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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 m ¼ P 0 ðd 0 Þ ½dB m 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 b SÀ1 ½hð^t Þ À b ^ 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 Þ; ^ ^ 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 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 . 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 kÞ 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 d x d; 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 k 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. 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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