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					                                            Computer Networks 51 (2007) 2529–2553
                                                                                                        www.elsevier.com/locate/comnet




                   Wireless sensor network localization techniques
                                             a,b,*                         b,c                                     b,c
                   Guoqiang Mao                    , Barıs Fidan
                                                         ß                   , Brian D.O. Anderson
                          a
                           School of Electrical and Information Engineering, The University of Sydney, Australia
                                                 b
                                                   National ICT Australia Ltd., Australia1
               c
                   Research School of Information Sciences and Engineering, The Australian National University, Australia

                      Received 27 June 2006; received in revised form 6 October 2006; accepted 15 November 2006
                                                    Available online 3 January 2007

                                                       Responsible Editor: E. Ekici




Abstract

   Wireless sensor network localization is an important area that attracted significant research interest. This interest is
expected to grow further with the proliferation of wireless sensor network applications. This paper provides an overview
of the measurement techniques in sensor network localization and the one-hop localization algorithms based on these
measurements. A detailed investigation on multi-hop connectivity-based and distance-based localization algorithms are
presented. A list of open research problems in the area of distance-based sensor network localization is provided with
discussion on possible approaches to them.
Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Wireless sensor networks; Localization; AOA; RSS; TDOA




1. Introduction                                                         tions and electronics have enabled the development
                                                                        of low-cost, low-power and multi-functional sensors
   Wireless sensor networks (WSNs) are a signifi-                        that are small in size and communicate in short dis-
cant technology attracting considerable research                        tances. Cheap, smart sensors, networked through
interest. Recent advances in wireless communica-                        wireless links and deployed in large numbers, pro-
                                                                        vide unprecedented opportunities for monitoring
                                                                        and controlling homes, cities, and the environment.
                                                                        In addition, networked sensors have a broad spec-
  *
    Corresponding author. Address: School of Electrical and
                                                                        trum of applications in the defence area, generating
Information Engineering, The University of Sydney, Australia.           new capabilities for reconnaissance and surveillance
Tel.: +61 2 93512962; fax: +61 2 93513847.                              as well as other tactical applications [1].
    E-mail address: g.mao@ieee.org (G. Mao).                               Self-localization capability is a highly desirable
  1
    National ICT Australia is funded by the Australian Govern-          characteristic of wireless sensor networks. In envi-
ment’s Department of Communications, Information Technol-
ogy and the Arts and the Australian Research Council through
                                                                        ronmental monitoring applications such as bush fire
the Backing Australia’s Ability initiative and the ICT Centre of        surveillance, water quality monitoring and precision
Excellence Program.                                                     agriculture, the measurement data are meaningless

1389-1286/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.comnet.2006.11.018
2530                            G. Mao et al. / Computer Networks 51 (2007) 2529–2553

without knowing the location from where the data             both 2-dimension (R2 ) and 3-dimension (R3 ), we
are obtained. Moreover, location estimation may              choose to focus on 2D localization problems for
enable a myriad of applications such as inven-               ease of explanation.
tory management, intrusion detection, road traffic                The rest of the paper is organized as follows. In
monitoring, health monitoring, reconnaissance and            Section 2, measurement techniques in WSN locali-
surveillance.                                                zation are discussed; these include angle-of-arrival
   Sensor network localization algorithms estimate           (AOA) measurements, distance related measure-
the locations of sensors with initially unknown loca-        ments and RSS profiling techniques. Distance
tion information by using knowledge of the absolute          related measurements are further classified into
positions of a few sensors and inter-sensor measure-         one-way propagation time and roundtrip propaga-
ments such as distance and bearing measurements.             tion time measurements, the lighthouse approach
Sensors with known location information are called           to distance measurements, received signal strength
anchors and their locations can be obtained by using         (RSS)-based distance measurements and time-differ-
a global positioning system (GPS), or by installing          ence-of-arrival (TDOA) measurements. In Section
anchors at points with known coordinates. In appli-          3, one-hop localization techniques based on these
cations requiring a global coordinate system, these          measurements are discussed. Section 4 discusses
anchors will determine the location of the sensor            nonline-of-sight error mitigation techniques in
network in the global coordinate system. In applica-         WSN localization. Sections 5 and 6 focus on
tions where a local coordinate system suffices (e.g.,          multi-hop localization techniques, in particular con-
smart homes), these anchors define the local coordi-          nectivity-based and distance-based multi-hop local-
nate system to which all other sensors are referred.         ization techniques. Section 7 discusses open research
Because of constraints on the cost and size of               problems in distance-based localization. Finally a
sensors, energy consumption, implementation envi-            summary is provided in Section 8.
ronment (e.g., GPS is not accessible in some
environments) and the deployment of sensors (e.g.,           2. Measurement techniques
sensor nodes may be randomly scattered in the
region), most sensors do not know their locations.             Measurement techniques in WSN localization
These sensors with unknown location information              can be broadly classified into three categories:
are called non-anchor nodes and their coordinates            AOA measurements, distance related measurements
will be estimated by the sensor network localization         and RSS profiling techniques.
algorithm.
   In this paper, we provide an overview of tech-            2.1. Angle-of-arrival measurements
niques that can be used for WSN localization.
Review of wireless network localization techniques              The angle-of-arrival measurement techniques can
can be found in [2–4]. The focus of these references         be further divided into two subclasses: those making
is on localization techniques in cellular network and        use of the receiver antenna’s amplitude response
wireless local area network (WLAN) environments              and those making use of the receiver antenna’s
and on the signal processing aspect of localization          phase response.
techniques. Sensor networks vary significantly from              Beamforming is the name given to the use of
traditional cellular networks and WLAN, in that              anisotropy in the reception pattern of an antenna,
sensor nodes are assumed to be small, inexpensive,           and it is the basis of one category of AOA measure-
cooperative and deployed in large quantity. These            ment techniques. The measurement unit can be of
features of sensor networks present unique chal-             small size in comparison with the wavelength of
lenges and opportunities for WSN localization. Pat-          the signals. The beam pattern of a typical aniso-
wari et al. described some general signal processing         tropic antenna is shown in Fig. 1. One can imagine
tools that are useful in cooperative WSN localiza-           that the beam of the receiver antenna is rotated elec-
tion algorithms [5] with a focus on computing the            tronically or mechanically, and the direction corre-
       ´
Cramer–Rao bounds for localization using a variety           sponding to the maximum signal strength is taken
of different types of measurements [5]. Our review in         as the direction of the transmitter. Relevant param-
contrast focuses on the measurement techniques               eters are the sensitivity of the receiver and the beam
and localization algorithms in WSNs. While many              width. A technical problem to be faced and over-
techniques covered in this paper can be applied in           come arises when the transmitted signal has a vary-
                                     G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                  2531




                                                                        Fig. 2. An antenna array with N antenna elements.
Fig. 1. An illustration of the horizontal antenna pattern of a
typical anisotropic antenna.
                                                                  a transmitter far away from the antenna array and
ing signal strength. The receiver cannot differentiate             the ith antenna element can be approximated by
the signal strength variation due to the varying
                                                                  Ri % R0 À id cos h;                                       ð1Þ
amplitude of the transmitted signal and the signal
strength variation caused by the anisotropy in the                where R0 is the distance between the transmitter and
reception pattern. One approach to dealing with                   the 0th antenna element and h is the bearing of the
the problem is to use a second non-rotating and                   transmitter with respect to the antenna array. The
omnidirectional antenna at the receiver. By normal-               transmitter signals received by adjacent antenna ele-
izing the signal strength received by the rotating                ments have a phase difference of 2p d cos h, which al-
                                                                                                           k
anisotropic antenna with respect to the signal                    lows us to obtain the bearing of the transmitter
strength received by the non-rotating omnidirec-                  from the measurement of the phase difference. This
tional antenna, the impact of varying signal strength             approach works quite well for high SNR but may
can be largely removed.                                           fail in the presence of strong co-channel interference
   Another widely used approach [6] to cope with                  and/or multipath signals [7].
the varying signal strength problem is to use a min-                  The accuracy of AOA measurements is limited by
imum of two (but typically at least four) stationary              the directivity of the antenna, by shadowing and by
antennas with known, anisotropic antenna patterns.                multipath reflections. How to obtain accurate AOA
Overlapping of these patterns and comparing the                   measurements in the presence of multipath and
signal strength received from each antenna at the                 shadowing errors has been a subject of intensive
same time yields the transmitter direction, even                  research. AOA measurements rely on a direct line-
when the signal strength changes. Coarse tuning is                of-sight (LOS) path from the transmitter to the recei-
performed by measuring which antenna has the                      ver. However a multipath component may appear as
strongest signal, and it is followed by fine tuning                a signal arriving from an entirely different direction
which compares amplitude responses. Because small                 and can lead to very large errors in AOA measure-
errors in measuring the received power can lead to a              ments. Multipath problems in AOA measurements
large AOA measurement error, a typical measure-                   can be addressed by using the maximum likelihood
ment accuracy for four antennas is 10–15°. With                   (ML) algorithms [7]. Different ML algorithms have
six antennas, this can be improved to about 5°,                   been proposed in the literature which make different
and 2° with eight antennas [6].                                   assumptions about the statistical characteristics of
   The second category of measurement techniques,                 the incident signals [8–10]. They can be classified into
known as phase interferometry [7], derives the AOA                deterministic and stochastic ML methods. Typically
measurements from the measurements of the phase                   ML methods will estimate the AOA of each separate
differences in the arrival of a wave front. It typically           path in a multipath environment. The implementa-
requires a large receiver antenna (relative to the                tion of these methods is computationally intensive
wavelength of the transmitter signal) or an antenna               and requires complex multidimensional search. The
array. Fig. 2 shows an antenna array of N antenna                 dimensionality of the search is equal to the total
elements. The adjacent antenna elements are sepa-                 number of paths taken by all the received signals
rated by a uniform distance d. The distance between               [7]. The problem is further complicated by the fact
2532                            G. Mao et al. / Computer Networks 51 (2007) 2529–2553

that the total number of paths is not known a priori         to [23] for a detailed technical discussion on AOA
and must be estimated. Different from the earlier             measurement techniques.
ML methods, which assume the incoming signal is
an unknown stochastic process, another class of              2.2. Distance related measurements
ML methods [11–13] assume that the structure of
the signal waveform is known to the receiver. This              Distance related measurements include propaga-
assumption is possible in some digital communica-            tion time based measurements, i.e., one-way propa-
tion systems because the modulation format is                gation time measurements, roundtrip propagation
known to the receiver and many systems are                   time measurements and time-difference-of-arrival
equipped with a known training sequence in the pre-          (TDOA) measurements, and RSS measurements.
amble. This extra information is exploited to                Another interesting technique measuring distance,
improve the accuracy of AOA measurements or sim-             which does not fall into the above categories, is
plify computation.                                           the lighthouse approach shown in [24]. In the
   Yet another class of AOA measurement methods              following paragraphs we provide further details of
is based on so-called subspace-based algorithms              these techniques.
[14–17]. The most well known methods in this cate-
gory are MUSIC (multiple signal classification) [14]          2.2.1. One-way propagation time and roundtrip
and ESPRIT (estimation of signal parameters by               propagation time measurements
rotational invariance techniques) [15,16]. These                One-way propagation time and roundtrip propa-
eigenanalysis based direction finding algorithms uti-         gation time measurements are also generally known
lize a vector space formulation, which takes advan-          as time-of-arrival measurements. Distances between
tage of the underlying parametric data model for the         neighboring sensors can be estimated from these
sensor array problem. They require a multi-array             propagation time measurements.
antenna in order to form a correlation matrix using             One-way propagation time measurements mea-
signals received by the array. The measured signal           sure the difference between the sending time of a sig-
vectors received at the M array elements is visual-          nal at the transmitter and the receiving time of the
ized as a vector in M dimensional space. Utilizing           signal at the receiver. It requires the local time at
an eigen-decomposition of the correlation matrix,            the transmitter and the local time at the receiver to
the vector space is separated into signal and noise          be accurately synchronized. This requirement may
subspaces. Then the MUSIC algorithm searches                 add to the cost of sensors by demanding a highly
for nulls in the magnitude squared of the projection         accurate clock and/or increase the complexity of
of the direction vector onto the noise subspace. The         the sensor network by demanding a sophisticated
nulls are a function of angle-of-arrival, from which         synchronization mechanism. This disadvantage
angle-of-arrival can be estimated. For linear arrays,        makes one-way propagation time measurements a
Root-MUSIC [18], a polynomial rooting version of             less attractive option than measuring roundtrip time
MUSIC, improves the resolution capabilities of               in WSNs. Roundtrip propagation time measure-
MUSIC. A weighted norm version of MUSIC,                     ments measure the difference between the time when
WMUSIC [19], also gives an extension in the resolu-          a signal is sent by a sensor and the time when the
tion capabilities compared to the original MUSIC.            signal returned by a second sensor is received at
ESPRIT [15,16] is based on the estimation of signal          the original sensor. Since the same clock is used to
parameters via rotational invariance techniques. It          compute the roundtrip propagation time, there is
uses two displaced subarrays of matched sensor               no synchronization problem. The major error source
doublets to exploit an underlying rotational invari-         in roundtrip propagation time measurements is the
ance among signal subspaces for such an array. A             delay required for handling the signal in the second
comprehensive experimental evaluation of MUSIC,              sensor. This internal delay is either known via a pri-
Root-MUSIC, WMUSIC, Min-Norm [20] and                        ori calibration, or measured and sent to the first sen-
ESPRIT algorithms can be found in [21]. A very               sor to be subtracted. A detailed discussion on
large number of AOA measurement techniques                   circuitry design for roundtrip propagation time mea-
have been developed which are based on MUSIC                 surements can be found in [25].
and ESPRIT, to cite but two, see e.g. [17,22]. Due              Time delay measurement is a relatively mature
to space limitations, we do not provide an exhaus-           field. The most widely used method for obtaining
tive list of them in this paper. Readers may refer           time delay measurement is the generalized cross-cor-
                                  G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                  2533

relation method [26,27]. A detailed discussion on the
cross-correlation method is given in Section 2.2.3.
    Based on the observation that the speed of sound
in the air is much smaller than the speed of light
(RF) in the air, Priyantha et al. developed a tech-
nique to measure the one-way propagation time
[28], which solved the synchronization problem. It
uses a combination of RF and ultrasound hardware.
On each transmission, a transmitter sends an RF
signal and an ultrasonic pulse at the same time.
The RF signal will arrive at the receiver earlier than
the ultrasonic pulse. When the receiver receives the
RF signal, it turns on its ultrasonic receiver and lis-
tens for the ultrasonic pulse. The time difference
between the receipt of the RF signal and the receipt
of the ultrasonic signal is used as an estimate of the
one-way acoustic propagation time. Their method
gave fairly accurate distance estimate at the cost
of additional hardware and complexity of the sys-
tem because ultrasonic reception suffers from severe
multipath effects caused by reflections from walls
and other objects.
    A recent trend in propagation time measurements
is the use of ultra wide band (UWB) signals for accu-
rate distance estimation [29,30]. A UWB signal is a
signal whose bandwidth to center frequency ratio
is larger than 0.2 or a signal with a total bandwidth
of more than 500 MHz. UWB can achieve higher
accuracy because its bandwidth is very large and               Fig. 3. An illustration of the lighthouse approach for distance
                                                               measurement.
therefore its pulse has a very short duration. This
feature makes fine time resolution of UWB signals
and easy separation of multipath signals possible.             The unknown angular velocity x can be derived
                                                               from the difference between the time instant when
2.2.2. Lighthouse approach to distance measurements            the optical receiver first detects the beam and the
   Another interesting approach to distance mea-               time instant when the optical receiver detects the
surements is the lighthouse approach [24] which                beam for the second time. Therefore the distance
derives the distance between an optical receiver               d1 can be derived from the time duration t1 that
and a transmitter of a parallel rotating optical beam          the optical receiver dwells in the beam.
by measuring the time duration that the receiver                  The lighthouse approach measures the distance
dwells in the beam. Fig. 3 illustrates the principle           between an optical receiver and the rotational axis
of the lighthouse approach. A transmitter located              of the optical beam generated by the transmitter.
at the origin is equipped with a parallel optical              A major advantage of the lighthouse approach is
beam, i.e., an optical beam whose beam width b is              the optical receiver can be of a very small size, thus
constant with respect to the distance from the rota-           making the idea of ‘‘smart dust’’ possible [24]. How-
tional axis of the beam. The optical beam rotates at           ever the transmitter may be large. The approach
an unknown angular velocity x around the Z axis.               also requires a direct line-of-sight between the
An optical receiver in the XY plane and at a dis-              optical receiver and the transmitter.
tance d1 from the Z axis detects the beam for a time
duration t1. From Fig. 3, it can be shown that                 2.2.3. Time-difference-of-arrival measurements
                                                                  There is a category of localization algorithms
            b             b                                    utilizing TDOA measurements of the transmitter’s
d1 %               ¼              :                  ð2Þ
       2 sinða1 =2Þ 2 sinðxt1 =2Þ                              signal at a number of receivers with known location
2534                                 G. Mao et al. / Computer Networks 51 (2007) 2529–2553

information to estimate the location of the transmit-             this increases differences between time-of-arrival.
ter. Fig. 4 shows a TDOA localization scenario with               Closely spaced multiple receivers may give rise to
a group of four receivers at locations r1, r2, r3, r4 and         multiple received signals that cannot be separated.
a transmitter at rt. The TDOA between a pair of                   For example, TDOA of multiple signals that are
receivers i and j is given by                                     not separated by more than the width of their
                                                                  cross-correlation peaks (whose location on the
                1
Dtij , ti À tj ¼ ðkri À rt k À krj À rt kÞ;     i 6¼ j;   ð3Þ     time-delay axis corresponds to TDOA) usually can-
                c                                                 not be resolved by conventional TDOA measure-
where ti and tj are the time when a signal is received            ment techniques [32]. Yet another factor affecting
at receivers i and j, respectively, c is the propagation          the accuracy of TDOA measurement is multipath.
speed of the signal, and kÆk denotes the Euclidean                Overlapping cross-correlation peaks due to multi-
norm.                                                             path often cannot be resolved. Even if distinct peaks
     Measuring the TDOA of a signal at two receivers              can be resolved, a method must be designed for
at separate locations is a relatively mature field [31].           selecting the correct peak value, such as choosing
The most widely used method is the generalized                    the largest or the first peak [7].
cross-correlation method, where the cross-correla-                   It is worth noting that Gardner and Chen pro-
tion function between two signals si and sj received              posed an approach in [32,33], which exploits the
at receivers i and j is given by integrating the lag              cyclostationarity property of a certain signal to
product of two received signals for a sufficiently                  obtain substantial tolerance to noise and interfer-
long time period T,                                               ence. The cyclostationarity property is a direct
            Z                                                     result of the underlying periodicities in the signal
           1 T                                                    due to periodic sampling, scanning, modulating,
qi;j ðsÞ ¼       si ðtÞsj ðt À sÞ dt:                ð4Þ
           T 0                                                    multiplexing, and coding operations employed in
                                                                  the transmitter. Both the frequency-shifted and
The cross-correlation function can also be obtained               time-shifted cross-correlations are utilized to exploit
from an inverse Fourier transform of the estimated                the unique cyclostationarity property of the signal.
frequency domain cross-spectral density function.                 Their method requires the signal of interest to have
Frequency domain processing is often preferred be-                a known analog frequency or digital keying rate
cause the signals can be filtered prior to computation             that is distinct from that of the interfering signal.
of the cross-correlation function. The cross-correla-
tion approach requires very accurate synchroniza-
tion among receivers but does not impose any                      2.2.4. Distance estimation via received signal strength
requirement on the signal transmitted by the trans-               measurements
mitter. The accuracy and temporal resolution capa-                   Another category of distance related measure-
bilities of TDOA measurements will improve when                   ment techniques estimates the distances between
the separation between receivers increases because                neighboring sensors from the received signal
                                                                  strength measurements [34–38]. These techniques
                                                                  are based on a standard feature found in most wire-
                                                                  less devices, a received signal strength indicator
                                                                  (RSSI). They are attractive because they require
                                                                  no additional hardware, and are unlikely to signifi-
                                                                  cantly impact local power consumption, sensor size
                                                                  and thus cost.
                                                                     In free space, other things being equal the RSS
                                                                  varies as the inverse square of the distance d
                                                                  between the transmitter and the receiver. Let us
                                                                  denote this received power by Pr(d). The received
                                                                  power Pr(d) is related to the distance d through
                                                                  the Friis equation [39]

                                                                              P t Gt Gr k2
Fig. 4. Localization using time-difference-of-arrival measure-     P r ðdÞ ¼         2
                                                                                             ;                       ð5Þ
ments.                                                                         ð4pÞ d 2
                                             G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                    2535

                                                                                                       À1=np             r2
where Pt is the transmitted power, Gt is the trans-                                           P ij                   À
                                                                          ^
                                                                          d ij ¼ d 0                             e       2g2 n2
                                                                                                                              p   :   ð10Þ
mitter antenna gain, Gr is the receiver antenna gain                                       P 0 ðd 0 Þ
and k is the wavelength of the transmitter signal in
meters.
     The free-space model however is an over-ideali-                      2.3. RSS profiling measurements
zation, and the propagation of a signal is affected
by reflection, diffraction and scattering. Of course,                           Yet another category of localization techniques,
these effects are environment (indoors, outdoors,                          i.e., the RSS profiling-based localization techniques
rain, buildings, etc.) dependent. However, it is                          [42–46], work by constructing a form of map of the
accepted on the basis of empirical evidence that it                       signal strength behavior in the coverage area. The
is reasonable to model the RSS Pr(d) at any value                         map is obtained either offline by a priori measure-
of d at a particular location as a random and log-                        ments or online using sniffing devices [44] deployed
normally distributed random variable with a dis-                          at known locations. They have been mainly used for
tance-dependent mean value [40,41]. That is,                              location estimation in WLANs, but they would
                                                                        appear to be attractive also for wireless sensor
                                                 d
P r ðdÞ ½dB mŠ ¼ P 0 ðd 0 Þ ½dB mŠ À 10np log10     þ X r;                networks.
                                                 d0                           In this technique, in addition to there being
                                                                ð6Þ       anchor nodes (e.g., access points in WLANs) and
                                                                          non-anchor nodes, a large number of sample points
where P0(d0) [dB m] is a known reference power va-
                                                                          (e.g., sniffing devices) are distributed throughout the
lue in dB milliwatts at a reference distance d0 from
                                                                          coverage area of the sensor network. At each sample
the transmitter, np is the path loss exponent that
                                                                          point, a vector of signal strengths is obtained, with
measures the rate at which the RSS decreases with
                                                                          the jth entry corresponding to the jth anchor’s trans-
distance and the value of np depends on the specific
                                                                          mitted signal. Of course, many entries of the signal
propagation environment, Xr is a zero mean Gauss-
                                                                          strength vector may be zero or very small, corre-
ian distributed random variable with standard devi-
                                                                          sponding to anchor nodes at larger distances (rela-
ation r and it accounts for the random effect of
                                                                          tive to the transmission range or sensing radius)
shadowing [39]. In this paper, we use the notation
                                                                          from the sample point. The collection of all these
[dBm] to denote that power is in dB milliwatts units.
                                                                          vectors provides (by extrapolation in the vicinity
Otherwise, it is in milliwatts.
                                                                          of the sample points) a map of the whole region.
    It is trivial to conclude from Eq. (6) that, given
                                                                          The collection constitutes the RSS model, and it is
the RSS measurement, Pij, between a transmitter i
                                                                          unique with respect to the anchor locations and
and a receiver j, a maximum likelihood estimate of
                                                                          the environment. The model is stored in a central
the distance, dij, between the transmitter and the
                                                                          location. By referring to the RSS model, a non-
receiver is
                                                                          anchor node can estimate its location using the
                      À1=np
               P ij                                                       RSS measurements from anchors.
^ij ¼ d 0
d                             :                    ð7Þ
            P 0 ðd 0 Þ                                                        In summary, a number of measurement tech-
                                                                          niques are available for WSN localization. Which
Note that Pij and P0(d0) in Eq. (7) are measured in                       measurement technique to use for location estima-
milliwatts instead of dB milliwatts. Using Eqs. (6)                       tion will depend on the specific application. Typi-
                                 ^
and (7), the estimated distance d ij can be related                       cally, localization algorithms based on AOA and
to the true distance                                                      propagation time measurements are able to achieve
                X                lnð10ÞX r          X
                                                                          better accuracy than localization algorithms based
^             À r            À                    À r
d ij ¼ d ij 10 10np ¼ d ij 10 10np lnð10Þ ¼ d ij e gnp ;        ð8Þ       on RSS measurements. However, that accuracy is
            10                                ^                           achieved at the expense of higher equipment cost.
where g ¼ lnð10Þ. The expected value of d ij is                                                           ´
                                                                          Patwati et al. gave the Cramer–Rao lower bounds
                Z 1                                 r2                  for location estimation using TOA, RSS and AOA
  ^         1               ÀXr À X r                 2 2
E d ij ¼ pffiffiffiffiffiffi      d ij e gnp e 2r2 dX r ¼ d ij e2g np :     ð9Þ       measurements respectively in [5]. However the
           2pr À1
                                                                                  ´
                                                                          Cramer–Rao lower bound may be too optimistic
Thus the maximum likelihood estimate in Eq. (7) is                        when the measurement error deviates from Gauss-
a biased estimate of the true distance and an unbi-                                                    ´
                                                                          ian. Moreover the Cramer–Rao bound assumes
ased estimate is given by                                                 the underlying estimator is an unbiased estimator.
2536                                     G. Mao et al. / Computer Networks 51 (2007) 2529–2553

This assumption may not be satisfied by many local-                     surement. Denote by h(x) = [h1(x), . . . , hN(x)]T the
ization techniques.                                                    bearings of a transmitter located at x = [x, y]T at
                                                                       the receiver locations, where hi(x), 1 6 i 6 N is
                                                                       related to x by
3. One-hop localization techniques
                                                                                    y À yi
                                                                       tan hi ðxÞ ¼        :                             ð11Þ
   In this section, we discuss the principles of one-                               x À xi
hop localization techniques in which the non-
anchor node to be localized is the one-hop neighbor                    The measured bearings of the transmitter consist of
of a sufficient number of anchors.                                       the true bearings corrupted by additive noises
                                                                       e = [e1, . . . , eN]T, which are assumed to be zero-mean
                                                                       Gaussian noises with N · N covariance matrices
3.1. AOA based one-hop localization techniques                         S ¼ diagfr2 ; . . . ; r2 g, i.e.,
                                                                                        1      N

    In the absence of noise and interference, bearing                  b ¼ hðxt Þ þ e:                                                          ð12Þ
lines from two or more receivers will intersect to
                                                                       When the receivers are identical and much closer to
determine a unique location, i.e., the location esti-
                                                                       each other than to the transmitter, the variances of
mate of the transmitter. In the presence of noise,
                                                                       bearing measurement errors are equal, i.e.,
more than two bearing lines will not intersect at a
                                                                       r2 ¼ Á Á Á r2 ¼ r2 . The ML estimator of the transmit-
                                                                         1         N
single point and statistical algorithms, sometimes
                                                                       ter location xt is given by
called triangulation or fixing methods, are required
in order to obtain the location estimate of the trans-                              1           T
                                                                       xt ¼ arg min ½hð^t Þ À bŠ SÀ1 ½hð^t Þ À bŠ
                                                                       ^                 x                 x                                    ð13Þ
mitter [47,48]. This is shown in Fig. 5.                                            2
                                                                                    1 X ðhi ð^t Þ À bi Þ
    Location estimation using bearing measurements                                     N                 2
                                                                                             x
is a well researched problem [47–52]. The pioneering                      ¼ arg min               2
                                                                                                           :                                    ð14Þ
                                                                                    2 i¼1      ri
work in the area is that of Stanfield [49]. His
approach has been further generalized in [50,52]                       The nonlinear minimization problem in Eq. (13) can
and has been implemented in many practical sys-                        be solved by a Newton–Gauss iteration [47,48]
tems. Another well-known approach is the maxi-
mum likelihood estimator [47,51].                                      xt;kþ1 ¼ xt;k þ ðhx ð^t;k ÞT SÀ1 hx ð^t;k ÞÞÀ1 hx ð^t;k ÞT SÀ1 ½b À hx ð^t;k ފ;
                                                                       ^        ^           x               x             x                    x
    The 2D localization problem using bearing mea-                                                                                              ð15Þ
surements can be formulated as follows. Let
xt = [xt, yt]T be the true coordinate vector of the                    where hx ð^t;k Þ denotes the partial derivative of h
                                                                                   x
transmitter to be estimated from bearing measure-                                                          ^
                                                                       with respect to x evaluated at xt;k . The use of
ments b = [b1, . . . , bN]T, where N is the total number               Eq. (15) requires an initial estimate close enough
of receivers. Let xi = [xi, yi]T be the known location                 to the true minimum of the cost function. Such an
of the receiver associated with the ith bearing mea-                   initial estimate may be obtained from prior infor-
                                                                       mation, or using a suboptimal procedure [48].
                                                                          The Stanfield approach assumes that the mea-
                                                                       surement error is small enough such that ei % sin ei,
                                                                       1 6 i 6 N. In that case, the cost function in Eq.
                                                                       (14) becomes

                                                                       1 X sin2 ðhi ð^t Þ À bi Þ
                                                                          N
                                                                                     x
                                                                                                 :                                              ð16Þ
                                                                       2 i¼1        r2i


                                                                       Using the relation
                                                                       sinðhi ðxt Þ À bi Þ ¼ sin hi ðxt Þ cos bi À cos hi ðxt Þ sin bi
                                                                                           ðy À y i Þ cos bi À ðxt À xi Þ sin bi
                                                                                      ¼ t                                        ;
                                                                                                                        ri
                                                                                 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
                                                                                                  2                       2
Fig. 5. In the presence of noise, bearing lines from three receivers   where ri ¼ ðxt À xi Þ þ ðy t À y i Þ , Eq. (16) becomes
will not interact at the same point.
                                         G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                      2537


1 X ½ðy t À y i Þ cos bi À ðxt À xi Þ sin bi Š
   N                                          2
                                                                      3.2. TDOA-based one-hop localization techniques
2 i¼1                  r2 r2
                         i i
                                                                          Given the TDOA measurement Dtij and the coor-
   1         T                                                        dinates of receivers i and j, Eq. (3) defines one
  ¼ ðAxt À bÞ RÀ1 SÀ1 ðAxt À bÞ;                           ð17Þ
   2                                                                  branch of a hyperbola whose foci are at the loca-
                                                                      tions of receivers i and j and on which the transmit-
where
                                                                      ter rt must lie. In R2 , measurements from a
      2                        3                                      minimum of three receivers are required to uniquely
          sin b1    À cos b1
  6                            7                                      determine the location of the transmitter. This is
A¼6          .
             .          .
                        .      7;                          ð18Þ
  4          .          .      5                                      illustrated in Fig. 6.
                                                                          In a system of N receivers, there are N À 1 line-
          sin bN   À cos bN                                           arly independent TDOA equations, which can be
     2                              3
          x1 sin b1 À y 1 cos b1                                      written compactly as
  6                                 7                                  2                                    3
  6                 .
                    .               7                                       kr1 À rt k À krN À rt k À cDt1N
b¼6                 .               7;                     ð19Þ
  4                                 5                                  6                                    7
                                                                       6                   .
                                                                                           .                7 ¼ 0:     ð22Þ
         xN sin b1 À y 1 cos bN                                        4                   .                5
R ¼ diagfr2 ; . . . ; r2 g:                                ð20Þ          krN À1 À rt k À krN À rt k À cDtN À1;N
          1            N


Stanfield implicitly assumes that even though R is                     In practice, Dtij is not available; instead we have the
not perfectly known, a rough estimate of R can be                     noisy TDOA measurement D~ij given by
                                                                                                      t
obtained. Since the cost function weakly depends
on R, the fact that the estimate is rough will not sig-               D~ij ¼ Dtij þ nij ;
                                                                       t                                                            ð23Þ
nificantly affect the solution. Under these assump-
tions, the minimization of Eq. (17) with respect to                   where nij denotes an additive noise, which is usually
xt is a well known problem and the solution is given                  assumed to be an independent zero-mean Gaussian
by                                                                    distributed random variable. Eq. (22) is a nonlinear
                                                                      equation that is difficult to solve, especially when
                   À1
xt ¼ ðAT RÀ1 SÀ1 AÞ AT RÀ1 SÀ1 b:
^                                                          ð21Þ       the receivers are arranged in an arbitrary fashion.
   Note that the closed form solution in the Stan-                    Moreover, in the presence of noise, Eq. (22) may
field approach depends on two assumptions: first,                       not have a solution.
the measurement error is small such that ei % sin ei,                    A noisy version of Eq. (22) can be written as
1 6 i 6 N; second, R is known. One may chose to                       2         3 2 kr1 Àrt kÀkrN Àrt k 3 2              3
accept the first assumption but reject the second                          D~1N
                                                                            t                    c                 e1N
                                                                      6 . 7 6                    .           7 6 . 7
assumption. In that case an iterative procedure                       6 . 7¼6                    .           7 þ 6 . 7:    ð24Þ
                                                                      4 . 5 4                    .           5 4 . 5
can be used to obtain the solution to the minimiza-                                   krN À1 Àrt kÀkrN Àrt k
                                                                        D~N À1N
                                                                          t                                       eN À1N
tion problem, which has no advantage over the ML                                                 c
technique [48].
   Analytical expressions for the bias and the
covariance matrix of the estimation errors associ-
ated with the ML approach and with the Stanfield
approach were given in [48]. It was shown that the
Stanfield approach provides biased estimates even
for a large number of bearing measurements and
the ML approach is asymptotically unbiased at a
large number of measurements. However the RMS
(root mean square) error of Stanfield approach is
not necessarily larger than that of the ML
approach. A quite different approach is referred to
at the end of Section 3.3, using a very recently intro-
duced method of exploiting the over-determined
nature of the noiseless problem.                                            Fig. 6. Intersecting hyperbolas from three receivers.
2538                                            G. Mao et al. / Computer Networks 51 (2007) 2529–2553

Denote by D~ the TDOA measurement vector
                     t                                                            method performs significantly better than the spher-
                         T
½D~1N ; . . . ; D~N À1N Š . Denote by f(rt) the vector
    t             t                                                               ical interpolation method and is more robust
                                                                  T
 1
½c ðkr1 À rt k À krN À rt kÞ;...; 1 ðkrN À1 À rt k À krN À rt kފ
                                  c
                                                                                  against noise than the divide and conquer method.
and denote by S the covariance matrix of the                                      The computational complexity of Chan’s method
TDOA measurement errors. The ML estimator                                         is comparable to the spherical interpolation method
minimizes the following quadratic function:                                       but substantially less than the Taylor-series method
            Â              ÃT Â             Ã                                     [47]. Recently, Dogancay and Drake developed a
Qð^t Þ ¼ D~ À fð^t Þ SÀ1 D~ À fð^tÞ ;
     r          t      r          t      r                   ð25Þ                 closed form solution for localization of distant
in which f(rt) is a nonlinear vector function. In order                           transmitters based on triangulation of hyperbolic
to obtain a reasonably simple estimator, f(rt) can be                             asymptotes [58,59]. The hyperbolic curves are
linearized using Taylor series around a reference                                 approximated by linear asymptotes. The solution
point r0                                                                          exhibits some performance degradation with respect
                                                                                  to the maximum likelihood estimator at low noise
fðrt Þ % fðr0 Þ þ f r ðr0 Þðrt À r0 Þ;                                  ð26Þ      levels but outperforms the maximum likelihood esti-
                                                  2
where fr(r0) is a (N À 1) · 2 (in R ) matrix of partial                           mator at medium to high noise levels.
derivative of f with respect to r evaluated at r0. A
recursive solution to the ML estimator can then                                   3.3. Distance-based one-hop localization techniques
be obtained [47]
                                            À1               Â             Ã        The most well-known distance-based localization
^t;kþ1 ¼ ^t;k þ f r ðrt;k ÞT S À1 f r ðrt;k Þ f r ðrt;k ÞT S À1 D~ À fðrt;k Þ :
r        r                                                       t
                                                                                  technique is based on use of GPS. The GPS space
                                                                        ð27Þ      segment consists of 24 satellites in the medium earth
                                                                                  orbit at a nominal altitude of 20,200 km with an
This method relies on a good initial guess of the
                                                                                  orbital inclination of 55°. Each satellite carries
transmitter location. Moreover, in some situations
                                                                                  several high accuracy atomic clocks and radiates a
this method can result in significant location estima-
                                                                                  sequence of bits that starts at a precisely known
tion errors due to geometric dilution of precision
                                                                                  time. The location of a GPS satellite at any particu-
(GDOP) effects. GDOP describes a situation in
                                                                                  lar time instant is known. A GPS receiver located on
which a relatively small ranging error can result in
                                                                                  the earth derives its distance to a GPS satellite from
a large location estimation error because the trans-
                                                                                  the difference of the time a GPS signal is received at
mitter is located on a portion of the hyperbola far
                                                                                  the receiver and the time the GPS signal is radiated
away from both receivers [7,53]. Fang [54] gave an
                                                                                  by the GPS satellite. Ideally, distance measurements
exact solution to the hyperbolic equations in
                                                                                  to three GPS satellites allow the GPS receiver to
Eq. (22) when the number of TDOA measurements
                                                                                  uniquely determine its position. In reality, four sat-
are equal to the number of unknown transmitter
                                                                                  ellites, rather than three, are required because of
coordinates. However his solution cannot make
                                                                                  synchronization error in the receiver’s clock. The
use of extra measurements. Other techniques that
                                                                                  fourth distance measurement provides information
can deal with the more general situation with extra
                                                                                  from which the synchronization error of the receiver
measurements include the spherical interpolation
                                                                                  can be corrected and the receiver’s clock can be syn-
method [55], which is derived from least-squares
                                                                                  chronized to an accuracy better than 100 ns.
‘‘equation-error’’ minimization, and the divide and
                                                                                      Generally in a WSN, for a non-anchor node at
conquer method [56]. The divide and conquer
                                                                                  unknown location xt with noise-contaminated dis-
estimate is formed by combining the maximum like-                                                         ~        ~
                                                                                  tance measurements d 1 ; . . . ; d N to N anchors at
lihood estimates using possibly overlapping subsec-
                                                                                  known locations x1, . . . , xN, the location estimation
tions of the measurement data vector. The divide
                                                                                  problem can be formulated using a maximum likeli-
and conquer method can achieve the optimum per-
                                                                                  hood approach as
formance but it requires that the Fisher information
matrix is sufficiently large. Chan and Ho [57] devel-                                                        T
                                                                                  xt ¼ arg min½dð^t Þ À ~ SÀ1 ½dð^t Þ À ~
                                                                                  ^              x      dŠ       x      dŠ;            ð28Þ
oped a closed form solution valid for an arbitrary
number of TDOA measurements and arbitrarily                                       where ~ is a N · 1 distance measurement vector,
                                                                                           d
distributed transmitters. The solution is an approx-                              dð^t Þ is also a N · 1 vector ½k^t À x1 k;... ;k^t À xN kŠ
                                                                                    x                             x               x
imation of the maximum likelihood estimator when                                  and S is the covariance matrix of the distance mea-
the TDOA measurement errors are small. Chan’s                                     surement errors. This minimization problem can be
                                        G. Mao et al. / Computer Networks 51 (2007) 2529–2553                             2539

solved using a similar procedure described in Sec-                   eT Ae þ eT b þ c ¼ 0;                                ð32Þ
tion 3.1 and Section 3.2.                                                                     T
                                                                     where e = [e1, e2, e3] , the matrix A, vectors b and c
   An interesting development in the area is the use
                                                                     can be expressed in the form of known inter-anchor
of the Cayley–Menger determinant [60,61] to reduce
                                                                     distances d12, d13, d23 and measured distances
the impact of distance measurement errors on the                     ~ ~ ~
                                                                     d t1 ; d t2 ; d t3 . Eq. (32) forms an additional equality
location estimate [62,63]. To illustrate the concept,
                                                                     constraint on the non-anchor node position. For a
consider a non-anchor node xt having distance mea-
                                                                     non-anchor node forming m quadrilaterals with
surements to three anchors x1, x2, x3 in R2 , which is
                                                                     neighboring anchors, there are m independent equa-
shown in Fig. 7.
                                                                     tions like Eq. (32). These equality constraints can be
   The Cayley–Menger determinant of this quadri-
                                                                     combined with Eq. (28) using Lagrange multipliers
lateral is given by
                                                                   [62]. Numerical methods, such as the gradient des-
                          0 d2 d2 d2 1                             cent algorithm, can be exploited to search for the
                                 12 13  t1  
                          2                                        solution, which gives a location estimate superior
                          d 12 0 d 2 d 2 1 
                          2         23  t2                         to that obtained using Eq. (28) only.
 Dðx1 ; x2 ; x3 ; xt Þ ¼  d 13 d 2
                                 23 0 d 2 1 :
                                         t3     ð29Þ
                          2                                             The essence of using the Cayley–Menger determi-
                          d t1 d 2 d 2  0 1                        nant to reduce the impact of distance measurement
                                 t2  t3     
                          1     1   1   1 0                        errors is: the six edges of a planar quadrilateral are
                                                                     not independent, instead they must satisfy the
A classical result on the Cayley–Menger determi-
                                                                     equality constraint in Eq. (30). This equality
nant is given by the following theorem:
                                                                     constraint can be exploited to reduce the impact
Theorem 1 (Theorem 112.1 in [61]). Consider an n-                    of distance measurement errors. This idea may also
tuple of points x1, . . . , xn in m-dimensional space with           extend to TDOA and AOA [64] based localization.
n P m + 1. The rank of the Cayley–Menger matrix
M(x1, . . . , xn) (defined analogously to the right side of           3.4. Lighthouse approach to one-hop localization
Eq. (29) but without the determinant operation) is at
most m + 1.                                                               The lighthouse approach uses a base station
                                                                     equipped with three mutually perpendicular parallel
    According to the above theorem, in R2 ,                          optical beams to locate all optical receivers within
Dðx1 ; x2 ; x3 ; xt Þ ¼ 0:                                  ð30Þ     the range and line-of-sight of the beams in R3 . In
                                                                     Section 2.2.2, we have described the principle of
Note that the distances between anchors d12, d13 and                 the lighthouse approach to measure the distance
d23 can be inferred from known anchor positions.                     of an optical receiver to the rotational axis of a par-
The true distances between the non-anchor node                       allel optical beam [24]. Without loss of generality,
and the anchors are related to the measured dis-                     assuming the rotational axes of the three mutually
tances by                                                            perpendicular parallel optical beams are X, Y, and
~
d ti ¼ d ti þ ei ;   1 6 i 6 3:                             ð31Þ     Z axes respectively. As shown in Fig. 8, ignoring
                                                                     the measurement errors, the unknown receiver loca-
Putting Eq. (31) into Eq. (30), it can be shown that                 tion xt = [xt, yt, zt]T is related to the distance mea-
[62]                                                                 surements to the X, Y, and Z axes, denoted by
                                                                     ~ ~           ~
                                                                     d x ; d y and d z , respectively, through the following
                                                                     equations:
                                                                     ~
                                                                     d 2 ¼ y 2 þ z2 ;                                     ð33Þ
                                                                       x     t    t
                                                                     ~
                                                                     d 2 ¼ x2 þ z2 ;                                      ð34Þ
                                                                       y    t    t
                                                                     ~
                                                                     d2    ¼   x2   þ   y2:                               ð35Þ
                                                                       z        t        t

                                                                     Solving the above equations gives eight solutions,
                                                                     each corresponding a point in one of the eight quad-
                                                                     rants in R3 . By using priori knowledge of which
                                                                     quadrant the receiver is located in, only one solution
Fig. 7. A fully connected planar quadrilateral in sensor networks.   is chosen.
2540                                    G. Mao et al. / Computer Networks 51 (2007) 2529–2553

                                                                      comparison with distance-estimation based tech-
                                                                      niques, the RSS-profiling based techniques produce
                                                                      relatively small location estimation errors [42]. In
                                                                      [35], Elnahrawy et al. proposed several area-based
                                                                      localization algorithms using RSS profiling; these
                                                                      algorithms are area based because instead of esti-
                                                                      mating the exact location of the non-anchor node,
                                                                      they simply estimate a possible area that should
                                                                      contain it. Two different performance parameters
                                                                      apply: accuracy, or the likelihood that an object is
                                                                      within the area, and precision, i.e., the size of the
                                                                      area. Ref. [35] also considered three different tech-
                                                                      niques for the area based algorithms, viz., single
Fig. 8. An illustration of the lighthouse approach to localization.
                                                                      point matching, area based probability and Bayes-
                                                                      ian networks. The performance of all three algo-
   A practical system using the lighthouse approach
                                                                      rithms was compared with the point based
for 2D (XY plane) localization was reported to have
                                                                      algorithm of [42]. The conclusion was that all algo-
aP mean relative error of 1.1% in x axis (i.e.,
1    M                                                                rithms performed similarly, with a fundamental
M    i¼1 j^t;i À xt;i j=xt;i , M is the total number of receiv-
          x
                                                                      limit existing in the case of the RSS-profiling based
                                 and
ers in the experiment)P a mean relative error of
                                                                      localization algorithms, a conclusion also consistent
2.8% in y axis (i.e., M M j^t;i À y t;i j=y t;i ) [24]. Tech-
                               1
                                  i¼1 y                               with that of [65]. A rule of thumb is provided in [35].
niques dealing with non-ideal situations such as mis-
                                                                      Using 802.11 technology, with dense sampling and a
alignment of the rotational axes of optical beams
                                                                      good algorithm, one can expect a median localiza-
and non-parallel beams were also discussed in [24].
                                                                      tion error (i.e., distance between the estimated loca-
                                                                      tion and the true location) of about 3 m and a 97th
3.5. RSS-profiling based localization                                  percentile error of about 9 m. With relatively sparse
                                                                      sampling, every 6 m, or 37 m2/sample, one can still
    Given the RSS model constructed using the pro-                    achieve a median error of 4.5 m and 95th percentile
cedure described in Section 2.3, each non-anchor                      error of 12 m.
node unaware of its location uses the signal strength                    In [66], Ni et al. presented a weighted version of
measurements it collects, stemming from the anchor                    the RSS-profiling based localization technique
nodes within its sensing region, and thus creates its                 which achieves a more accurate location estimate.
own RSS finger print, which is then transmitted to                     Denote by c the signal strength vector of the non-
the central station. Then the central station matches                 anchor node. Denote by bi and xi the signal strength
the presented signal strength vector to the RSS                       vector and the location vector of the ith sample
model, using probabilistic techniques or some kind                    point respectively. In the weighted version of the
of nearest neighbor-based method, which chooses                       RSS-profiling based localization algorithm, the
the location of a sample point whose RSS vector                       location estimate of the non-anchor point is given
is the closest match to that of the non-anchor node                   by
to be the estimated location of the non-anchor node                                  1
                                                                             X
                                                                             N
                                                                                  kcÀbi k2
[42]. In this way, an estimate of the location of the                 xt ¼
                                                                      ^          PN              xi ;                   ð36Þ
                                                                                          1
non-anchor node can be obtained. The estimate is                             i¼1  i¼1 kcÀbi k2
transmitted to the non-anchor node from the central
station. Obviously, a non-anchor node could also                      where kc À bik denotes the Euclidean distance be-
obtain the full RSS model from the central station                    tween the two vectors c and bi, and N is the total
and perform its own location estimation.                              number of sample points. Experimental evaluation
    The accuracy of this technique depends on two                     showed that Ni’s approach achieves a median local-
distinct factors: the particular technique used to                    ization error of 1 m and a maximum localization er-
build the RSS model, with the resultant quality of                    ror of 2 m, which appears to be better than those
that model, and the technique used to fit the mea-                     reported in [67].
sured signal strength vector from a non-anchor                           The major practical obstacle in the RSS-profiling
node into the appropriate part of the model. In                       based localization is the extensive amount of profil-
                               G. Mao et al. / Computer Networks 51 (2007) 2529–2553                           2541

ing data required. Substantial and possibly costly          be the most attractive for a WSN because of its
initial experiments are needed to establish the             relatively simple hardware requirement.
model. Subsequent changes in the environment
(e.g., inside a building, office occupancy can change)        4. Nonline-of-sight error mitigation
can affect the model, and so a static model derived
from a single-shot experiment may be inadequate                A common problem in many localization tech-
in some applications. Recently, there has been pro-         niques is the nonline-of-sight (NLOS) error miti-
posed a method of online profiling, which would              gation. NLOS errors between two sensors can
reduce or possibly eliminate the amount of profiling         arise when either the line-of-sight between them is
required before deployment, but at the expense of           obstructed, perhaps by a building, or the line-of-
deploying a large number of additional devices              sight measurements are contaminated by reflected
(termed ‘‘sniffers’’) at known locations [36,44].            and/or diffracted signals. As NLOS error mitigation
Together with a large number of stationary emitters         in AOA based localization [75–77] and distance
(anchor nodes) deployed at known locations, the             based localization [78–81] share some degree of com-
‘‘sniffers’’ can be used to construct and update the         monality, we review them together in this section.
RSS model online.                                           Most NLOS error mitigation techniques assume that
                                                            NLOS corrupted measurements only constitute a
3.6. Localization based on hybrid measurements              small fraction of the total measurements. Since
                                                            NLOS corrupted measurements are inconsistent
   There are a number of other localization algo-           with LOS expectations, they can be treated as outli-
rithms based on data fusion [68] of hybrid measure-         ers. A typical approach is to assume that the mea-
ments. McGuire et al. explored data fusion of RSS           surement error has a Gaussian distribution, then
and TOA measurements for mobile terminal locali-            the least-squares residuals are examined to deter-
zation in a CDMA cellular network [69]. Li and              mine if NLOS errors are present [76,80,81] (by
Zhuang considered mobile user localization using            regarding any large residual as due to the NLOS sig-
hybrid TDOA/AOA measurements in a macrocell                 nals). Unfortunately, this approach fails to work
wideband CDMA system with frequency division                when multiple NLOS measurements are present as
duplex [70]. Gu and Gunawan considered mobile               the multiple outliers in the measurement tend to bias
user localization in a CDMA cellular network using          the final estimate decision and reduce the residuals.
hybrid AOA/TOA measurements [71]. Kleine-Ost-               This behavior motivates the use of deletion diagnos-
mann and Bell [72] presented a data fusion architec-        tics. In deletion diagnostics, the effects of eliminating
ture for combining TDOA and TOA measurements.               various measurements from the total set are com-
Thomas et al. considered the fusion of TDOA and             puted and ranked [80,82].
AOA measurements [73]. Catovic and Sahinoglu                   Some other approaches are proposed to reduce
                         ´
[74] computed the Cramer–Rao bounds on the loca-            estimation errors for time-of-arrival (TOA) [79,83]
tion estimation accuracy of two different hybrid             and TDOA [75] respectively when the majority of
schemes, i.e., TOA/RSS and TDOA/RSS, and                    the measurements are NLOS measurements. In
found that hybrid schemes offer improved accuracy            [79], Venkatraman et al. employed a constrained
with respect to conventional TOA and TDOA                   nonlinear optimization approach for TOA NLOS
schemes. Fundamentally, localization based on               error mitigation in a cellular network. Bounds on
hybrid measurements can achieve a performance               the NLOS error and the relationship between the
improvement over that based on a single measure-            true ranges are extracted from the geometry of
ment type because measurement noise for different            the cell layout and the measured range circles to
types of measurements comes from different sources.          serve as constraints. Wang et al. proposed an algo-
Therefore errors in the location estimate for each          rithm which attempts to mitigate NLOS error effect
measurement type are at least partially independent.        in a TOA based location system, utilizing the infor-
This independence between different types of mea-            mation that NLOS error causes the measured
surements can be exploited by data fusion tech-             distance to be greater than the true distance. A qua-
niques [68] to create estimators that have better           dratic programming approach is used to solve for
accuracy than estimators based on single measure-           an ML estimate of the source location [84]. Li
ment types. Among those hybrid techniques, the              and Zhuang proposed two NLOS error mitigation
fusion of RSS and TOA measurements appears to               algorithms assuming a full knowledge of NLOS
2542                            G. Mao et al. / Computer Networks 51 (2007) 2529–2553

error distribution (i.e., the probability that each          falls within 30% of the separation distance between
measurement is NLOS and the probability distribu-            two adjacent reference points.
tion of NLOS error) and a partial knowledge of                   The ‘‘DV (distance vector)-hop’’ approach devel-
NLOS error distribution (i.e., the probability that          oped by Niculescu and Nath [87] starts with all
each measurement is NLOS and the mean value                  anchors flooding their locations to other nodes in
of the probability distribution of NLOS error)               the network. The messages are propagated hop-
respectively [75]. However this prior information            by-hop and there is a hop-count in the message.
may be difficult to obtain in a WSN.                           Each node maintains an anchor information table
                                                             and counts the least number of hops that it is away
5. Connectivity based multi-hop localization                 from an anchor. When an anchor receives a message
algorithms                                                   from another anchor, it estimates the average dis-
                                                             tance of one hop using the locations of both anchors
   In the following sections, we shall review                and the hop-count, and sends it back to the network
multi-hop localization techniques in which the               as a correction factor. When receiving the correc-
non-anchor nodes are not necessarily the one-hop             tion factor, a non-anchor node is able to estimate
neighbors of the anchors. In particular, we focus            its distance to anchors and performs trilateration
on connectivity-based and distance-based multi-              to estimate its location. The algorithm was tested
hop localization algorithms due to their prevalence          using simulation with a total of 100 nodes uniformly
in multi-hop WSN localization techniques.                    distributed in a circular region of diameter 10. The
   There is a distinct category of localization algo-        average node degree, i.e., average number of neigh-
rithms, called connectivity-based or ‘‘range free’’          bors per node, is 7.6. Simulation results showed that
localization algorithms, which do not rely on any            the algorithm has a mean error of 45% transmission
of the measurement techniques in the earlier                 range with 10% anchors; and has a reduced mean
sections. Instead they use the connectivity informa-         error of about 30% transmission range when the
tion, i.e., ‘‘who is within the communications range         percentage of anchors increases above 20%.
of whom’’ [85] to estimate the locations of the non-             Shang et al. [85] developed a centralized algo-
anchor nodes. The principle of these algorithms is: a        rithm by using multi-dimensional scaling (MDS).
sensor being in the transmission range of another            MDS was originally used in psychometrics and psy-
sensor defines a proximity constraint between both            chophysics and it is a set of data analysis techniques
sensors, which can be exploited for localization.            that displays the structure of distance-like data as a
Bulusu et al. [86] and Niculescu and Nath [87] devel-        geometric picture. In their algorithm, the shortest
oped distributed connectivity-based localization             paths (i.e., the number of hops) between all pairs
algorithms; Shang et al. [85] and Doherty et al.             of nodes are first computed, which are used to con-
[88] developed centralized connectivity-based locali-        struct a distance matrix for MDS. Then MDS is
zation algorithms.                                           applied to the distance matrix and an approximate
   In [86], Bulusu et al. defined a connectivity met-         value of the relative coordinates of each node is
ric, which is the ratio of the number of transmitter         obtained. Finally, the relative coordinates are trans-
signals successfully received to the total number of         formed to the absolute coordinates by aligning the
signals from that transmitter, to measure the quality        estimated relative coordinates of anchors with their
of communication for a specific transmitter-receiver          absolute coordinates. The location estimates
pair. A receiver at an unknown location uses the             obtained using earlier steps can be refined using a
centroid of its reference points as its location esti-       least-squares minimization. Simulation was con-
mate, where a reference point is a transmitter with          ducted using 100 nodes uniformly distributed in a
a known location and whose connectivity metric               square of size 10 · 10 and four anchors were ran-
exceeds a certain threshold (90% in [86]). An exper-         domly placed in the region. The average node
iment was conducted in a 10 m · 10 m outdoor                 degree is 10. Simulation results showed a localiza-
parking lot using four reference points placed at            tion error of 0.35. Shang et al. further improved
the four corners of the 10 m · 10 m square. The              their algorithm in [89] by dividing the entire sensor
10 m · 10 m square was subdivided into 100 smaller           network into overlapping local regions. Localiza-
1 m · 1 m grids and the receivers were placed at the         tion is performed in individual regions using the ear-
grid points. Experimental results showed that for            lier described procedures. Then these local maps are
over 90% of the data points the localization error           patched together to form a global map by using
                                  G. Mao et al. / Computer Networks 51 (2007) 2529–2553                          2543

common nodes shared between adjacent regions.                  anchors increases to 18; it reduces to 0.5R when
The improved algorithm can achieve better perfor-              the number of anchors increases to 50.
mance on irregularly shaped networks by avoiding                  In comparison with other localization algo-
the use of distance information between far away               rithms, the most attractive feature of the connectiv-
nodes. The improved algorithm can also be imple-               ity-based localization algorithms is their simplicity.
mented in a distributed fashion.                               However they can only provide a coarse grained
   In the centralized algorithm of Doherty et al.              estimate of each node’s location, which means that
[88], the connectivity-based localization problem is           they are only suitable for applications requiring an
formulated as a convex optimization problem and                approximate location estimate only. Also the local-
solved using existing algorithms for solving linear            ization error is highly dependent on the node den-
programs and semidefinite programming (SDP)                     sity of the network, the number of anchors and
algorithms. Semidefinite programs are a generaliza-             the network topology. The location error will be lar-
tion of the linear programs and have the form                  ger in a network with a smaller node density, fewer
                                                               anchors, or irregular network topology.
Minimize     cT x                                   ð37Þ
Subject to   FðxÞ ¼ F0 þ x1 F1 þ Á Á Á þ xn Fn ;    ð38Þ       6. Distance-based multi-hop localization algorithms
             Ax < b;                                ð39Þ
                                                                  The core of distance-based localization algo-
             Fi ¼ FT ;
                   i                                ð40Þ       rithms is the use of inter-sensor distance measure-
                                                               ments in a sensor network to locate the entire
where x = [x1, x2, . . . , xn]T and xi represents the          network.
coordinate vector of node i, i.e., xi = [xi, yi]. The             Based on the approach of processing the individ-
quantities A, b, c and Fi are all known. The inequal-          ual inter-sensor distance data, distance-based local-
ity (39) is known as a linear matrix inequality. A             ization algorithms can be considered in two main
connection between node i and j can be represented             classes: centralized algorithms and distributed algo-
by a ‘‘radial constraint’’ on the node locations:              rithms. Centralized algorithms use a single central
kxi À xjk 6 R, where R is the transmission range.              processor to collect all the individual inter-sensor
This constraint is a convex constraint and can be              distance data and produce a map of the entire sen-
transformed into a LMI using Schur complements                 sor network, while distributed algorithms rely on
[88]. A solution to the coordinates of the non-an-             self-localization of each node in the sensor network
chor nodes satisfying the radial constraints can be            using the distances the node measures and the local
obtained by leaving the objective function cTx blank           information it collects from its neighbors. Next we
and solving the problem. Because there may be                  review the main characteristics as well as relevant
many possible coordinates of the non-anchor nodes              studies in the literature for each of the two classes
satisfying the constraints, the solution may not be            and compare them at the end of the section.
unique. If we set the element of c corresponding to
xi (or yi) to be 1 (or À1) and all other elements of           6.1. Centralized algorithms
c to be zero, the problem becomes a constrained
maximization (or minimization) problem. A lower                   In certain networks where a centralized informa-
bound or an upper bound on xi (or yi) satisfying               tion architecture already exists, such as road traffic
the radial constraints can be computed, from which             monitoring and control, environmental monitoring,
a rectangular box bounding the location estimates              health monitoring, and precision agriculture moni-
of the non-anchor nodes can be obtained. The                   toring networks, the measurement data of all the
algorithm was tested using simulation with a total             nodes in the network are collected in a central pro-
of 200 nodes randomly placed in a square of size               cessor unit. In such a network, it is convenient to
10R · 10R and the average node degree is 5.7 [88].             use a centralized localization scheme.
Simulation results showed that the mean location                  Once feasible to implement, the main motive
error is a monotonically decreasing function of the            behind the interest in centralized localization
number of anchors. When the number of anchors                  schemes is the likelihood of providing more accurate
is small, the estimated location is as poor as a ran-          location estimates than those provided by distrib-
dom guess of the node’s coordinates. The mean                  uted algorithms. In the literature, there exist three
location error reduces to R when the number of                 main approaches for designing centralized
2544                            G. Mao et al. / Computer Networks 51 (2007) 2529–2553

distance-based localization algorithms: multidimen-          method is its robustness against converging to a
sional scaling (MDS), linear programming and sto-            false local minimum. In order to apply this tool to
chastic optimization approaches.                             the problem of localizing a sensor network with m
    The MDS approach used in the connectivity-               anchor nodes numbered from 1 to m and n non-
based localization algorithms mentioned in Section           anchor nodes numbered from m + 1 to m + n, the
5, e.g. [85], can be readily extended to incorporate         location estimation problem is reformulated in an
distance measurements into the corresponding opti-           optimization framework as minimization of the cost
mization problem. Such an extension of the algo-             function
rithm in [85] using the MDS approach can be
                                                                   X XÀ
                                                                   mþn
                                                                                                    Á2
found in [90]. In this work, the whole sensor net-           J¼                              ~
                                                                                k^i À ^j k À d ij
                                                                                 x x                             ð41Þ
work is divided into smaller groups where adjacent                i¼mþ1 j2N i
groups may share common sensors. Each group
                                                                                                              ~
                                                             over f^i jm þ 1 6 i 6 m þ ng, where Ni, ^i and d ij de-
                                                                    x                                x
contains at least three anchors or sensors whose
locations have already been estimated. MDS is used           note, respectively, the neighborhood of node i, the
to estimate the relative locations of sensors in each        estimate of the location xi of node i, and the mea-
group and build local maps. Local maps are then              sured distance between nodes i and j.
stitched together to form an estimated global map               An algorithm to solve the above optimization
of the network by utilizing common sensors                   problem using the SA method is provided in [93].
between adjacent local maps. The estimated loca-             The performance of this algorithm is improved in
tions of the anchors in this estimated global map            [94] utilizing the information about the sensor loca-
are later iteratively aligned with the true locations        tions hidden in the knowledge of whether a given
of anchors to obtain the final estimated global               pair of sensors are neighbors and mitigating a
map. Although this algorithm appears to have a dis-          certain kind of localization error caused by flip
tributed architecture, since a large number of itera-        ambiguity, a concept which is described in detail
tions (implies a high communication cost) are                in Section 7. The effectiveness of the enhanced algo-
required for the algorithm to converge, it is more           rithm in [94] is also demonstrated via simulations
appropriate to be implemented using a centralized            where the relation between the actual value dij and
                                                                                     ~
                                                             the measured value d ij of the distance between
architecture. Ji’s algorithm was tested using a total
                                                                                              ~
                                                             nodes i and j is assumed to be d ij ¼ d ij ð1 þ 0:1nij Þ,
of 400 nodes (10% anchors) uniformly distributed
in a square of 100 · 100 and a transmission range            where nij is a zero-mean Gaussian noise of unit var-
of R = 10. The distance measurement error was                iance. The simulations were performed in a sensor
assumed to be uniformly distributed in the range             network of 200 nodes uniformly distributed in a
[0, g]. It was shown that when g is 0, 0.05R, 0.25R          square of size 10 by 10. The results of these simula-
and 0.5R, the localization error is 0.1R, 0.15R,             tions were compared with the ones obtained using
0.3R and 0.45R respectively.                                 the SDP approach with gradient search improve-
    Similarly to the MDS approach, the semi-definite          ment [92] in Fig. 9, where the location estimation
programming (SDP) approach used for connec-                  error is normalized by the transmission range. As
tivity-based localization algorithms can also be             can be seen in the figure, the SA algorithm has bet-
extended to incorporate distance measurements                ter accuracy than the SDP algorithm with gradient
[88]. In [91] the distance-based sensor network local-       search. This is an expected result of robustness of
ization problem is formulated in a quadratic form            SA against convergence to false local minima; how-
and solved using SDP; and in [92] the result in              ever it is worth noting that the computational cost
[91] is improved using a gradient search procedure           of the SA approach is higher.
to fine-tune the initial solution obtained using SDP.
    The stochastic optimization approach suggests            6.2. Distributed localization
an alternative formulation and solution of the dis-
tance-based localization problem using combinato-               Similarly to the centralized ones, the distributed
rial optimization notions and tools. The main tool           distance-based localization approaches can be
used in this approach is the simulated annealing             obtained as an extension of the distributed connec-
(SA) technique [93], which is a generalization of            tivity-based localization algorithms in Section 5 to
the well known Monte Carlo method in combinato-              incorporate the available inter-sensor distance infor-
rial optimization. One particular property of the SA         mation. In [87], after developing the ‘‘DV-hop’’
                                                                                 G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                2545

                                                250                                                               range in the presence of 5% distance measurement
 Localization Error (% of Transmission Range)



                                                                                     SDP – 5% Anchor              error (normalized by the transmission range).
                                                                                     SDP – 10% Anchor
                                                200
                                                                                                                     The four-stage scheme of [96] is based on intro-
                                                                                     SA – 5% Anchor
                                                                                                                  duction of the notion of ‘‘tentative uniqueness’’,
                                                                                     SA – 10% Anchor
                                                                                                                  where a node is called ‘‘tentatively uniquely’’ local-
                                                150                                                               izable if it has at least three neighbors that are either
                                                                                                                  non-collinear anchors or ‘‘tentatively uniquely’’
                                                100
                                                                                                                  localizable nodes. In the first stage, the ‘‘tentatively
                                                                                                                  uniquely’’ localizable nodes are selected. The loca-
                                                                                                                  tions of these tentatively uniquely localizable nodes
                                                50                                                                are estimated in stages two and three. In the second
                                                                                                                  stage, each non-anchor node produces its estimated
                                                 0
                                                                                                                  distances to at least three anchors using a ‘‘DV-dis-
                                                 1.1   1.2   1.3   1.4   1.5   1.6     1.7    1.8       1.9   2   tance’’ like algorithm. An estimated distance to an
                                                                    Transmission Range
                                                                                                                  anchor node allows the location of the non-anchor
Fig. 9. Performance of SA algorithm with flip ambiguity mitiga-                                                    node to be constrained inside a square centered at
tion and SDP algorithm with gradient search improvement.                                                          that anchor node. In comparison with a circle, the
                                                                                                                  use of a square may simplify computation. The esti-
algorithm described in Section 5, a modified version                                                               mated distances to more than three anchors allow
of this algorithm which includes distance measure-                                                                the location of the non-anchor node to be confined
ments into the localization process, the ‘‘DV-dis-                                                                inside a rectangular box, which is the intersection of
tance’’ algorithm, is presented as well. The main                                                                 the squares corresponding to each of these anchors.
idea in the ‘‘DV-distance’’ algorithm as compared                                                                 The location of the non-anchor node is estimated to
to the ‘‘DV-hop’’ algorithm is propagation of mea-                                                                be at the center of the rectangular box. The initial
sured distance among neighboring nodes instead of                                                                 location estimates obtained in the second stage are
hop count.                                                                                                        refined in the third stage by a least-squares trilater-
   Two similar approaches are the two-stage locali-                                                               ation using the location estimates of the neighboring
zation scheme of Savarese and Rabaey [95] and the                                                                 nodes and the measured distances. In the final stage
four-stage algorithm of Savvides et al. [96]. In the                                                              of the algorithm, the location of each node deemed
first stage of the scheme in [95], a ‘‘hop-terrain’’                                                               not tentatively uniquely localizable in stage one is
algorithm, which is similar to the ‘‘DV-hop’’ algo-                                                               estimated using the location estimates of its tenta-
rithm, is used to obtain an initial estimate of the                                                               tively uniquely localizable neighbors.
node locations. In the second stage, the measured                                                                    All of the above three algorithms [87,96,95] have
distances between neighboring nodes are used to                                                                   three phases [97]: (a) determination of the distances
refine the initial location estimates iteratively. At                                                              between non-anchor nodes and anchor nodes; (b)
each iteration step, each node updates its location                                                               derivation of the location of at least some non-
estimate by a least-squares trilateration using the                                                               anchor nodes from their distances to the anchors;
location estimates of its neighbors and the measured                                                              (c) refinement of the location estimates using mea-
distances. To mitigate location estimate errors                                                                   sured distances between neighbors. In [97] the per-
caused by error propagation and unfavorable net-                                                                  formances of the three algorithms and some
work topologies, a confidence value is assigned to                                                                 variants of them were compared and it was con-
each node’s location, where an anchor has a higher                                                                cluded that the algorithms have comparable perfor-
confidence value (close to 1) and a non-anchor node                                                                mance and which algorithm has better accuracy
with few neighbors and poor constellation has lower                                                               depends on the specific application conditions such
confidence value (close to 0). The proposed algo-                                                                  as distance measurement error, vertex degree and
rithm is tested via simulation as well in [95] using                                                              percentage of anchors. The algorithm proposed by
a sample network with 400 nodes, 5% of which                                                                      Nagpal et al. [98] more recently can be classified into
are anchor nodes, uniformly placed in a 100 by                                                                    the same category as the above three algorithms.
100 square and an average node degree greater than                                                                   There exists another category of distributed
7. The simulation results demonstrated that the                                                                   localization algorithms in the literature, where local
algorithm is able to achieve an average location esti-                                                            maps are constructed using distance measurements
mation error of less than 33% of the transmission                                                                 between neighboring nodes first and then common
2546                              G. Mao et al. / Computer Networks 51 (2007) 2529–2553

nodes between local maps are used to stitch them               worth noting that decentralized localization is
together to form a global map. The localization                strictly harder than centralized, i.e., any algorithm
                                         ˇ
algorithms by Ji and Zha [90] and Capkun et al.                for decentralized localization can always be applied
[99] are typical examples of this category. In the             to centralized problems, but not the reverse.
               ˇ
algorithm of Capkun et al. [99], each node builds                 From the perspective of location estimation accu-
its local coordinate system and the locations of its           racy, centralized algorithms are likely to provide
neighbors are calculated in the local coordinate sys-          more accurate location estimates than distributed
tem. Then the directions of the local coordinate sys-          algorithms. However centralized algorithms suffer
tems are aligned to be the same using common                   from the scalability problem and generally are not
nodes between adjacent local coordinate systems.               feasible to be implemented for large scale sensor
Finally, the local coordinate systems are reconciled           networks. Other disadvantages of centralized algo-
into a global coordinate system using linear transla-          rithms, as compared to distributed algorithms, are
tion. Error propagation and the large number of                their requirement of higher computational complex-
iterations required for the algorithm to converge              ity and lower reliability due to accumulated infor-
are the major problems in these algorithms.                    mation inaccuracies/losses caused by multi-hop
    A recent direction of research in distributed sen-         transmission over a wireless network.
sor network localization is the use of particle filters            On the other hand, distributed algorithms are
[100]. Particle filters have been used in navigation            more difficult to design because of the potentially
and tracking [101]. In [102], Ihler et al. formulated          complicated relationship between local behavior
the sensor network localization problem as an infer-           and global behavior, e.g., algorithms that are locally
ence problem on a graphical model and applied a                optimal may not perform well in a global sense.
variant of belief propagation (BP) techniques, the             Optimal distribution of the computation of a cen-
so-called nonparametric belief propagation (NBP)               tralized algorithm in a distributed implementation
algorithm, to obtain an approximate solution to                in general is an unsolved research problem. Error
the sensor locations. In [102], the NBP idea is imple-         propagation is another potential problem in distrib-
mented as an iterative local message exchange algo-            uted algorithms. Moreover, distributed algorithms
rithm, in each step of which each sensor node                  generally require multiple iterations to arrive a
quantifies its ‘‘belief’’ about its location estimate,          stable solution which may cause the localization
sends this belief information to its neighbors,                process to take longer time than the acceptable in
receives relevant messages from them, and then iter-           some cases.
atively updates its belief. The iteration process is ter-         To compare the centralized and distributed dis-
minated only when some convergence criterion is                tance-based localization algorithms from the com-
met about the beliefs and location estimates of the            munication energy consumption perspective, one
sensors in the network. The main advantages of                 needs to consider the individual amounts of energy
the NBP algorithm are its easy implementation in               required for each type of operation in the localiza-
a distributed fashion and sufficiency of a small num-            tion algorithm in the specific hardware and the
ber of iterations to converge. Furthermore it is               transmission range setting. Depending on the set-
capable of providing information about location                ting, the energy required for transmitting a single
estimation uncertainties and accommodating non-                bit could be used to execute 1000–2000 instructions
Gaussian distance measurement errors. It is demon-             [103]. Centralized algorithms in large networks
strated via simulations [102] that the overall perfor-         require each sensor’s measurements to be sent over
mance of NBP is comparable to that of a centralized            multiple hops to a central processor, while distrib-
MAP (maximum a posteriori) estimate. Some future               uted algorithms require only local information
research directions to further improve the NBP                 exchange between neighboring nodes but many such
approach can be found in [102].                                local exchanges may be required, depending on the
                                                               number of iterations needed to arrive at a stable
6.3. Centralized versus distributed algorithms                 solution. A comparison of the communication
                                                               energy efficiencies of centralized and distributed
   Centralized and distributed distance-based locali-          algorithms can be found in [104]. It was concluded
zation algorithms can be compared from perspectives            in [104] that in general, if in a given sensor network
of location estimation accuracy, implementation and            and distributed algorithm, the average number of
computation issues, and energy consumption. It is              hops to the central processor exceeds the necessary
                                  G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                          2547

number of iterations, then the distributed algorithm           i, j 2 V, which are connected by an edge in E, also
will be more energy-efficient than a typical central-            satisfies the equality in (ii) for any other vertex pairs
ized algorithm.                                                that are not connected by a single edge.
                                                                    If a framework ðG; pÞ is rigid but not globally
7. Graph theoretic research problems in distance-              rigid, like the ones in Fig. 10, there exist two types
based sensor network localization                              of discontinuous deformations that can prevent a
                                                               representation of G consistent with p, i.e., a repre-
     Despite a significant number of approaches                 sentation ðG; p1 Þ satisfying kpðiÞ À pðjÞk ¼ kp1 ðiÞÀ
developed for WSN localization, there are still many           p1 ðjÞk for any vertex pair i, j 2 V, which are con-
unsolved problems in the area. The challenges to be            nected by an edge in E, from being unique (in the
addressed are both in analytical characterization of           sense that it differs from other such representations
the sensor networks (from the aspect of localization)          at most by translation, rotation or reflection)
and development of (efficient) localization algo-                [107,110]: flip and discontinuous flex ambiguities. In
rithms for various classes of sensor networks under            flip ambiguities in Rd , (d 2 {2, 3}), at least a vertex
a variety of conditions. In this section, we present           (sensor node) v has a set of neighbors which span
some of these research problems with a discussion              a (d À 1)-dimensional subspace, which leads to the
on possible approaches to them. Although these                 possibility of the neighbors forming a mirror
problems may also exist in localization using other            through which v can be reflected. Fig. 10a depicts
types of measurement techniques (e.g., TDOA and                an example of flip ambiguity. In discontinuous flex
AOA), we focus on distance-based sensor network                ambiguities, the temporary removal of an edge or,
localization.                                                  in some cases, a set of edges allows the remaining
     A fundamental problem in distance-based sensor            part of the graph to be flexed to a different realiza-
network localization is whether a given sensor net-            tion (which cannot be obtained from the original
work is uniquely localizable or not. A framework               realization by translation, rotation or reflection)
that is useful for analyzing and solving the problem           and the removed edge reinserted with the same
is graph theory [105–109]. In a graph theoretical              length. Fig. 10b depicts an example.
framework, a sensor network can be represented by                   Use of graph rigidity and global rigidity notions
a graph G = (V, E) with a vertex set V and an edge             in sensor network localization are well described
set E, where each vertex i 2 V is associated with a            and their importance is well demonstrated from
sensor node si in the network, and each edge                   both the algorithmic and the analytic aspects in
(i, j) 2 E corresponds to a sensor pair si, sj for which       the recent literature [106,98,108,111]. Particularly,
the inter-sensor distance dij is known. The location           it is established in [105,106] that a necessary and suf-
information about the sensors corresponds to a rep-            ficient condition for unique localization of a d-
resentation of the representative graph. In general, a         dimensional sensor network is global rigidity of
d-dimensional (d 2 {2, 3}) representation of a graph           any d-dimensional representation ðG; pÞ, where G
G = (V, E) is a mapping p : V ! Rd , which assigns             is the representative graph of the sensor network
a location in Rd to each vertex in V. Given a graph            and the edge lengths kpðiÞ À pðjÞk imposed by p
G = (V, E) and a d-dimensional representation of it,
the pair ðG; pÞ is called a d-dimensional framework.
     A particular graph property associated with
                                                                                                                             5
unique localizability of sensor networks is global                        3                 3        1
                                                                                                                 4                    4
                                                                                                                         1
rigidity [106,108,109]. A framework ðG; pÞ is glob-             1             4
                                                                                  1    4
                                                                                                         5
                                                                                                 2
ally rigid if every framework ðG; p1 Þ satisfying
kpðiÞ À pðjÞk ¼ kp1 ðiÞ À p1 ðjÞk for any vertex pair                                                                2
                                                                                                             3                    3
i, j 2 V, which are connected by an edge in E, also                      2                 2

satisfies the same equality for any other vertex pairs          Fig. 10. An illustration of discontinuous deformations on non-
that are not connected by a single edge. A relaxed             globally rigid frameworks: (a) Flip ambiguity: vertex 4 can be
form of global rigidity is rigidity: A framework               reflected across the edge (2, 3) to a new location without violating
                                                               the distance constraints. (b) Discontinuous flex ambiguity:
ðG; pÞ is rigid if there exists a sufficiently small posi-
                                                               removing the edge (1, 4), flexing the edge triple (1, 5), (1, 2),
tive constant e such that every framework ðG; p1 Þ             (2, 3), and reinserting the edge (1, 4) so that the distance
satisfying (i) kpðiÞ À p1 ðiÞk < e for all i 2 V and (ii)      constraints are not violated in the end, we obtain a new
kpðiÞ À pðjÞk ¼ kp1 ðiÞ À p1 ðjÞk for any vertex pair          framework.
2548                                     G. Mao et al. / Computer Networks 51 (2007) 2529–2553

are equal to the corresponding known inter-sensor                      distance measurements. We have little knowledge in
distances dij, assuming that the absolute positions                    this area. For example, it is a common knowledge
of at least three sensors in R2 (which do not lie on                   that in the presence of noisy distance measurements,
the same line) or four sensors in R3 (which do not                     a node (in R2 ) is likely to have flip ambiguity prob-
lie on the same plane) are known. Formal statement                     lem if its neighbors are nearly collinear. However
of this relation can be found in [105,106].                            there is little work in quantifying this relationship.
    Note that the necessity of global rigidity for                     A recent work focusing on robust distributed local-
unique localization as stated is valid for general sit-                ization of sensor networks with certain distance
uations where other a priori information is not help-                  measurement errors and ambiguities caused by
ful. Rigidity is needed, in any case, to have a finite                  these errors is presented in [107]. In this paper, cer-
number of solutions. However, in some cases where                      tain criteria are provided in selection of the sub-
ðG; pÞ is rigid but not globally rigid, some additional                graphs of the representative graph of a network to
a priori information may compensate the need for                       be used in a localization algorithm robust against
global rigidity. For example, consider a sensor net-                   such errors. The analysis in [107], however, is not
work that can be represented by a unit disk graph,                     complete and there may be other criteria that may
where there is an edge between representative verti-                   better characterize robustness of a given sub-net-
ces of two sensor nodes if and only if the distance                    work against distance measurement errors.
between them is less than a certain threshold                              Another relevant research problem is under-
R > 0, which is called the transmission range or                       standing and utilizing the error propagation charac-
sensing radius [106]. Then the ambiguities due to                      teristics in a sensor network. This issue emerges
the non-globally rigid nature of the representative                    especially in estimation of the location of non-
graph may sometimes be eliminated using the unit                       immediate neighbors of anchor sensors, i.e., k-hop
disk graph properties as demonstrated in Fig. 11.                      neighbors of anchor nodes with k P 2. Other things
In practice, a wireless sensor always has a limited                    being equal, a node further away from anchor nodes
transmission range, which implies that a WSN                           is likely to have a larger location estimation error,
may have the property of a unit disk graph. There-                     because its location estimation error is not only
fore global rigidity is only a sufficient condition for                  affected by the distance measurement errors to its
unique localization of a WSN; the necessary condi-                     neighbors but also affected by the location estima-
tion for unique localization is still an open research                 tion errors of its neighbors using which the node’s
problem.                                                               location is estimated. Numerous simulations and
    A more challenging research problem is analyz-                     experimental studies suggested that in addition to
ing the characteristics of wireless sensor networks                    distance measurement error, error propagation (as
(from the aspect of localization) in the case of noisy                 well as location estimation error) may be affected
                                                                       by node degree, network topology, and the distribu-
                                    1                                  tion of both non-anchor and anchor nodes. Some of
                                                                       the related tools can be seen in [102]. Other relevant
                                          R                            work includes the papers by Niculescu and Nath
               2                                                       [112] and by Savvides et al. [113]. In [112], Niculescu
                                        4 (Case 2)                     and Nath used a combination of the Cramer–Rao   ´
                                                                       bound and simulations to investigate the error char-
                                                                       acteristics of four classes of multi-hop APS (Ad Hoc
                                                                       positioning systems) algorithms. In [113] Savvides
                                    3                                  et al. also used a combination of the Cramer–Rao´
                                                                       bound and simulations to investigate the error char-
                       4 (Case 1)                                      acteristics for a specific scenario in which anchors
Fig. 11. Localization of a non-globally rigid unit-disk framework      are located near the boundary of the region and
in R2 : assume that the location of vertices 1, 2, 3 and the lengths   non-anchor nodes are located inside the region.
of the edges (2, 4) and (3, 4) are known and that there is no edge     Some qualitative trends on how localization error
between 1 and 4. There exist two possible locations for vertex 4 in    varies with average node degree, number of anchors
general: Case 1 and Case 2. We can eliminate Case 2 using the
unit-disk property since for this case there had to be an edge
                                                                       and distance to anchors are observed. A potential
between vertices 1 and 4 since d14 < R. Hence Case 1 gives the                                             ´
                                                                       problem with using the Cramer–Rao bound to
correct unique location of vertex 4.                                   study the performance of a localization algorithm
                                 G. Mao et al. / Computer Networks 51 (2007) 2529–2553                                     2549

                  ´
is that the Cramer–Rao bound assumes the underly-             sented because of their popularity in wireless sensor
ing estimator is unbiased. This assumption needs to           network localization. Despite significant research
be validated with the estimators used in various              developments in the area, there are still quite many
localization algorithms, and in particular, the class         unsolved problems in wireless sensor network local-
of algorithms which minimize the sum of the square            ization. A discussion on some fundamental research
of the difference between measured distances and               problems in distance-based location and possible
estimated distances. The aforementioned flip ambi-             approaches to these problem was also presented in
guity and discontinuous flex ambiguity problems                this paper.
make such validation particularly difficult. An esti-
mator may produce an unbiased estimate in one
                                                              Acknowledgment
topology (e.g., a dense network with high node
degree) but give a biased estimate in another topol-
                                                                 The authors would like to thank the anonymous
ogy (e.g., a spare network with low node degree).
                                                              reviewers for their helpful comments which have
We are yet to develop more comprehensive knowl-
                                                              significantly improved the quality of the paper.
edge in this large and fascinating area.
    Still another research area is concerned with
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                                                                     drive servo systems.

                        Guoqiang Mao received the Bachelor
                        degree in electrical engineering, Master                                Brian D.O. Anderson was born in Syd-
                        degree in engineering and Ph.D. in tele-                                ney, Australia, and received his under-
                        communications engineering in 1995,                                     graduate education at the University of
                        1998 and 2002, respectively. After grad-                                Sydney, with majors in pure mathemat-
                        uation from Ph.D., he worked in                                         ics and electrical engineering. He subse-
                        ‘‘Intelligent Pixel Incorporation’’ as a                                quently obtained a Ph.D. degree in
                        Senior Research Engineer for one year.                                  electrical engineering from Stanford
                        He joined the School of Electrical and                                  University. Following completion of his
                        Information Engineering, the University                                 education, he worked in industry in Sil-
                        of Sydney in December 2002 where he is                                  icon Valley and served as a faculty
a Senior Lecturer now. His research interests include wireless                                  member in the Department of Electrical
sensor networks, wireless localization techniques, network QoS,      Engineering at Stanford. He was Professor of Electrical Engi-
telecommunications traffic measurement, analysis and modeling,         neering at the University of Newcastle, Australia from 1967 until
and network performance analysis.                                    1981 and is now a Distinguished Professor at the Australian
                                                                     National University and Chief Scientist of National ICT Aus-
                                                                     tralia Ltd. His interests are in control and signal processing. He is
                         Barıs Fidan received the B.S. degrees in
                              ß                                      a Fellow of the IEEE, Royal Society London, Australian Acad-
                         electrical engineering and mathematics      emy of Science, Australian Academy of Technological Sciences
                         from Middle East Technical University,      and Engineering, Honorary Fellow of the Institution of Engi-
                         Turkey in 1996, the M.S. degree in elec-    neers, Australia, and Foreign Associate of the US National
                         trical engineering from Bilkent Univer-     Academy of Engineering. He holds doctorates (honoris causa)
                         sity, Turkey in 1998, and the Ph.D.                               ´
                                                                     from the Universite Catholique de Louvain, Belgium, Swiss
                         degree in Electrical Engineering-Systems    Federal Institute of Technology, Zurich, Universities of Sydney,
                                                                                                           ¨
                         at the University of Southern California,   Melbourne, New South Wales and Newcastle. He served a term
                         Los Angeles, USA in 2003. After work-       as President of the International Federation of Automatic Con-
                         ing as a postdoctoral research fellow at    trol from 1990 to 1993 and as President of the Australian
                         the University of Southern California for   Academy of Science between 1998 and 2002. His awards include
one year, he joined the Systems Engineering and Complex Sys-         the IEEE Control Systems Award of 1997, the 2001 IEEE James
tems Program of National ICT Australia and the Research              H. Mulligan, Jr. Education Medal, and the Guillemin-Cauer
School of Information Sciences and Engineering of the Austra-        Award, IEEE Circuits and Systems Society in 1992 and 2001, the
lian National University, Canberra, Australia in 2005, where he is   Bode Prize of the IEEE Control System Society in 1992 and the
currently a researcher. His research interests include autonomous    Senior Prize of the IEEE Transactions on Acoustics, Speech and
formations, sensor networks, adaptive and nonlinear control,         Signal Processing in 1986.
switching and hybrid systems, mechatronics, and various control

				
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