Efficient bias reduction approach of time‐of‐flight‐based wireless localisation networks in NLOS states
2018; Institution of Engineering and Technology; Volume: 12; Issue: 11 Linguagem: Inglês
10.1049/iet-rsn.2018.5167
ISSN1751-8792
AutoresHailiang Xiong, Meixuan Peng, Kongfan Zhu, Yang Yang, Zhengfeng Du, Hongji Xu, Shu Gong,
Tópico(s)Target Tracking and Data Fusion in Sensor Networks
ResumoIET Radar, Sonar & NavigationVolume 12, Issue 11 p. 1353-1360 Research ArticleFree Access Efficient bias reduction approach of time-of-flight-based wireless localisation networks in NLOS states Hailiang Xiong, Hailiang Xiong The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorMeixuan Peng, Meixuan Peng The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorKongfan Zhu, Corresponding Author Kongfan Zhu sfbxxzx@139.com Ministry of Justice of the People's Republic of China, Beijing, 100020 People's Republic of ChinaSearch for more papers by this authorYang Yang, Yang Yang The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorZhengfeng Du, Zhengfeng Du The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorHongji Xu, Hongji Xu The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorShu Gong, Shu Gong The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this author Hailiang Xiong, Hailiang Xiong The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorMeixuan Peng, Meixuan Peng The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorKongfan Zhu, Corresponding Author Kongfan Zhu sfbxxzx@139.com Ministry of Justice of the People's Republic of China, Beijing, 100020 People's Republic of ChinaSearch for more papers by this authorYang Yang, Yang Yang The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorZhengfeng Du, Zhengfeng Du The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorHongji Xu, Hongji Xu The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this authorShu Gong, Shu Gong The School of Information Science and Engineering, Shandong University, Qingdao, 266237 People's Republic of ChinaSearch for more papers by this author First published: 17 September 2018 https://doi.org/10.1049/iet-rsn.2018.5167Citations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract An efficient bias mitigation algorithm based on time of flight is proposed for positioning the target location and reducing the non-line-of-sight (NLOS) error and clock jitter error in three-dimensional wireless cooperative localisation networks. Through linearising the range-based expressions and utilising novel three-step weighted linear least squares algorithm, an algebraic solution of target can be derived, in which the clock jitter error and NLOS error can be alleviated effectively. Meanwhile, the Cramer–Rao lower bound (CRLB) is derived for the standard of performance evaluation. The location accuracy of the proposed algorithm is analysed and compared with the conventional methods through simulation experiment. The simulation results indicate that the precision of the proposed algorithm can approach the CRLB, what is more, the proposed algorithm can provide obvious improvements in positioning accuracy compared to the state-of-the-art approaches. 1 Introduction In the last few decades, wireless positioning algorithm has become an important research topic, since location information is the crucial necessity for cooperative wireless sensors networks (CWSNs) [1-6]. Recently, all sorts of location services have been presented and researched in a large number of documents, including urgency positioning, earthquake surveillance, precision navigation, target tracking, and many military applications [7-10]. Owing to a wide range of applications for location services, the high-accuracy position estimation requires to be provided. Most of studies in the documents assumed that there were a quantity of sensors that were also regarded as reference points. Sensors will be utilised to estimate the geographical location of the unknown target point by receiving the signals emitted by the target nodes [11-13]. As the distribution of the sensors are more random, the positioning will be more complex. At present, there are many methods to locate the target nodes. Radio ranging techniques are the most common used, which can be realised by time of arrival (TOA) estimation, time difference of arrival (TDOA) estimation, received signal strength (RSS) estimation, angle of arrival (AOA) estimation, and some combinations of these methods [2, 14-18]. In TOA estimation method, one sets up the geometric relationship through measuring the time of flight (TOF) from the target node to the sensors. Synchronisation between the transmitter and the receiver is necessary; meanwhile, there is clock jitter error caused by frequency change in TOA estimation [19-22]. In TDOA estimation method, one establishes a series of range equations via measuring the time difference of flight from the target node to the sensors. Commonly, there is a compensation in all the measured ranges, which is caused by the clock disalignment between the receiver clock and the synchronised transmitter clocks [15]. In RSS estimation method, the range equations are built by the RSS at the receiver. Undoubtedly, RSS estimation is the most popular measurement method in radio ranging techniques for its simplicity [14]. However, it is extremely sensitive to channel condition and leads to inaccurate measurement. In AOA estimation method, we estimate the azimuth and elevation angle between reference node and target node in order to derive the location of target [23]. However, the acquisition of the angle measurement information demands antenna beamforming technologies with a mass of complex computation [24]. According to the measured TOA, TDOA, RSS, or AOA information at the receiving terminal and the geometric relation between the sensors and the target node, range equations can be established. At the same time, the range equations containing the unknown coordinates of the target are non-linear, so localisation of target is not an easy task. In general, the probability distribution of errors are known, we adopt non-linear approaches [such as non-linear least squares (NLS) algorithm or maximum likelihood estimation (MLE)] [4] to resolve the cooperative positioning problem. In these two non-linear methods of solving the cooperative positioning, MLE is regarded as the weighted form of the NLS algorithm. Nevertheless, neither NLS algorithm nor the MLE algorithm can assure global solution. Therefore, numerical approximation method is a common consideration. Iterative algorithm (represented by the Newton iteration, Taylor series iteration, steepest descent method etc.) with perfect initial three-dimensional (3D) position coordinates [25] is an effective numerical approximation method. In order to avoid the disadvantages of the non-linear iteration approaches, one can use linearisation technology to derive the linear pseudorange equations for the location problem. It needs to choose the known sensors as reference nodes for acquiring a series of linear expressions [26]. In [16, 27-31], only line-of-sight (LOS) propagation paths between the sensors and the target node were taken into consideration. However, non-line-of-sight (NLOS) paths inevitably appear in practical environments, which will have a serious impact on the positioning accuracy. In NLOS propagation environment, the NLOS error caused by reflections, diffractions, and scatterings of signals is the considerable obstacle in signal propagation [32-34]. Although it is difficult to separate each error from the total signal, we know that the NLOS error is the major part of the bias errors; furthermore, it can be regulated as the additional TOF and is always positive for delay-based TOF positioning networks. As a result, the bias errors always make the measured TOA, TDOA, RSS, or AOA larger than the true value [35]. Inescapable NLOS propagation interference has a considerable influence on positioning accuracy of the target node [7, 36-38]; hence, elimination of NLOS errors becomes a troublesome task [39]. To alleviate the impact of the bias errors, some bias mitigation algorithms have been proposed. In [40], residual weighting algorithm was proposed. In the positioning process, the algorithm did not require the statistical characteristics of the errors, only asked to measure a large number of intermediate estimations. No doubt, the proposed algorithm required heavy calculation loads. In [41], the two-step least squares method was proposed. In the algorithm, the author assumed that NLOS errors were random variables; furthermore, they were alleviated by the second-step least squares. Unfortunately, the algorithm was only used under the conditions of NLOS paths were rare. In [42, 43], the geometry-constrained position estimation (GPE) algorithm was proposed. Owing to the influence of the NLOS error, the target node was located in the overlapping area of the range circles generated by the ranges between the sensors and the target node. It solved a linear equation with linear constraints. Whereas when the NLOS error was very large, the range circles did not have overlapping parts, so that the position of the target node could not be determined. In [44], LOS time-of-arrival measurements reconstruction algorithm was proposed, which identified NLOS paths and reconstructing LOS measurements through subtracting NLOS measurements from all measurements. Unluckily, the algorithm also has a drawback, it is challenging to distinguish the LOS and NLOS measurements results. Combining all of the above algorithms, it is not difficult to find that these algorithms cannot eliminate the NLOS error perfectly. In this article, to reduce the impact of deviation, an efficient bias mitigation algorithm of delay-based TOF CWSNs in NLOS environment is presented. We use linearisation techniques to deal with non-linear equations and solve the proposed positioning problem by weighted linear least squares (WLLS) algorithm. Meanwhile, we acquire the position of unknown target node via TOF estimation between the sensors and the target node. In the position estimation model, due to the change of transmit frequency, there is also clock jitter error in addition to the NLOS error and noise. Accordingly, we propose a novel three-step WLLS algorithm to mitigate the clock jitter error, NLOS error, and noise bias. In the first step, we reduce the clock jitter error and noise, and derive the target location through WLLS algorithm in the presence of NLOS error. In the second step, according to the solution of the first step, we reduce the NLOS error, and derive the position of target, but there is an unknown parameter in the position of target. Hence, in the third step, we solve the unknown parameter and get the final solution of the target location. In addition, to appraise the performance in different conditions, we obtain the CRLB and the root mean square error (RMSE) of the proposed reduction algorithm, the results are compared with RMSEs of the conventional methods. The simulation results validate that the proposed TOF-based bias reduction algorithm is superior to state-of-the-art algorithms, it can obtain more perfect positioning accuracy. The remainder of this paper is structured as follows. In Section 2, a 3D cooperative positioning measurement model is presented. In Section 3, an efficient three-step cooperative positioning estimation algorithm based on TOF ranging is proposed in the NLOS environment. The CRLB of TOF ranging method is derived in Section 4. The simulation results are presented and the performance analysis and comparison are discussed in Section 5. Finally, we draw the conclusions of the paper in Section 6. 2 Three-dimensional cooperative positioning measurement model In this section, a 3D cooperative positioning measurement model is established. The fundamental framework of the presented localisation system is outlined in Fig. 1, where KRi denotes the known reference node, and UT denotes the unknown target node. The unknown position of target is determined by the TOF measurements from reference nodes to target node [15]. In TOF-based cooperative positioning networks under NLOS conditions, the clock jitter error and NLOS error have a dramatically influence on the position estimation of the target node. Hence, we propose a three-step weighted linear least squares (T-SWLLS) location algorithm to alleviate the above biases and noise error. We consider a 3D wireless positioning network with N reference nodes. To simplify the discussion, only one target node is depicted in Fig. 1. The reference nodes are denoted by , , the coordinates of are , which are known, and the target node is denoted by , which is unknown. According to the geometric relationship between reference nodes and target node in Fig. 1, we can derive the geometric relationship expressions about the location coordinates of the target node in the presence of NLOS error and clock jitter error. The wireless positioning problem can be expressed by the following non-linear equation (1) where denotes the TOF measured value from the target node to the reference node in NLOS condition, c is the speed of electromagnetic wave with , , the TOF measured value from the target node to the reference node in the LOS environment, the error caused by NLOS environment, and it always takes positive value in delay-based TOF systems. Commonly, the NLOS error is assumed to be exponential distribution with parameter , its mean is , and its variance is . denotes the distance between the target node and the reference node. denotes the location of the target node in the presence of NLOS error, denotes Euclidian norm of , denotes the clock jitter error caused by change of transmission frequency in the signal transmission process, and is assumed to be zero mean Gaussian random variable with covariance , and obeys Gaussian distribution with mean zero and covariance , and are independent of each other, so also obeys Gaussian distribution with zero mean and covariance . Fig. 1Open in figure viewerPowerPoint Fundamental positioning framework with N known reference sensors and one unknown target in wireless cooperative localisation networks According to (1), (2) can be obtained (2) 3 Bias elimination position estimation In order to locate the target node, we need to establish the equations based on TOF estimation. Commonly, these equations are non-linear. In this section, we use the linear method to solve the positioning problem. We deal with non-linear equations and linearise them; meanwhile, we introduce additional parameters. To estimate the target location through solving the linear equations, we present a T-SWLSS algorithm, in which the noise, NLOS error, and clock jitter error in wireless positioning networks will be taken into account by the use of weighted matrix. In the first step, we derive a rough solution of , in which the clock jitter error and noise error can be reduced. Squaring the fourth equation of (1), we can get (3) where (4) (5) Neglecting high-order items, we can obtain (6) According to the characteristic of the equation, the following formula can be obtained by identical deformation of (6) (7) where denotes error term. Equation (7) can be converted into the matrix form as (8) where (9) (10) (11) (12) (13) (14) and denotes . Then, we define (15) According to the weighted least squares criterion, we can derive the cost function with respect to (16) where is the weighted matrix, then (17) where is the covariance matrix of . Then, the value of corresponds to the minimum value of (18) The cost function is a quadratic function; thus, the value of can be derived by differential as (19) Hence, according to , we can obtain a rough estimation of . In the second step, using the solution of the least squares in the first step, we reduce the NLOS error, at the same time, derive target position with an unknown parameter by the weighted least squares (WLS) algorithm. We square both sides of (2) and transform (2) into a linear equation (20) Through transforming (20), we can draw (21) Converting (21) into the matrix form, we have (22) where (23) (24) (25) (26) (27) (28) (29) we should note that represents all error terms of (21). denotes multiplication of the corresponding elements of the two matrices with the same dimension. The cost function of can be given by (30) where is the weighted matrix. It can be derived by the inverse of the covariance of the error terms; hence, the weighted matrix can be written as (31) (32) (33) and (34) where denotes the correlation matrix of the NLOS error, the origin moments of , and the correlation matrix of . Hence, according to , we can obtain the estimated solution of the target (35) There is a difference between the estimated value of the target location and the true value of the target location , the difference is represented by . So, the following equation can be obtained (36) There is an unknown parameter in this solution, and we can solve this parameter by the WLS algorithm again. In the third step, we obtain the unknown parameter by the WLS algorithm. According to (35) and (36), (22) can be rewritten as (37) Equation (37) is equivalent to (38) is represented by ; hence, the following equation can be obtained (39) where (40) Equation (39) is a linearised equation about ; therefore, the cost function of can be written as (41) where is the weighted matrix, and . In order to obtain , we let (42) Consequently, the optimal estimate of can be expressed as (43) Substituting (43) into (35), one can get final solution as follows: (44) Here, we should note that the first element of the matrix represents x, the second element of the matrix represents y, and the third element of the matrix represents z. 4 Cramer–Rao lower bound The precision of a specific cooperative localisation system is determined by a lot of factors, such as the algorithm utilised to cope with the measurements, accuracy of measurement, and the geometry structure of the nodes in the positioning system. To evaluate the precision of miscellaneous localisation algorithms, some methods of estimating precision have been introduced. The CRLB of the algorithm is an important index of the performance of the algorithm [4]. CRLB can be acquired by known location coordinates of the sensors and the standard deviation of bias errors. In most cases, we cannot determine the distribution of errors precisely, so it is tough to attain CRLB in the actual estimation. The CRLB is a pivotal performance evaluation in the location system [34, 45]. In this section, to identify the CRLB of unbiased estimate under the proposed TOF-based cooperative localisation model, we assume that obeys an exponential distribution of parameter , with mean and the variance , and the probability density function (PDF) is given by (45) According to (2), we derive that . Let , hence, the PDF of the NLOS error is (46) Taking the natural logarithm of (47) Then, the CRLB of becomes (48) (49) (50) where denotes the row column element of the inverse of Fisher information matrix , is Fisher information matrix, which can be expressed as (51) The detailed derivation of the Fisher information matrix is explained in the Appendix. 5 Simulations results and performance assessment The errors of cooperative positioning based on TOF estimation model primarily brought out by measurement errors (including clock jitter error and noise) and NLOS errors. The measurement errors are Gaussian-distributed random variables with covariance , the NLOS errors . We assume that they are identical for all measurements, that is, to say, , . In the simulations, we assume there are one target and ten sensors, the positions of the sensors are set arbitrarily in the cubic area, and their true positions are listed in Table 1. Table 1. True positions of target and sensors Targets x, m y, m z, m 1 10 10 10 Sensors , m , m , m 1 50 80 30 2 0 0 0 3 40 60 20 4 20 40 80 5 70 30 40 6 70 50 80 7 20 50 30 8 40 20 60 9 30 30 30 10 10 80 20 In order to prove the validity of the proposed algorithm, we carried out the simulation. We can derive the estimation of the target position. To evaluate the performance of varying conditions, the RMSE is considered. RMSE is defined as (52) where and represent the true target position and the estimation of the target position in the ith experiment, respectively. In the simulation, we set , L indicates the number of experiments performed, we can obtain all the positions by 1000 independent experiments. Fig. 2 shows the RMSEs of the proposed new algorithm, the CRLB, the proposed method in [41], the proposed method in [42], and linear least squares (LLS) algorithm in the case of fixed mean of NLOS error. NLOS error is considered to be in meter level and it is set to . The horizontal ordinate represents the standard deviation of the measurement errors , and the vertical ordinate represents the RMSE. We assume the covariance matrix of the measurement errors . The measurement errors are independent of each other; therefore, we set . becomes an diagonal matrix with diagonal elements equal to . Fig. 2Open in figure viewerPowerPoint RMSE as a function of for the proposed new algorithm, the algorithm proposed in [41], the algorithm proposed in [42], and LLS algorithm In Fig. 2, we note that as standard deviation of measurement error increases, the RMSE also increases. The simulation result indicates that the large measurement error influences the positioning accuracy. There are five curves in the figure, comparing the five curves, it is not difficult to find that the proposed algorithm can basically approximate to the CRLB and the accuracy of the methods proposed in [41, 42], and LLS algorithm is worse than the CRLB. Moreover, the RMSE of the proposed algorithm is less than the RMSEs of the algorithms proposed in [41, 42], and LLS algorithm. For the method proposed in [42], we only consider the effect of NLOS error on the performance of the algorithm; therefore, it is a horizontal line in Fig. 2. As a result, the performance of the proposed new algorithm is usually better than the methods proposed in [41, 42], and LLS algorithm. The traditional LLS algorithm only alleviates the measurement noise error and does not involve the NLOS error, thus we ignore the impact of NLOS errors on its performance. Fig. 3 shows the RMSEs of the proposed new algorithm, the algorithm proposed in [41], and the algorithm proposed in [42] with fixed standard deviation of measurement error . We let . The horizontal ordinate represents the mean of NLOS error , and the vertical ordinate represents the RMSE. As the mean of NLOS error increases, RMSE also increases. When is equal to 0.2 m, there is no significant difference between RMSE of the proposed new algorithm and the RMSE of the method proposed in [42]. However, as grows, the advantages of the proposed algorithm become more apparent. Compared with the method proposed in [41], the RMSE of the proposed new algorithm is obviously lower than the RMSE of the method proposed in [41]. In general, the RMSE of the proposed algorithm is the lowest, and the performance of the proposed algorithm is more perfect than that of the methods proposed in [41, 42]. Therefore, it alleviates the NLOS error to a certain degree and makes positioning accuracy higher. Fig. 3Open in figure viewerPowerPoint RMSE as a function of for the proposed new algorithm, the algorithm proposed in [41], and the algorithm proposed in [42] Fig. 4 is shown to compare the RMSEs of the presented algorithm based on four sensors, six sensors, eight sensors, and ten sensors, respectively. Here, we let . In Fig. 4, the horizontal ordinate represents the mean of NLOS error . As increases, RMSE also increases. Comparing the four curves in Fig. 4, we can clearly see that the larger the number of used sensors is, the better the performance of the proposed algorithm will be. In addition, when , the RMSE of the proposed algorithm for the case of ten sensors used in the communication network is 2.2 m; the RMSE of the proposed algorithm for the case of four sensors used in the communication network is 6.5 m. Hence, it can be concluded that the effect of the different number of sensor nodes on the performance of the proposed algorithm is significant. However, in order to maintain low implementation costs, the number of sensors used needs to be limited. The most suitable number of sensors used requires further study. Fig. 4Open in figure viewerPowerPoint RMSE as a function of for the proposed new algorithm based on four sensors, six sensors, eight sensors, and ten sensors Overall, from the simulation results, we can notice that the presented algorithm can achieve the CRLB accuracy in the presence of the NLOS error, clock jitter error, and noise. However, as the error increases, the positioning accuracy will gradually decrease. 6 Conclusion In this article, an effective three-step bias reduction algorithm was presented in TOF-based wireless sensors localisation network. We accomplished the objective of mitigating the influence of NLOS error, clock jitter, and noise. In the proposed positioning approach, a set of non-linear equations from TOF measurements is established to solve the wireless location problem with errors. Through the T-SWLLS algorithm, target position coordinates were obtained, NLOS error, clock jitter error, and noise bias were mitigated to a certain degree, and the location accuracy was improved. The simulation results showed that the proposed bias reduction method can achieve the CRLB in the presence of Gaussian noise, NLOS bias, and clock jitter error; moreover, the proposed method can obtain better localisation performance. 7 Acknowledgments This work was supported in part by the Natural Science Foundation of China under grants no. 61401253, in part by the Key Research & Development Program of Shandong Province under grant no. 2017GGX201003, in part by the Natural Science Foundation of Shandong Province under grant nos. ZR2016FM29 and ZR2015FM026. 9 Appendix The logarithm of is described in (47), its derivatives are easily found as (53) (54) (55) (56) (57) (58) (59) (60) (61) so, we can obtain (62) (63) (64) (65) (66) (67) where (62)–(67) are the components of the Fisher information matrix. denotes the row column element of Fisher information matrix . So, the CRLB of the 3D positioning can be given by (68) (69) (70) where is equivalent to . Also, represents the determinant of the matrix , it can be derived by (71) 8 References 1Georges, H.M., Wang, D., Xiao, Z. et al: ‘Hybrid global navigation satellite systems, differential navigation satellite systems and time of arrival cooperative positioning based on iterative finite difference particle filter’, IET Commun., 2015, 9, (14), pp. 1699– 1709 2Shen, Y., Win, M.Z.: ‘Fundamental limits of wideband localization – part I: a general framework’, IEEE Trans. Inf. 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