-didate positions of the swept node are enumerated
for
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Yunhao Liu et al.: Location, Localization, and Localizability
283
the location computation of the target node. Moreover,after each
bilateration, Sweeps checks the consistencyamong the candidate
position sets and deletes those in-compatible items. Under this
mechanism, Sweeps canlocate a large proposition of theoretically
localizablenodes in a network. However, the worst case computa-tion
complexity of this design grows exponentially withthe number of
nodes.
Fig.10. Bilateration. (a) Measuring distance to 2 reference
nodes.
(b) Bilateration creates two possible locations.
(b) Coordinate System StitchingCoordinate system stitching is a
different way to ad-
dress the same problem. It has attracted a lot of re-search
efforts recently[48,52,54]. It works in a bottom-upmanner, in which
localization is originated in a localgroup of nodes in relative
coordinates. By graduallymerging such local maps, it finally
achieves entire net-work localization in global coordinates, as
illustrated inFig.11.
Fig.11. Coordinate system stitching.
Coordinate system stitching works as follows:1) Split the
network into small overlapping sub-
regions. Very often each sub-region is simply a singlenode and
its one-hop neighbors.
2) For each sub-region, compute a “local map”,which is
essentially an embedding of the nodes in thesub-region into a
relative coordinate system.
3) Finally, merge sub-regions using a coordinate sys-tem
registration procedure. Coordinate system regis-tration finds a
rigid transformation that maps pointsin one coordinate system to a
different coordinate sys-tem. Thus, step 3) places all the
sub-regions into asingle global coordinate system. Many algorithms
dothis step suboptimally, since there is a closed-form, fastand
least-square optimal method of registering coordi-nate system.
Fig.12. Robust quadrilateral.
Moore et al.[52] outline an approach that producesmore robust
local maps. Rather than using three ar-bitrary nodes, they use
“robust quadrilateral” (robustquads) to define a map. As shown in
Fig.12, a robustquad consists of four subtriangles (∆ABC , ∆ADC
,∆ABD , ∆BCD) that satisfy:
b× sin2(θ) > dmin,
where b is the length of the shortest side, θ is the small-est
angle, and dmin is a predetermined constant basedon average
measurement error. The idea is that thepoints of a robust quad can
be placed correctly withrespect to each other (i.e., no
“flips”[55]). Moore et al.demonstrate that the probability of a
robust quadrila-teral experiencing internal flips given zero mean
Gaus-sian measurement error can be bounded by setting
dminappropriately. In effect, dmin filters out quads that havetoo
much positional ambiguity to be localized with con-fidence. The
appropriate level of filtering is based onthe amount of uncertainty
in distance measurements.Unfortunately, coordinate system stitching
suffers fromerror propagation caused by local map stitching.
Mooreet al. prove the probability of their algorithm construct-ing
correct local maps and prove error lower bound onthe local map
positions. Furthermore, these techniqueshave a tendency to orphan
nodes, either because theycould not be added to a local map or
because their lo-cal map failed to overlap sufficiently with
neighboringmaps. Moore et al. argue that this is acceptable
be-cause the orphaned nodes are the nodes most likely todisplay
high error. They point out that “for many ap-plications, missing
localization information for a knownset of nodes is preferential to
incorrect information foran unknown set”. However, this answer may
not besatisfactory for some applications, many of which can-not use
nodes without locations for sensing, routing,
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284 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
target tracking, or other tasks.A more general form of
coordinate system stitch-
ing is the component based localization[56]. A compo-nent is
defined as a group of nodes that form a rigidstructure. Using rigid
components as basic units, thealgorithm[56] merges and localizes
components throughinter-component distance measurements and anchor
in-formation.
As shown in Fig.13 three inter-component distancemeasurements
constrain the relative geometric relation-ship between two
components A and B, both of whichare adjacent to two anchors. From
the perspective ofeach single node, none of them has (at least) two
neigh-boring anchors. In contrast, from the perspective
ofcomponents, component A and component B can bemerged into a
bigger component, which is localizableby referring to the four
anchors. Next, all nodes in thetwo components are localized. The
component-basedlocalization algorithms are applicable for sparse
net-works. Similar to Sweeps, this design cannot
guaranteeterminating in polynomial time either, which is a
majordrawback.
Fig.13. Component-based localization.
Coordinate system stitching techniques are quitecompelling. They
are inherently distributed, since sub-region and local map
formation can trivially occur inthe network and stitching is easily
formulated as a peer-to-peer algorithm.
2.2.4 Comparative Study and Directions of FutureResearch
(a) Beacon NodesBeacon nodes (a.k.a. seeds or anchors) are
necessary
for localizing a network in the global coordinate system.Beacon
nodes have no difference from ordinary networknodes except knowing
their global locations as a priori.This knowledge can be
hard-coded, or acquired throughsome extra hardware like a GPS
receiver.
Beacon configuration has significant impact on lo-calization.
Existing work finds that higher localizationaccuracy can be
achieved if beacons are placed in a con-vex hull around the
network. Placing additional bea-cons in the center of the network
is also helpful. Thus,it is necessary for system designers to plan
the beaconlayout before deploying a network.
(b) Node DensityMany localization algorithms are sensitive to
node
density. For instance, hop-count-based schemes gene-rally
require high node density so that the hop countapproximation for
distance is accurate. Similarly, al-gorithms that depend on beacon
nodes fail when thebeacon density is not sufficiently high in a
specific re-gion. Thus when designing or analyzing an algorithm,it
is important to consider its requirement on node den-sity, since
high density may not be always true.
(c) AccuracyGiven a localization algorithm, location
accuracy
shows how well the computed locations match with thephysical
positions of the nodes. To be specific, locationaccuracy is defined
as the expected Euclidean distancebetween the location estimate and
the actual locationof an unknown node, while location precision
indicatesthe percentage of the results satisfying a
pre-definedaccuracy requirement.
For a given localization result, location accuracytrades off
with location precision. If we relax the ac-curacy requirement, we
can increase precision, and viceversa. Thus, we must put these two
metrics in a com-mon framework for comparison. We can fix
locationprecision, say 95%, and evaluate the localization
algo-rithms based on the corresponding accuracy achieve-ments.
The error propagation demonstrates how locationaccuracy varies
with the increase of measurement er-ror. Intuitively, localization
error is linear with mea-surement error. However, it is not true
for many local-ization systems, especially for those sequential
locali-zation algorithms, such as trilateration and bilatera-tion.
Nodes with large location errors would contami-nate their
neighbors’ estimates. In this scenario, mea-surement error is no
longer the only factor contributingto localization error.
(d) CostIn general, the cost of a localization system
includes
hardware cost and energy cost. Hardware cost consistsof three
parts: node density, beacon density, and mea-surement equipment.
Usually, expensive equipmentsprovide more accurate measurements. A
localizationprocedure often involves inter-node measurement,
com-putation and communication, among which communi-cation consumes
most energy. This is why distributedalgorithms are often more
compelling than centralizedalgorithms.
After years of extensive study on this topic, manylocalization
solutions are presented. Table 2 presentsan overview of typical
approaches in terms of accu-racy, node density, beacon percentage,
computationcost, communication cost, and error propagation.
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Yunhao Liu et al.: Location, Localization, and Localizability
285
Table 2. Comparative Study of Localization Algorithms
Localization Algorithm Accuracy Node Beacon Computation
Communication Error
Density Percentage Cost Cost Propagation
Centralized MDS High Low Low High High Low
SDP High High Median High High Low
Distributed Beacon based Low High High Low Low High*
Coordinate stitching Low High Low Median Median High
∗: in case of iterative localization
In conclusion, a number of typical localization ap-proaches are
surveyed and evaluated with various me-trics in this section. All
approaches have their ownmerits and drawbacks, making them suitable
for differ-ent applications. Hence, the design of a localization
al-gorithm should sufficiently investigate application pro-perties,
as well as take into account algorithm gener-ality and flexibility.
In present and foreseeable futurestudy, obtaining a Pareto
improvement is a major chal-lenge. That is, increasing the
performance of one of themetrics without degradation on others.
In all localization algorithms discussed above, nodesshould
participate actively during a localization pro-cess, i.e., sending
or receiving radio signals, or mea-suring physical data. For some
applications, however,the to-be-locate objects cannot participate
in localiza-tion and it is also difficult to attach networked
nodesto them. One typical application is intrusion detection,in
which it is impossible and unreasonable to equip in-truders with
locating devices. To tackle this issue, re-cently a novel concept
of Device-Free Localization, alsocalled Transceiver-Free
Localization, is proposed[57-58].Device-free localization is
envisioned to be able to de-tect, localize, track, and identify
entities free of devices,and works by processing the environment
changes col-lected at scattering monitoring points. Existing
workfocuses on analyzing RSSI changes, and often suffersfrom high
false positives. How to design a device-freelocalization system
which can provide accurate loca-tions is a challenging and
promising research problem.
2.3 Error Control for Network Localization
2.3.1 Noisy Distance Measurement
Many localization algorithms are range-based andadopt distance
ranging techniques, in which measur-ing errors are inevitable.
Generally, these errors canbe classified into two categories:
extrinsic and intrin-sic. The extrinsic error is attributed to the
physicaleffects on the measurement channel, such as the pre-sence
of obstacles, multipath and shadowing effects,and the variability
of the signal propagation speed dueto environmental dynamics. On
the other hand, theintrinsic error is caused by limitations of
hardware and
software. While the extrinsic one is more unpredictableand
challenging during real deployments, the intrin-sic one causes many
complications when using multi-hop measurements to estimate node
locations. Resultsfrom field experiments demonstrated that even
rela-tively small ranging errors can significantly amplify theerror
of location estimates[52]; thus, dealing with sucherrors is an
essential issue for high-accuracy localizationalgorithms.
(a) Errors in Distance MeasurementsTable 3 lists the typical
measuring (intrinsic) error
of a range of nowadays ranging techniques: TDoA, RSSin
AHLoS[25], Ultra Wideband system[59], RF Time ofFlight ranging
systems[60], and Elapsed Time betweenthe two Time of Arrival (EToA)
in BeepBeep[30]. Ingeneral, RF-based techniques, e.g., RSS, UWB and
RFToF, can achieve the meter-level accuracy in a rangeof tens of
meters. Time-related methods have more ac-curate results in the
order of centimeters, but requireextra hardware and energy
consumption.
On the other hand, extrinsic errors are caused by en-vironmental
factors or unexpected hardware malfunc-tion, leaving difficulties
on characterizing them. We willbriefly discuss the state-of-the-art
works on controllingthe intrinsic and extrinsic errors in the
following sub-sections of location refinement and robust
localization,respectively.
Table 3. Measurement Accuracy of Different
Ranging Techniques
Technology System Accuracy Range
TDoA AHLoS 2 cm 3∼10mRSS AHLoS 2∼4m 30∼100mUWB PAL UWB 1.5m
N/A
RF ToF RF ToF 1∼3m 100mEToA BeepBeep 1∼2 cm 10m
(b) Negative Impact of Noisy Ranging ResultsErrors in distance
ranging make localization more
challenging in the following three aspects:• Uncertainty. Fig.14
illustrates an example of tri-
lateration under noisy ranging measurements. Trila-teration
often meets the situation that the three circlesdo not intersect at
a common point. In other words,
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286 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
Fig.14. Trilateration under noisy ranging measurements.
there does not exist any position satisfying all
distanceconstraints.• Non-Consistency. In many cases, a single
node
has many reference neighbors. Any subgroup of them(no less than
three) can locate this node by multilat-eration. The computed
result, however, is varying ifdifferent groups of references are
chosen, resulting innon-consistency. Thus, when alternative
references areavailable, it is a difficult task to determine which
com-bination of references provides the best results.• Error
Propagation. The results of a multihop lo-
calization process are based on a series of single
hopmultilaterations in an iterative manner[25]. In such aprocess,
errors, coming from each step of multilatera-tion, propagate and
accumulate[50-51].
2.3.2 Error Characteristics of Localization
Localization error is a function of a wide range ofnetwork
configuration parameters, including the num-bers of beacons, the
density of node deployment, net-work topology, etc., which
constitute a complicated sys-tem. Understanding the error
characteristics of loca-lization is one essential step towards
controlling errors.The Cramer Rao Lower Bound (CRLB)[61] providesa
means for computing a lower bound on the covari-ance of any
unbiased location estimate that uses dis-tance measurements. In
addition, CRLB can serve as abenchmark for localization algorithms:
if the bound isclosely achieved, there is little gain to continue
workingon improving the algorithm accuracy. Furthermore,
thedependence of CRLB on network parameters helps tounderstand the
error characteristics of network local-ization.
(a) What is CRLBThe Cramer Rao Lower Bound (CRLB) is a clas-
sic result from statistics that gives a lower boundon the error
covariance for an unbiased estimate ofparameter[61]. This bound
provides a useful guidelineto evaluate various estimators. One
important andsurprising advantage of CRLB is that we can calcu-late
the lower bound without ever considering any par-ticular estimation
method. The only thing needed is
the statistical model of the random observations, i.e.,f(X|θ),
where X is the random observation, and θ isthe parameter to be
estimated. Any unbiased estimatorθ̂ must satisfy
Cov(θ̂) > {−E[∇θ(∇θ ln f(X|θ))T]}−1 (1)
where Cov(θ̂) is the error covariance of the estimator,E[·]
indicates expected value, and ∇θ is the gradientoperator with
respect to θ.
The CRLB is limited to unbiased estimators thatprovide estimates
equal to the ground truth if averagedover enough realizations. In
some cases, however, bi-ased estimators can achieve both a variance
and a meansquared error that are below the CRLB.
(b) CRLB for Multihop LocalizationIn network localization, the
parameter vector θ of
interest consists of the coordinates of nodes to be local-ized,
given by θ = [x1, y1, x2, y2, . . . , xL, yL]T, where Lis the
number of nodes to be localized. The observationvector X is formed
by stacking the distance measure-ments d̂ij . Let M denote the size
of X. We assume thedistance measurement are Gaussian[52,62], so the
pdf ofX is vector Gaussian. According to (1), we find thatCRLB = {
1σ2 [G′(θ)]T[G′(θ)]}−1, where σ2 is the vari-ance of each distance
measurement error, and G′(θ) isthe M × 2L matrix whose mn-th
element is
G′(θ)mn =
xi − xjdij
, if θn = xi,
xj − xidij
, if θn = xj ,
yi − yjdij
, if θn = yi,
yj − yidij
, if θn = yj ,
0, otherwise.
(2)
The above result on CRLB is with the assumptionthat the location
information of beacons is exact. Whenthe beacon nodes have location
uncertainty, we canalso characterize localization accuracy using a
covari-ance bound that is similar to CRLB. Both the twobounds are
tight in the sense that localization algo-rithms achieve these
bounds in case of highly accurateranging measurements. In addition,
according to (2),the CRLB can be computed analytically and
efficiently,avoiding the need for expensive Monte-Carlo
simula-tions. The computational efficiency facilitates to
studylocalization performance of large-scale networks.
(c) CRLB for One-Hop LocalizationOne-hop multilateration is the
source of the location
error that could be amplified by the iterative fashion of
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Yunhao Liu et al.: Location, Localization, and Localizability
287
network localization. The CRLB for multilateration ex-actly
demonstrates how measurement errors and nodegeometry affect
location accuracy.
Consider the one-hop localization problem: there arem reference
nodes v1, v2, . . . , vm and one node v0 to belocalized. From (1)
and (2), we obtain
σ20 = σ2m
[ m−1∑
i=1
m∑
j>i
sin2 αij]−1
(3)
where σ20 is the variance of the estimate location of v0,αij is
the angle between each pair of reference nodes(i, j). According to
(3), the uncertainty of location es-timate consists of two parts:
the ranging error (σ2)and the geometric relationship of references
and to-be-localized nodes (αij). Eliminating the impact ofranging
errors, the error amplification effect caused bythe node geometry
has been demonstrated as the Geo-graphic Dilution of Precision
(GDoP)[63], which is de-fined as σ0/σ.
To gain more insights of GDoP, we consider a sim-plified case of
multilateration, where the to-be-localizednode v0 is put at the
center of a circle and m = 3 ref-erence nodes v1, v2, v3 lie on the
circumference of thatcircle, setting all references the same
distance to v0, asshown in Fig.15(a). Fixing v1 at β1 = 0, GDoP
be-comes a function of the locations of v2 and v3, denoted
Fig.15. The impact of node geometry on the accuracy
multilat-
eration. (a) Design of experiments. (b) 3D plot of 1/GDoP.
by β2, β3 ∈ [0, 2π] respectively. We plot the GDoP inFig.15(b)
and conclude that different geometric formsof multilateration
provide different levels of localiza-tion accuracy. In particular,
in this circular trilatera-tion, the highest location accuracy
would be achieved ifreference nodes are evenly separated, namely,
β1 = 0,β2 = 2π/3, and β3 = 4π/3.
2.3.3 Location Refinement
Since localization is often conducted in a distributedand
iterative manner, error propagation is considered asa serious
problem, in which nodes with inaccurate lo-cation estimates
contaminate the following localizationprocess based on them.
(a) Framework of Location RefinementTo deal with error
propagation, a number of location
refinement algorithms have been proposed. In general,they are
composed of three major components[62]:• Node Registry. Each node
maintains a registry
which contains the node location estimate and the cor-responding
estimate confidence (uncertainty).• Reference Selection. When
redundant references
are available, based on an algorithm-specified strategy,each
node selects the reference combination achievingthe highest
estimate confidence (lowest uncertainty) tolocalize itself.•
Registry Update. In each iteration, if higher esti-
mate confidence (lower uncertainty) is achieved, a nodeupdates
its registry and broadcasts this information toits neighbors.
Algorithm 1 outlines a framework of location refine-ment, in
which how to select appropriate reference com-binations is
critical. Different strategies lead to differ-ent location
refinement algorithms.
Algorithm 1. A Framework of Location Refinement
1: Each node holds the tuple (p, e), where p is the nodelocation
estimate, e is the corresponding estimateconfidence
(uncertainty).
2: Initialization step (optional):
Each node computes an initialized location estimate.
3: In each iteration, nodes update their registries.
do
for all to-be-localized node t doexamine local neighborhood
N(t)select the best reference combination and computethe estimate
location p̂t and confidence êt decidewhether to update the
registry of t with the newtuple
while the termination condition is not met.
(b) Metrics for Location RefinementAlthough GDoP characterizes
the effects of node
geometry on location estimate, it cannot be directly
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288 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
applied due to the need of the ground truth location ofeach
node. This is a challenging issue and has attracteda lot of
attentions.
Savarese et al.[64] introduce a confidence associatedwith each
node’s location, and weights multilaterationresults based on such
confidence in the one-hop loca-lization procedure. The estimate
confidence is definedas follows. Beacons immediately start off with
confi-dence 1; to-be-localized nodes begin with a low con-fidence
and raise their confidences at subsequent re-finement iterations.
In each iteration, a node choosesthose reference nodes that will
raise its confidence tolocalize itself, and sets its confidence to
the average ofthose references’ confidences after a successful
multi-lateration. This strategy is based on the intuition thatthe
estimated locations of nodes close to beacons aremore reliable, but
puts littler emphasis on the effects ofgeometry on location
estimate.
Through analyzing how ranging errors and referencelocation
errors affecting localization, Liu et al.[62] de-sign a location
refinement scheme with error manage-ment. Each node maintains
information (p, e), wherep is the estimated location, and e is the
correspond-ing estimate error, a metric reflecting the level of
un-certainty. At the first beginning, each beacon is ini-tialized
with a registry (beacon loc, 0), and the to-be-localized nodes are
initialized as (unknown loc, ∞). In-stead of using the traditional
Least Square (LS) solu-tion (ATA)−1ATb mentioned previously, a
robust LS(RLS) solution is adopted, p̂ = (ATA + CA)−1ATb,where CA =
E[∆AT ·∆A] is the covariance matrix ofperturbation of ∆A (the
perturbation of A). Based onthe RLS solution, the estimated
location error causedby noisy distance measurements can be
expressed by
E‖e∆b‖2 = E‖(ATA + CA)−1AT∆b‖2 (4)
where ∆b is the perturbation of b due to ranging er-rors.
Similarly, the error due to location uncertainty ofreferences
is
E‖e∆a‖2 = E‖(ATA + CA)−1B∆a‖2 (5)
where a = (a11, a21, . . . , an1, a12, a22, . . . , an2)T, a
vec-tor rearranging elements in matrix A, ∆a is the per-turbation
of a because of location uncertainty, and Bis a matrix satisfying
ATb = Ba. The total locationerror is the summation of these two
terms, as theyare assumed to be uncorrelated, i.e., ê = E‖e∆b‖2
+βE‖e∆a‖2, where β is a parameter to compensate forthe
over-estimation of the error due to a. Small valueof β works well
in practice[62].
By defining Quality of Trilateration (QoT)[55], theaccuracy of
trilateration can be characterized, enabling
the comparison and selection among various geomet-ric forms of
trilateration. Assuming some probabilitydistribution of ranging
errors, probability tools are ac-cordingly applied to quantify
trilaterations. The largevalue of QoT indicates the estimate
location is, withhigh probability, sufficiently close to the real
location.Similar to [64], each node maintains a confidence
asso-ciated with its location estimate. Let t = Tri(s, {si,i = 1,
2, 3}) denote a trilateration for s based on threereference nodes
si and Q(t) be the quality of t. Theconfidence of s (based on t) is
computed according tothe confidences of references C(si):
Ct(s) = Q(t)3∏
i=1
C(si). (6)
In each iteration, a to-be-localized node chooses
thetrilateration that achieves the highest confidence to lo-calize
itself. Different from the design in [64], whichonly takes the
reference nodes reliability into account,QoT also considers the
geometry of nodes when com-puting confidence. Compared to least
square based ap-proaches, QoT provides additional information that
in-dicates how accurate a particular trilateration is.
Suchdifference enables QoT the ability of distinguishing
andavoiding poor trilaterations that are of much
locationuncertainty.
2.3.4 Robust Localization
Compared with intrinsic errors, extrinsic errorsare more
unpredictable and usually caused by non-systematic factors.
Especially in some cases, the errorscan be extremely large due to
hardware malfunction orfailure and adversary attacks. These severe
errors canbe seen as outliers of measurements that
significantlydeteriorate localization accuracy. Dealing with
outliersrecently becomes a hot research topic.
We classify existing outlier-resistant works into twomajor
categories: explicitly sifting and implicitly de-emphasizing. The
explicitly sifting methods are gene-rally based on the intuition
that normal ranging mea-surements are compatible while an outlier
is likely tobe inconsistent with other normal and outlier
rangings.Through examining the inconsistency, we can identifyand
reject outlier measurements. In contrast, the im-plicitly
de-emphasizing methods do not accept or rejecta localization result
by fixing a threshold; instead theyemploy robust statistics
methods, such as high break-down point estimators and influence
functions, to miti-gate the negative effects of outliers.
Based on the understanding that when there are re-dundant
geometric constraints, there must be some in-consistency between
outlier ranging results and normal
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Yunhao Liu et al.: Location, Localization, and Localizability
289
ranging results, Liu et al.[65] use the mean square errorζ2 of
the distance measurements based on the estimatedlocations as an
indicator of the level of inconsistency,i.e.,
ζ2 =1m
m∑
i=1
(δi − ‖p̄0 − pi‖)2 (7)
where m is the number of location references {(pi, δi),i = 1, 2,
. . . , m} and p̄0 is the estimated location. Itfirst estimates the
location with the LS-based method,and then assesses whether the
estimated location couldbe derived from a set of consistent
location referencesbased on ζ2. If positive, it accepts the
estimation result;otherwise, it identifies and removes the most
inconsis-tent location reference and repeats the above process.
According to robust statistics, the least squares al-gorithm is
sensitive to outliers, since its breakdownpoint is zero. One of the
most commonly used highbreakdown point (close to 50%) statistics
algorithms isthe method of least median of squares (LMS), which
isadopted in [66]. It estimates the location using
p̄0 = arg minp0
med i(δi − ‖p0 − pi‖)2. (8)
LMS-based localization first randomly draws M sub-sets of size m
from the available location references (Mand m are specified by
application requirements), andthen computes the estimated location
and the residuecorresponding to each subset. The final estimated
lo-cation is selected according to (8).
Also deriving inspiration from robust statistics, therecent work
SISR[67] uses a residual shaping influencefunction to de-emphasize
the “bad nodes” and “badlinks” during the localization procedure.
To overcomethe non-robustness to outliers of LS-based methods,
in-stead of optimizing the sum of squared residues (as il-lustrated
in Fig.16(a)), i.e., F =
∑i,j r(i, j)
2, wherer(i, j) is the residue, SISR solves the optimization
prob-lem of F =
∑i,j s(i, j), where
s(i, j) ={
αr(i, j)2, if |r(i, j)| < τ ;ln(|r(i, j)| − u)− v, otherwise,
(9)
and u = τ − 12ατ , v = ln(
12ατ
) − ατ2, α and τ are pa-rameters to be configured to control the
shape of theinfluence function. Fig.17 plots a comparison
betweenthe standard squared residual used in conventional
leastsquares and the shaped residual defined by (9). We cansee that
the shaped residual function is shaped like a“U” near 0 and like
“wings” for value greater than τ ,the reason why SISR can discount
outliers from thelocalization procedure (as illustrated in
Fig.16(b)).
Although many noise-resistant solutions have been
Fig.16. Two possible localization solutions of nodes A, B, C,
D,
and E, where E has large measurement errors. Squares
indicate
the ground truth location, and circles the computed
localization
solution. (a) Even; the measurement error from E is
amortized
over A, B, C, and D. (b) Uneven; solutions for A, B, C, and
D
are accurate, but that for E is very inaccurate.
Fig.17. Comparison between the standard squared residual
used
in conventional least squares and the shaped residuals used
in
SISR with α = 4 and τ = 1.
proposed, they often emphasize on dealing with noisydistance
rangings, ignoring the use of multimodal mea-surements. In a design
based on multimodal measure-ments, the measuring data of distance,
angle, and/or
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290 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
time difference are used simultaneously, providing morerobust
localization results. Recently, some progressfrom computational
geometry suggests an efficient al-gorithm that can realize graphs
with provable errorbounds based on distance and angle information.
Thisresult reveals the great potential of multimodal mea-surements
to control localization error, although its per-formance in
practical systems is still unknown. Withthe rapid development of
integrated circuits, multi-modal measurement has been available on
many wire-less devices, especially sensor motes. Designing
robustalgorithms based multimodal measurements is anotherpromising
research issue.
Another direction of future research is detecting andsifting
outlier ranging results. Typically, the quadri-lateral structure is
proposed for outlier detection[52].But such a method is infeasible
for sparse or mode-rate networks. What is the theoretical
foundation ofquadrilateral-based methods? Can we find any otherkind
of structure being able to detect outliers? Howto sufficiently
exploit redundant information from dis-tance constraints[68]? To
the best of our knowledge, allthese issues are not
well-studied.
3 Localizability
3.1 Network Localizability
Based on distance ranging techniques, the groundtruth of a
wireless ad-hoc network can be modeled bya distance graph G = (V,
E), where V is the set ofwireless communication devices (e.g.,
laptops, RFIDtags, or sensor nodes) and there is an un-weighted
edge(i, j) ∈ E if the distance between vertices i and j canbe
measured or both of them are at known locations,e.g., beacon nodes.
Associated with each edge (i, j), weuse a function d(i, j): E → R
to denote the measureddistance value between i and j.
An essential question occurs as to whether or not anetwork is
localizable given its distance graph. This iscalled the network
localizability. A graph G = (V, E)with possible additional
constraints I (such as theknown locations of beacon nodes) is
localizable if thereis a unique location p(v) of every node v such
thatd(i, j) = ‖p(i)−p(j)‖ for all links (i, j) in E and the
con-straint I is preserved, where ‖ · ‖ denotes the
Euclideandistance in the 2D plane. Different from localizationthat
determines locations of wireless nodes, localizabi-lity focuses on
the location-uniqueness of a network.
On one hand, localizability assists localization fun-damentally
and importantly. As previously mentioned,localization often
consumes a large amount of compu-tational resource and makes sense
only when networksare localizable. Hence, testing localizability
before
localization can save unnecessary and meaningless com-putation,
as well as accompanying power consumption.
On the other hand, being aware of localizability is ofgreat
benefit to many aspects of network operation andmanagement,
including topology control, network de-ployment, mobility control,
power scheduling, and geo-graphic routing, as shown in Fig.18.
Taking deploymentadjustment as an example, many measurements
(e.g.,augmenting communication range, increasing node orbeacon
density, etc.) can be taken to improve thosenon-localizable
networks to be localizable, which can beeffectively guided by the
results of localizability testing.
Fig.18. Localizability can assist network operation and
manage-
ment.
Although the network localizability is given birth bythe
proliferation of wireless ad-hoc/sensor networks, theproblem of
unique graph realization has attracted a lotof efforts made by
researchers from different literaturesover 30 years. An obvious
requirement for a networkto be localizable is the network
connectivity: when anetwork is densely connected, it is more likely
to beuniquely localizable; otherwise, it may fail the
localiz-ability testing. Besides dense connectivity,
researchersalso point out other requirements for localizable
net-works, which will be discussed in the next subsection.
3.2 Graph Rigidity
3.2.1 Globally Rigid Graphs
Previous studies have shown that the networklocalizability
problem is closely related to graphrigidity[69-72].
A realization of a graph G is a function p thatmaps the vertices
of G to the points in Euclideanspace (this study assumes
2-dimension space). Gener-ally, realizations are referred to the
feasible ones thatrespect the pairwise distance constraints between
a pairof vertices i and j if the edge (i, j) ∈ E. That is to
say,d(i, j) = ‖(p(i) − p(j)‖ for all (i, j) ∈ E. Two realiza-tion
of G are equivalent if they are identical under tri-vial variation
in 2D plane: translations, rotations, andreflections. A distance
graph G has at least one fea-sible realization which represents the
ground truth ofthe corresponding network. Formally, G is
embeddablein 2D space and all pairwise distances are
compatible,
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Yunhao Liu et al.: Location, Localization, and Localizability
291
i.e., satisfying the triangle inequality. We assume G
isconnected and has at least 4 vertices in the
followinganalysis.
A graph is called generically rigid if one cannot con-tinuously
deform any of its realizations in the planewhile preserving
distance constraints[70,72]. A graph isgenerically globally rigid
if it is uniquely realizable un-der translations, rotations, and
reflections. A realiza-tion is said to be generic if the vertex
coordinates arealgebraically independent[70]. Since the set of
genericrealizations is dense in the space of all realizations,
weomit this word for simplicity hereafter.
There are several distinct manners in which thenon-uniqueness of
realization can appear, as shown inFig.19. A graph that can be
continuously deformedwhile still satisfying all the constraints is
said to beflexible; otherwise it is rigid. Hence, rigidity is a
nec-essary condition for global rigidity. Rigid graphs, how-ever,
are still susceptible to discontinuous deformation.Specially, they
may be subject to flip ambiguities inwhich a set of nodes have two
possible configurationscorresponding to a “reflection” across a set
of mirrornodes (e.g., v and w in the flip example in Fig.19).
Thistype of ambiguity is not possible in 3-connected graphs.Fig.19
further provides a 3-connected and rigid graphwhich becomes
flexible upon removal of an edge. Af-ter the removal of the edge
(u, v), a subgraph can swinginto a different configuration in which
the removed edgeconstraint is satisfied and then reinserted. Such a
typeof ambiguity, called flex deformation, is eliminated
byredundant rigidity, the property that a graph remainsrigid upon
removal of any single edge.
Summarizing the conditions for eliminating ambigui-ties in graph
realization, Jackson and Jordan providethe necessary and sufficient
condition for global rigid-
ity in the following theorem.Theorem 1[71]. A graph with n >
4 vertices is glob-
ally rigid in 2 dimensions if and only if it is 3-connectedand
redundantly rigid.
Based on Theorem 1, the property of global rigi-dity can be
tested in polynomial time by combiningthe Pebble game algorithm[73]
and the network flowalgorithms[70,74] for rigidity and
3-connectivity, respec-tively.
3.2.2 Conditions for Network Localizability
The locations of all vertices in a globally rigid graphcan be
uniquely determined if fixing any group of 3vertices to avoid
trivial variation in 2D plane, such astranslation, rotation, or
reflection. Hence, for wirelessad-hoc networks, Eren et al. present
the following con-clusion that perfectly bridges the theory of
graph rigid-ity and the application of network localizability, as
il-lustrated in Fig.20.
Theorem 2[69]. A network is uniquely localizableif and only if
its distance graph is globally rigid and itcontains at least three
anchors.
Fig.21 shows the relationship between network con-nectivity and
localizability (global rigidity) through ex-tensive simulations. We
generate networks of 400 nodesrandomly, uniformly deployed in a
unit square [0, 1]2.The unit disk model with a radius is adopted
for com-munication and distance ranging. For each evaluation,we
integrate results from 100 network instances. Thecurve ri denotes
the percentage of i-connected networksin varied radius while rg
denotes globally rigid net-works. Like many other properties for
random geomet-ric graphs, both connectivity and rigidity have
transi-tion phenomena. It can be seen that rg lies between
Fig.19. Graph deformation and solutions.
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292 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
Fig.20. Connection between theory and application.
Fig.21. The relationship between connectivity and rigidity.
r3 and r6 and is closer to r3. This observation reflectsthe
theoretical conclusion that 3-connectivity is a nec-essary
condition while 6-connectivity is a sufficient onefor global
rigidity[75].
3.3 Inductive Construction of Globally RigidGraphs
Inductive construction of globally rigid graphs in-spires
localizability testing in a distributed manner,which is highly
appreciated by wireless ad-hoc/sensornetworks since centralized
approaches often consumelarge communication resource on data
transmission anddevice synchronization.
3.3.1 Trilateration
Nowadays, trilateration is an important and well ac-cepted
scheme to inductively construct localizable net-works. The basic
principle of trilateration is that theposition of an object can be
uniquely determined bymeasuring the distances to three reference
positions.It is widely-used in many real-world
applications[25,27]
as it is computationally efficient, fully distributed, andeasy
to implement. Importantly, the networks that canbe constructed by
iterative trilateration are localizable.
Theoretically, a trilateration ordering of a graphG = (V, E) is
an ordering (v1, v2, . . . , vn) of V for whichthe first 3 vertices
are pairwise connected and at least3 edges connect each vertex vj ,
4 6 j 6 n, to the setof the first j − 1 vertices. A graph is a
trilaterationextension if it has a trilateration ordering. It is
shownthat trilateration extensions are globally rigid[69,76].
Trilateration based approaches, however, recognizeonly a subset
(called trilateration extension) of globally
rigid graphs. In Fig.22(a), two globally rigid compo-nents are
connected by nodes i (i = 1, 2, . . . , 7). Sup-pose the nodes 1,
2, 3, and 4 in the left componentare known as localizable. The
localizability informa-tion, however, cannot propagate to the other
part bytrilateration since none of the nodes 5, 6, and 7 con-nects
to three localizable nodes. Obviously, trilate-ration wrongly
reports that nodes in the right com-ponent are not localizable,
ignoring the fact that theentire graph is globally rigid.
Fig.22. Deficiency of trilateration. (a) Geographical gap.
(b) Border nodes.
A similar situation recurs for the border nodes, asillustrated
in Fig.22(b). In this case, the border nodes 1and 2 cannot be
localized by trilateration even thoughnodes 3, 4, and 5 know their
locations. Actually, theentire graph in Fig.22(b) is globally rigid
and thus loca-lizable. Discarding locating border nodes is
unaccept-able, as border nodes often play critical roles in
manyapplications. For example, a sensor network for for-bidden
region monitoring has special interests on whenand where intruders
crash into, which are collected byborder nodes only.
3.3.2 WHEEL
The limitations of trilaterations motivate anothermethod to
construct localizable networks based wheelgraphs. A wheel graph Wn
is a graph with n vertices,formed by connecting a single vertex to
all vertices ofan (n − 1)-cycle. The vertices in the cycle will be
re-ferred to as rim vertices, the central vertex as the hub,an edge
between the hub and a rim vertex as a spoke,and an edge between two
rim vertices as a rim edge.Fig.23 shows a particular realization of
a wheel graphW6, in which node 0 is the hub and others are
rims.
Fig.23. Wheel graphs. (a) W4. (b) W6. (c) W9.
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Yunhao Liu et al.: Location, Localization, and Localizability
293
The wheel graph has many good properties. Fromthe standpoint of
the hub vertex, all elements, includ-ing vertices and edges, are in
its one-hop neighborhood,which indicates that the wheel structure
is fully in-cluded in the neighborhood graph of the hub
vertex.Furthermore, wheel graphs are important for localiz-ability
because they are globally rigid in 2D space[77],which indicates an
approach to identifying localizablevertices. Similar to the
trilateration extension, thewheel extension is defined as
follows.
Definition 1. A graph G is a wheel extension ifthere are
(a) three pairwise connected vertices, say v1, v2, andv3;
and
(b) an ordering of remaining vertices as v4, v5,v6, . . ., such
that any vi is included in a wheel graph(a subgraph of G)
containing three early vertices in thesequence.
Theorem 3[77]. The wheel extension is globallyrigid.
Fig.24 shows an example that is a wheel extensionbut not a
trilateration extension. A particular iterativelocalization process
on this example graph is illustratedin Fig.24. At beginning, three
beacons are available atthe bottom left. In the first iteration,
nodes in thebottom left hexagon are identified because they are
in-cluded in a wheel graph with 3 beacons. Such a proce-dure
continues until all localizable nodes are marked.
Fig.24. Wheel extension graph.
Compared to the previous trilateration (TRI) basedmethods, the
advantages of the proposed method(WHEEL) lie in:
1) Capability: recognizing a superset of localizablenodes, as
shown in Fig.25.
2) Efficiency: taking O(n) running time for sparsegraphs and
O(n3) for dense ones, where n is the net-work size.
3) Low cost: introducing no extra wireless commu-nication cost
by using only localized information.
Fig.25. Trilateration extension is a subset of WHEEL
extension.
Three examples are further provided to show howWHEEL outperforms
TRI does. In Fig.26, a particu-lar network with an “H” hole is
generated in which 400nodes are randomly distributed. The blue dots
denotethe nodes marked by TRI, while reds denote the nodesmarked by
WHEEL but not by TRI. Neither TRI norWHEEL can mark the remaining
blacks. WHEEL caneasily step over gaps, such as borders or
barriers, andrecognize more nodes than TRI does. The same
phe-nomenon recurs in all three network instances.
3.4 Node Localizability
Due to hardware or deployment constraints, for someapplications,
the networks are almost always not en-tirely localizable[78].
Indeed, theoretical analyses indi-cate that, in most cases, it is
unlikely that all nodes in anetwork are localizable, but a (large)
portion of nodescan be uniquely located[78]. Thus, the network
loca-lizability testing often fails unless networks are highlydense
and regular.
On the other hand, nodes are not equally importantsince they
play different roles in a network. Such dif-ferentiation can be
application specific. For example, asensor network for monitoring
forbidden regions has
Fig.26. Networks with “H” holes. (a) Case 1. (b) Case 2. (c)
Case 3.
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294 J. Comput. Sci. & Technol., Mar. 2010, Vol.25, No.2
special interest in when and where intruders enter,which are
collected by border nodes only. In addi-tion, many applications can
function properly as longas a sufficient number of nodes are aware
of theirlocations[78]. These observations motivate researchersto
consider the localizability problem beyond the net-work
localizability.
Although the theory of network localizability is com-plete, what
we real desire is to answer the following twofundamental questions
which cannot be solved by exi-sting methods:
1) Given a network configuration, whether or not aspecific node
is localizable?
2) How many nodes in a network can be located andwhich are
them?
Answering the above questions not only benefits lo-calization,
but also provides instructive directions toother location-based
services. Therefore, the node lo-calizability is addressed[78-79],
which focuses on thelocation-uniqueness of every single node.
Clearly, net-work localizability is a special case of node
localizabilityin which all nodes are localizable. Thus, node
localiz-ability is a more general issue.
The first major challenge for studying node local-izability is
to identify uniquely localizable nodes. Fol-lowing the results for
network localizability, an obvioussolution is to find a localizable
subgraph from the dis-tance graph, and identify all the nodes in
the subgraphlocalizable. Unfortunately, such a straightforward
at-tempt misses some localizable nodes and wrongly iden-tifies them
as non-localizable, since some conditions(e.g., 3-connectivity)
essential to network localizabilityare no longer necessary to node
localizability. As shownin Fig.27(a), the node u can be uniquely
located un-der this network configuration but not included in
the3-connected component of beacons. The uniquenessof u′s location
is explained in Figs. 27(b) and 27(c)where we decompose the network
into 2 subgraphs. Asu co