location, localization, and localizability

Liu Y, Yang Z, Wang X et al. Location, localization, and localizability. JOURNAL OF COMPUTER SCIENCE AND

TECHNOLOGY 25(2): 274297 Mar. 2010

Location, Localization, and Localizability

Yunhao Liu (), Member, ACM, Senior Member, IEEE, Zheng Yang ( ), Student Member, ACM, IEEEXiaoping Wang (), Student Member, IEEE, and Lirong Jian (), 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.

Survey2010 Springer Science+Business Media, LLC & Science Press, China

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.

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 unknownnodes 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 di 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 i = di di. Owingto ranging noises in di, i is often non-zero in practice.

The least squares method is used to assign a value to(x0, y0) that minimizes

ni=1

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 x2ny2nd2i +d2n = 2(xixn)x0+2(yiyn)y0

which yields the linear relationship

Ax = B

where A is an (n1)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 di substituting fordi. Now the least square solution is an estimate forx that minimizes Ax B2, 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


a power law. A generally employed model for wirelessradio propagation is as follows[26]:

P (d) = P (d0) 10 log( dd0

)+X

where P (d) is the received power at distance d, P (d0)the received power at some reference distance d0, the path-loss exponent, and X a log-normal randomvariable with variance 2 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 relativelycheap 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,


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 (34) 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 nodes coverage isan annulus, as illustrated in Fig.6(c); when an angu-lar sector (min, max) and a maximum range Rmax canbe determined for some radio antennas, the shape for asingle nodes 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


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+enlocal

11

e(nlocal/pi)arccos tt1t2dt

).

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


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


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 500MHz. 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 2.45GHz ISM band and achieves amaximum data rate of 2Mbps. Each symbol is trans-mitted with a chirp pulse that has a bandwidth of80MHz and a fixed duration of 1s.

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 Floyds all-pairs shortest-path algorithm).

2) Apply classical metric-MDS on M to determinea map that gives the locations of all nodes in relative


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


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() > dmin,

where b is the length of the shortest side, 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,


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.


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 310mRSS AHLoS 24m 30100mUWB PAL UWB 1.5m N/A

RF ToF RF ToF 13m 100mEToA BeepBeep 12 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,


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|), where X is the random observation, and isthe parameter to be estimated. Any unbiased estimator must satisfy

Cov() > {E[( ln f(X|))T]}1 (1)

where Cov() is the error covariance of the estimator,E[] indicates expected value, and is the gradientoperator with respect to .

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 of

interest consists of the coordinates of nodes to be local-ized, given by = [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 dij . 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 = { 12 [G()]T[G()]}1, where 2 is the vari-ance of each distance measurement error, and G() isthe M 2L matrix whose mn-th element is

G()mn =

xi xjdij

, if n = xi,

xj xidij

, if n = xj ,

yi yjdij

, if n = yi,

yj yidij

, if n = 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


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

20 = 2m[m1i=1

mj>i

sin2 ij]1

(3)

where 20 is the variance of the estimate location of v0,ij 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 (2)and the geometric relationship of references and to-be-localized nodes (ij). 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 0/.

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 1 = 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 2, 3 [0, 2pi] 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, 1 = 0,2 = 2pi/3, and 3 = 4pi/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 pt and confidence et 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


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 nodes 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, p = (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

Eeb2 = E(ATA+CA)1ATb2 (4)

where b is the perturbation of b due to ranging er-rors. Similarly, the error due to location uncertainty ofreferences is

Eea2 = E(ATA+CA)1Ba2 (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., e = Eeb2 +Eea2, where is a parameter to compensate forthe over-estimation of the error due to a. Small valueof 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)3i=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


ranging results, Liu et al.[65] use the mean square error2 of the distance measurements based on the estimatedlocations as an indicator of the level of inconsistency,i.e.,

2 =1m

mi=1

(i p0 pi)2 (7)

where m is the number of location references {(pi, i),i = 1, 2, . . . ,m} and p0 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 2. 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

p0 = arg minp0

med i(i p0 pi)2. (8)

LMS-based localization first randomly drawsM 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) ={r(i, j)2, if |r(i, j)| < ;ln(|r(i, j)| u) v, otherwise, (9)

and u = 12 , v = ln(

12

) 2, and 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 aU near 0 and like wings for value greater than ,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 = 4 and = 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


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,


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 localizable

if 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.


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.


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 if

there are(a) three pairwise connected vertices, say v1, v2, and

v3; 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 globally

rigid.Fig.24 shows an example that is a wheel extension

but 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.


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 us location is explained in Figs. 27(b) and 27(c)where we decompose the network into 2 subgraphs. Asu connects 2 beacons in the right component, it has 2possible locations denoted by u and u. If we adopt u asits location, it is impossible to embed the left subgraphinto the plane. Specifically, the left subgraph has 2

Fig.27. An example showing that the result from network localiz-

ability fails to identify node u as localizable. (a) u is localizable.

(b) Graph decomposing. (c) 2 realizations of the left subgraph.

realizations, but neither of them is compatible withu. Hence, u is uniquely localizable, although the 3-connectivity property does not hold. Motivated by theexample, it is clear that the results derived for networklocalizability cannot be directly applied and we have toreconsider the conditions for node localizability.

The big picture of the state-of-the-arts of node lo-calizability is shown in Fig.28. An obvious conditionfor a vertex to be localizable is that it must connect atleast three neighbor vertices, i.e., having a degree of atleast three. Goldenberg et al.[78] propose the first non-trivial necessary c

location, localization, and localizability

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