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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 1, JANUARY 2012
Non-Line-of-Sight Node Localization Based on Semi-Definite Programming in Wireless Sensor Networks Hongyang Chen, Gang Wang, Zizhuo Wang, H. C. So, and H. Vincent Poor, Fellow, IEEE
Abstract—An unknown-position sensor can be localized if there are three or more anchors making time-of-arrival (TOA) measurements of a signal from it. However, the location errors can be very large due to the fact that some of the measurements are from non-line-of-sight (NLOS) paths. In this paper, a semidefinite programming (SDP) based node localization algorithm in NLOS environments is proposed for ultra-wideband (UWB) wireless sensor networks. The positions of sensors can be estimated using the distance estimates from location-aware anchors as well as other sensors. However, in the absence of line-ofsight (LOS) paths, e.g., in indoor networks, the NLOS range estimates can be significantly biased. As a result, the NLOS error can remarkably decrease the location accuracy, and it is not easy to accurately distinguish LOS from NLOS measurements. According to the information known about the prior probabilities and distributions of the NLOS errors, three different cases are introduced and the respective localization problems are addressed. Simulation results demonstrate that this algorithm achieves high location accuracy even for the case in which NLOS and LOS measurements are not identifiable. Index Terms—Wireless sensor networks, non-line-of-sight (NLOS), time-of-arrival (TOA), semi-definite programming (SDP).
I. I NTRODUCTION
D
EVELOPMENT of localization algorithms for wireless sensor networks (WSNs) to find node positions is an important research topic because position information is a major requirement in many WSN applications. Examples include animal tracking, earthquake monitoring and locationaided routing.
Manuscript received October 4, 2010; revised March 30, 2011 and August 17, 2011; accepted October 4, 2011. The associate editor coordinating the review of this paper and approving it for publication was S. Ghassemzadeh. This research was supported in part by the National Natural Science Foundation of China under Grant No. 61071107, and the U. S. Office of Naval Research under Grant N00014-09-1-0342. The authors would like to thank Dr. W.-K. Ma from the Chinese University of Hong Kong, Dr. Kenneth W. K. Lui and the anonymous reviewers for their valuable suggestions concerning this work. H. Chen was with the Institute of Industrial Science, the University of Tokyo, Tokyo, Japan. He is now with Fujitsu Labs. Ltd, Japan (e-mail:
[email protected]). G. Wang is with the Institute of Communication Technology, Ningbo University, Ningbo 315211, China (e-mail: xd
[email protected]). Z. Wang is with the Department of Management Science and Engineering, Stanford University, Stanford, CA, USA (e-mail:
[email protected]). H. C. So is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2011.110811.101739
Based on the type of information provided for localization, sensor protocols can be divided into two categories: rangebased and range-free. Due to the coarse location accuracy of range-free schemes, solutions based on range-based localization are often preferable. Range estimates from anchors can be obtained using received signal strength (RSS), angleof-arrival (AOA) or time-of-arrival (TOA) observations of transmitted calibration signals. Impulse-based ultra-wideband (UWB) is a promising technology that allows precise ranging to be embedded into data communication. It is robust in dense multipath environments and it is able to provide accurate position estimation with low-data-rate communication. In this paper, we focus on the investigation of range-based localization algorithms for UWB WSNs. One of the main challenges for accurate node localization in range-based localization algorithms is non-line-of-sight (NLOS) propagation due to obstacles in the direct paths of beacon signals. NLOS propagation results in unreliable localization and significantly decreases the location accuracy if its effects are not taken into account. This often occurs in an urban or indoor environment. Some localization algorithms that cope with the existence of NLOS range measurement have been proposed in [1], [2] and [3], mostly for cellular networks. In those works, there are two approaches to deal with the localization problem in the presence of NLOS propagation [7], [17], [18]. The first approach identifies LOS and NLOS information and discards the NLOS range information. The second approach uses all NLOS and LOS measurements and provides weighting or scaling to reduce the adverse impact of NLOS range errors on the accuracy of position estimates. In both methodologies, it is assumed that the NLOS range estimates have been identified. In [4], Mazuelas et al. proposed the prior NLOS measurement correction (PNMC) method to mitigate the effects of NLOS propagation in cellular networks. Furthermore, Bahillo et al. implemented this interesting idea in a real indoor environment to alleviate the effect of severe NLOS propagation on distance estimates [5]. In the sensor network localization problem, the number of anchors is typically limited by practical considerations. Therefore, it is a waste of resources to discard NLOS range measurements. To make best use of all range measurements, we propose in this paper a computationally efficient semidefinite programming (SDP) approach for this problem which effectively incorporates both LOS and NLOS range information into the estimates. In the ideal scenario, by assuming that the NLOS er-
c 2012 IEEE 1536-1276/12$31.00 ⃝
CHEN et al.: NON-LINE-OF-SIGHT NODE LOCALIZATION BASED ON SEMI-DEFINITE PROGRAMMING IN WIRELESS SENSOR NETWORKS
ror is exponentially distributed with known parameters, the sum of the NLOS error and the measurement noise can be approximated as a Gaussian random variable. Given this fact, the NLOS measurement model can be approximately expressed in a form similar to the LOS measurement model. Combining the LOS and the approximate NLOS models, the approximate maximum likelihood (AML) estimation problem can be naturally formulated. However, the AML formulation is non-convex. We then relax this problem to an SDP and thus solve the relaxed problem. For the case in which only the probability of NLOS propagation and distribution parameters are known, the AML formulation is quite complex. To facilitate SDP relaxation, we further approximate the estimation problem. After this step, the SDP relaxation technique can be easily applied to estimate the node positions in this NLOS environment. The last case we focus on is the problem of NLOS mitigation without the requirement of accurately distinguishing between LOS and NLOS range estimates and without prior NLOS error information. Given a mixture of LOS and NLOS range measurements, our method is applicable in both cases without discarding any range information. To our knowledge, this method is the first SDP based approach to reduce the impact of NLOS in WSNs. The main advantages of this approach are given as follows. 1) The statistics of the NLOS bias errors are not assumed to be known a priori for our method. NLOS range estimates are not required to be readily distinguished from LOS range estimates through channel identification. Thus, it makes use of all measurements. 2) No range information is discarded. 3) SDP is efficiently applied to address the NLOS node localization problem which achieves excellent localization accuracy. In our proposed approach, we assume the following features of UWB TOA-based range estimation: the range bias errors in NLOS conditions are always positive and significantly larger in magnitude than the range-measurement noises. We will show that the node localization problem, given range information, can be cast into a nonlinear programming formalism. We then use SDP relaxation techniques and rely on both LOS and NLOS range estimates to estimate sensors’ positions in the NLOS environment. The rest of the paper is organized as follows. Section II introduces some technical preliminaries and Section III derives the SDP based localization algorithms for NLOS environments. In Section IV, simulation results are reported. Section V includes our conclusions. II. BACKGROUND In this section, we introduce some technical preliminaries. The basic setting of this paper is as follows. We assume an asynchronous UWB sensor network. Each node has the ability to achieve TOA estimation based on UWB channel model. We use a complex baseband-equivalent channel model which is adopted by the IEEE 802.15.4a working group. As described in [6], the modeling and characteristics of the employed channel models are available for residential, office,
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outdoor, and industrial environments. There are 𝑛 unknownposition sensors in R2 and 𝑚 anchor sensors whose positions are known a priori. We use x𝑖 ∈ R2 , 𝑖 = 1, 2, ..., 𝑛, to denote the unknown-position sensors and x𝑗 ∈ R2 , 𝑗 = 𝑛 + 1, 𝑛 + 2, ..., 𝑛 + 𝑚, to denote the anchors. We use 𝑟𝑖,𝑗 to denote the actual distance between the 𝑖th sensor and the 𝑗th sensor or anchor, i.e., 𝑟𝑖,𝑗 = ∥x𝑖 − x𝑗 ∥ ,
∀ 𝑖 = 1, 2, ..., 𝑛, 𝑗 = 1, 2, ..., 𝑚 + 𝑛. (1)
In practice, we get measurement information for a subset of pairs of nodes, which is denoted by ℰ. We use ℰ𝑙𝑜𝑠 (and respectively ℰ𝑛𝑙𝑜𝑠 ) to denote the set of index pairs such that the measurement between the nodes is LOS (and respectively NLOS). Similarly, we use ℰ1 to denote the set of index pairs such that the measurements come from unknown-position sensors and anchors, and ℰ2 to denote the set of index pairs such that the measurements come ∪ from unknown-position ∪ sensors [19]. By definition, ℰ = ℰ𝑙𝑜𝑠 ℰ𝑛𝑙𝑜𝑠 = ℰ1 ℰ2 . Note that only the index pairs of the nodes that can communicate with each other are included in ℰ. Notice that each of the measurements could be either LOS or NLOS. In this paper, we assume that the LOS range measurements are of the form 𝑑𝑖,𝑗 = 𝑟𝑖,𝑗 + 𝑛𝑖,𝑗 ,
(2)
2 where 𝑛𝑖,𝑗 ∼ 𝒩 (0, 𝜎𝑖,𝑗 ) is the measurement error which follows a zero-mean Gaussian distribution with standard deviation 𝜎𝑖,𝑗 . Similarly, the NLOS range measurement is assumed to be of the form
𝐷𝑖,𝑗 = 𝑟𝑖,𝑗 + 𝑛𝑖,𝑗 + 𝛿𝑖,𝑗 ,
(3)
where 𝛿𝑖,𝑗 is the error of the NLOS measurement. III. NLOS L OCALIZATION U SING SDP R ELAXATION According to the information we have on NLOS error prior probabilities and distributions, three different cases are introduced and the related problems are addressed respectively in the following subsections: 1) The ideal scenario in which we know which ranges are in NLOS conditions and the distribution parameters for NLOS error; 2) The scenario with limited prior information in which we do not know which ranges are in NLOS conditions, but we have a priori information on the NLOS probability and the distribution parameters at each anchor; and 3) The worst case in which we do not know any a priori information on NLOS errors, but we know the measurement noise power. This latter scenario is obviously the most interesting and practical case, but also the most difficult one for which to address the effect of NLOS errors. A. Known NLOS status and Distribution Parameters In this subsection, we consider the ideal scenario in which we know which ranges are in NLOS conditions and the distribution parameters for the NLOS error. We regard the
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sum of the NLOS error and the measurement noise as a new “measurement noise”. Given this fact, the NLOS measurement model can be approximately written in a form similar to the LOS measurement model. Unfortunately, the distribution of the new “measurement noise” is generally difficult to obtain. However, the mean and variance may be easily obtained. Based on this fact, we propose an AML formulation to estimate the sensor node positions. The non-convexity nature of the AML formulation makes the problem very difficult to deal with. Similarly to an approach used in the literature [11], we relax the AML problem to an SDP, which can be solved very efficiently and can achieve reasonable estimation accuracy. The NLOS measurements are given by 𝐷𝑖,𝑗 = ∥x𝑖 − x𝑗 ∥ + 𝑛𝑖,𝑗 + 𝛿𝑖,𝑗 , 𝑖 < 𝑗, (𝑖, 𝑗) ∈ ℰ𝑛𝑙𝑜𝑠 .
(4)
Without loss of generality, we here assume that the NLOS error, 𝛿𝑖,𝑗 , is exponentially distributed with mean 𝜆𝑖,𝑗 and variance 𝜆2𝑖,𝑗 [20]. Let 𝑐𝑖,𝑗 = 𝑛𝑖,𝑗 + 𝛿𝑖,𝑗 , which has mean 𝜆𝑖,𝑗 and variance 2 𝜆2𝑖,𝑗 + 𝜎𝑖,𝑗 by assuming the independence between 𝑛𝑖,𝑗 and 𝛿𝑖,𝑗 [15]. Then (4) can be written as 𝐷𝑖,𝑗 =∥x𝑖 − x𝑗 ∥ + 𝑐𝑖,𝑗 = ∥x𝑖 − x𝑗 ∥ + 𝜆𝑖,𝑗 + 𝑣𝑖,𝑗 , 𝑖 < 𝑗, (𝑖, 𝑗) ∈ ℰ𝑛𝑙𝑜𝑠 ,
(5)
where a new random variable 𝑣𝑖,𝑗 with zero mean and variance ′2 2 𝜎𝑖,𝑗 = 𝜆2𝑖,𝑗 + 𝜎𝑖,𝑗 is introduced. In this ideal scenario (known NLOS status), we can subtract the mean of the NLOS error from the measurements such that (5) is equivalent to ′ = ∥x𝑖 − x𝑗 ∥ + 𝑣𝑖,𝑗 , 𝑖 < 𝑗, (𝑖, 𝑗) ∈ ℰ𝑛𝑙𝑜𝑠 , 𝐷𝑖,𝑗
(6)
′ = 𝐷𝑖,𝑗 − 𝜆𝑖,𝑗 . where 𝐷𝑖,𝑗 On the other hand, the LOS measurements are given by
𝑑𝑖,𝑗 = ∥x𝑖 − x𝑗 ∥ + 𝑛𝑖,𝑗 , 𝑖 < 𝑗, (𝑖, 𝑗) ∈ ℰ𝑙𝑜𝑠 .
(7)
The joint likelihood function of X𝑠 can be written as ∏ ′ 𝑝(𝑑𝑖,𝑗 , 𝐷𝑖,𝑗 ∣X𝑠 ) = 𝑝𝑛𝑖,𝑗 (𝑑𝑖,𝑗 − ∥x𝑖 − x𝑗 ∥) 𝑖
