無線感測網路之合作式定位研究

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(1)國立交通大學電信工程學系碩士論文. 無線感測網路之合作式定位研究 Cooperative Localization in Wireless Sensor Networks. 研究生：王瑋萱指導教授：謝世福. 教授. 中華民國九十九年十一月.

(2) 無線感測網路之合作式定位研究 Cooperative Localization in Wireless Sensor Networks 研究生：王瑋萱. Student：W.S. Wang. 指導教授：謝世福. Advisor：S. F. Hsieh. 國立交通大學電信工程學系碩士論文. A Thesis Submitted to Department of Electrical Engineering Institute of Communications Engineering National Chiao Tung University In Partial Fulfillment of the Requirements For the Degree of Master of Science In Communication Engineering. Hsinchu 2010 Hsinchu, Taiwan, Republic of China. 中華民國九十九年十一月.

(3) 無線感測網路之合作式定位研究學生:王瑋萱. 指導教授:謝世福. 國立交通大學電信工程研究所. 中文摘要定位的應用非常廣泛，如:追蹤目標、航海定位和救援…等。在定位系統中，藉由感測器對待測目標的量測資料(角度或距離)估計待測目標的位置。當網路中多個待測目標可相互通訊時，合作式定位被提出:藉由待測目標之間(合作)的量測資料可以有效的提升定位精準度。在合作式定位中，最佳的演算法為最大概似法，其方程式為非線性方程式，可使用聯合牛頓法來實現。但是當待測目標數目增加時，運算量會變的相當大。因此我們提出兩種降低運算量的方式：第一種方式是藉由過去非合作線性化的經驗，我們成功推導出聯合泰勒展開演算法。第二種方法，我們提出分割疊代演算法來降低運算複雜度，疊代可提升定位準確性。也將原本的非線性方程式做雙曲線、新增變數和泰勒展開三種線性化處理。在效能評估中，我們討論三種非合作線性演算法的定位效能，進一步在分割演算法裡，針對泰勒展開線性化方法，設定在兩個待測目標情況下，推導其收斂的 MSE 理論值。另外，合作式的 Cramer-Rao Lower Bound (CRLB) 藉由 Fisher. Information Matrix (FIM) 的反矩陣求得，然而矩陣的大小隨著待測目標增加而變大，導致求反矩陣的運算非常的複雜。因此我們提出遞迴方塊反矩陣的特性，推導一個運算簡單的近似等 FIM，並且利用近似結果，進一步去推導近似的合作 CRLB。最後，利用電腦模擬驗證推導的結果。. i.

(4) Cooperative Localization in Wireless Sensor Networks Student: W. S. Wang. Advisor：S. F. Hsieh. Department of Communication Engineering National Chiao Tung University Abstract There are a lot of applications of localization including tracking, search, navigation and rescue. We can estimate position of mobile (object) by measurements (angles or distances) in sensor network. When mobiles can communicate to each other, cooperative localization has been proposed to improve the localization accuracy. In cooperative localization, the optimal ML estimator is nonlinear. It can be solved by Newton’s method, but the cost of computation increases when the number of mobiles increases. Therefore, we propose two methods to reduce the computation cost, joint Taylor-series expansion algorithm and divide-and-conquer algorithm. In divide-and-conquer algorithm, we use recursive method to enhance localization accuracy and simplify the nonlinear function by three linearization methods. Next, we compare the MSE performance of three linearized algorithms and derive the theoretical converged mean-square-error for divided Taylor-series expansion algorithm. Besides, cooperative Cramer-Rao Lower Bound (CRLB) is derived by Fisher Information Matrix (FIM) inverse, but the size of matrix increases when the number of mobiles increases. Then, we propose recursive block matrix inversion to derive a simple Approximated Equivalent FIM (AEFIM) and we further utilize the result to derive the Approximate Cooperative CRLB (AC-CRLB). Simulations are performed to support the theoretical results. ii.

(5) Acknowledgment 在碩士班的兩年期間學習做研究，起初對我而言既陌生而且艱難。但是在指導教授謝世福老師耐心的指引與教導，讓我在面對問題的時，能以正確的邏輯觀念與嚴密的思考模式，嚴謹的推敲出解決方法。學習的過程雖然辛苦，但是我很感激老師的栽培；也很感謝實驗室的學長、學姐、同學的幫忙與鼓勵。我很珍惜並感謝在這些日子裡接受到的磨練，讓我能勇敢並樂觀的接受未來的種種挑戰。. iii.

(6) Contents 中文摘要…………………………………………………………………………...i English Abstract…………………………………………………………….....ii Acknowledgments………………………………………………….…………iii Contents………………………………………………………………….…….....iv List of Figures………………………………………………………….………vii 1. Introduction …………………………………………………………………1 2. TOA Localization System………………………………….…………..4 2.1 Measurement Characterization……………......…………………………………...5 2.2 Basic Localization Algorithm……………………………………………….……..7 2.2.1 ML Estimator………………………………………………………………...8 2.2.2 Linearization of Least-Squares Estimator………………………………….10 2.2.2a Taylor-Series Expansion…........................…….………….………....10 2.2.2b Distance-Augmented………........................................……………...14 2.2.2c Hyperbolic-Canceled……….…………….……..…….......………....16 2.2.2d Summary of Three Linearization Algorithms..……………………...19. 3. Cooperative Localization System……………………..……………24 3.1 Joint Method……………………………………………………………………..27 3.1.1 Newton’s Algorithm…………………………….….……………….……..27 3.1.2 Taylor-Series Expansion Algorithm…….…………………………………29 3.1.3 Other Linearization Algorithms……………………………………………33. iv.

(7) 3.2 Divide-and-Conquer Method….…………….……………………………...……34 3.2.1 Two Category of Update Sequence………………………………………...36 3.2.1a Jacobi method………………………………………………………..36 3.2.1b Gauss-Seidel method………………………………………………...37 3.2.2 Divided individual localization: Newton’s Algorithm and Three Linearization Algorithms.............................................................................38 3.2.2a Newton’s Algorithm…………………...………………….…………38 3.2.2b Taylor-Series Expansion…………….……..………………………..40 3.2.2c Distance-Augmented……………………..……..…………………...42 3.2.2d Hyperbolic-Canceled..……………………..………………………..44 3.3 Compensation of Uncertain Virtual Sensor……………………………………...46. 4. Theoretical Analysis of Mean Square Error……..…………...49 4.1 Cramer-Roa Lower Bound of TOA Localization…….…………………............49 4.1.1 Traditional Localization CRLB…………………..........………….…….....50 4.1.2 Cooperative Localization CRLB…………………………………………...51 4.1.3 Cooperative Fisher Information for Two Mobiles based on Eigen Decomposition.............................................................................................52 4.1.4 Approximation of Equivalent FIM Matrix Based on Recursive Block Matrix Inversion………………………………............................................……..55 4.1.5 Approximated Cooperative CRLB in Case of a Central Mobile…..…….....61 4.2 Converged Theoretical Mean Square Error for Divided Linearized Algorithms...65 4.2.1 Taylor-Series Expansion Algorithm….………......................................…...66 4.2.1a Weighted Estimation for Two Mobiles………………………………67 4.2.1b Unweighted Estimation for Two Mobiles.…………………………..71 4.2.2 Distance-Augmented………………….................................................…...75. v.

(8) 4.2.3 Hyperbolic-Canceled……………..……………….............................……..76. 5. Computer Simulations…….…..…...……..……………….……………78 5.1 The Theoretical MSE of Three Uncooperative Linearized Algorithms……….....79 5.2 Comparison of Joint Newton’s Algorithm and Joint Taylor-Series Expansion Algorithm………………………………………………………………………...88 5.3 Comparison of Joint Algorithms and Divide-and-Conquer Algorithms…………94 5.3.1 Newton’s Method…………………………………………………………..95 5.3.2 Taylor-Series Expansion Method…………………………………………..99 5.4 Comparison of Divide-and-Conquer Algorithms…………………………….....101 5.4.1 Newton’s Method…………………………………………………………102 5.4.2 Taylor-Series Expansion Method…………………………………………103 5.4.3 Distance-Augmented Algorithm………………………………………….106 5.4.4 Hyperbolic-Canceled Algorithm………………………………………….108 5.5 Theoretical Analysis of Mean-Square-Error……………………………………114 5.5.1 Approximation of CRLB………………………………………………….115 5.5.2 Divided Taylor-series Expansion for Twins Mobiles…………………….117. 6. Conclusions and Future Work ………………………….………...118 Bibliography……………………….………………………….………………119. vi.

(9) List of Figures Figure 2.1 A basic localization system in wireless sensor network………………….. 4 Figure 2.2 A basic localization scenario with TOA measurement…………………….8 Figure 3.1 Cooperative localization system with cooperative TOA measurement…. 25 Figure 3.2 The joint localization method…………………………………………….34 Figure 3.3 A part of divide-and-conquer method…………………………………….34 Figure 3.4 Another part of divide-and-conquer method……………………………...35 Figure 3.5 The Jacobi method diagram for divide-and-conquer method…………….37 Figure 4.1 The angle relationship between sensors and mobile 1……..………….….61 Figure 4.2 Omni-direction type with same reliable cooperative positions…………...62 Figure 4.3 Beam type with same reliable cooperative positions……………………..62 Figure 4.4 Beam type with difference reliable cooperative positions………………..62 Figure 5.1 Positions of the sensors and mobiles……………………………………..80 Figure 5.2 The theoretical MSE versus noise variance for three unweighted linearized methods…………………………………………………………………..81 Figure 5.3 The theoretical MSE versus noise variance for three weighted linearized methods and CRLB.………………………………………………...……82 Figure 5.4 The simulated MSE versus noise variance for three weighted linearized methods and nonlinear algorithm in MATLAB………………………….84 Figure 5.5 The theoretical MSE of Taylor-series expansion method for different position of mobile………………………………………………………...85 Figure 5.6 The theoretical MSE of weighted distance-augmented algorithm for different position of mobile………………………………………………..86 Figure 5.7 The theoretical MSE of weighted hyperbolic-canceled method for different vii.

(10) position of mobile………………………………………………………..87 Figure 5.8 The MSE vs. convergence rate for joint Newton’s method and joint TS with good initial guess set……………………………………………….89 Figure 5.9 The MSE vs. convergence rate for joint Newton’s method and joint TS with bad initial guess set…………………………………………………..90 Figure 5.10 The converged MSE vs. The variance of noise…………………………91 Figure 5.11 MSE vs. the number of mobile for joint Newton’s and TS…………..…92 Figure 5.12 The computation cost vs. the number of mobile for joint TS and joint Newton on once global iteration………………………………………..93 Figure 5.13 The MSE vs. noise variance for joint and divided Newton’s algorithm...95 Figure 5.14 The MSE vs. number of mobile for joint Newton’s algorithm, divided Newton’s algorithm and cooperative CRLB……………………………96 Figure 5.15 CDF comparison of joint and divided Newton’s algorithm……………..97 Figure 5.16 The computation time cost vs. the number of mobile for joint and divided Newton’s algorithm on once global iteration…………………………...98 Figure 5.17 The MSE vs. noise variance for divided TS, joint TS and cooperative CRLB…………………………………………………………………...99 Figure 5.18 The CDF comparison of joint and divided TS algorithms………..……100 Figure 5.19 The MSE vs. Global iteration for Compare Jacobi and Gauss-Seidel…102 Figure 5.20 The MSE vs. global iteration for divided TS………………………..…103 Figure 5.21 The error and MSE vs. global iteration for divided TS………………..104 Figure 5.22 The CDF comparison of divided TS and uncooperative TS…………...105 Figure 5.23 The error and MSE vs. global iteration for divided DA……………….106 Figure 5.24 The CDF comparison of divided DA and uncooperative DA………….107 Figure 5.25 The error and MSE vs. global iteration for divided HC…….…………108 Figure 5.26 The CDF comparison of divided HC and uncooperative HC.…………109 viii.

(11) Figure 5.27 The MSE vs. global iteration for three divided linearized algorithms…110 Figure 5.28 The MSE vs. noise variance for three linearized divided algorithms….111 Figure 5.29 The MSE vs. number of mobile for three divided linearized algorithms…………………………………………………………..….112 Figure 5.30 The MSE vs. number of mobile for three divided linearized algorithms with fixed 14 mobiles………...………………………………………..113 Figure 5.31 The MSE vs. number of mobile for difference value of J xi .….…….. 115 Figure 5.32 The MSE vs. cooperative angles (alpha) for AC-CRLB and joint TS…116 Figure 5.33 The MSE vs. global iteration of divided TS for two mobiles………….117. ix.

(12) Chapter 1 Introduction In recent years there has been interest in wireless sensor networks for variety of applications [2, 3 ,4]. Among those are health, commercial, environmental, public safety, home applications. In literature there are many methods to provide the localization estimation for wireless sensor network. Classify these localization methods as the deterministic [1, 12, 27] and probabilistic approached [22, 23]. Typical positioning parameters include time-of-arrival (TOA) [29, 45], time-difference-of arrival (TDOA) [5, 6], angle-of-arrival (AOA) [7] and received signal strength (RSS) [8, 9], and hybrid TDOA/AOA of mixture method [10, 11]. In this thesis, we only consider the TOA localization algorithms. However, in cooperative localization system, cooperative connection or Ad Hoc short-range communication among the terminals will be supported [41, 42]. Because they consider small-scale information between unknown positions can improve the localization accuracy. Therefore they focus on the data fusion of large-scale and small-scale. Cooperative Localization with Optimum Quality of Estimate (CLOQ) is proposed in [26] which takes advantage of the behavior of the channel to provide accurate indoor positioning. But, the estimated positions are fixed when they are estimated. Therefore, [15] devised an error propagation aware algorithm to update the unknown positions. Note that cooperative localization is not a well solved problem because the distance measurements between any pairs of unknown positions are utilized to aid in the location estimation. Then, [12] devised three novel subspace methods to solve that problem. Assuming the range measurements error are Gaussian distributed, the ML estimator [42] is another localization method and it is a nonlinear least squares problem [30]. It can be solved 1.

(13) by joint Newton’s algorithm, but the computation cost becomes more complex when the numbers of unknown positions increases. The algorithm of [15] tracks the extent of the uncertain virtual position error, but this algorithm has trade-off between computation cost and localization accuracy. Then, we propose two methods to reduce the computation cost, joint Taylor-series expansion (TS) algorithm and divided-and-conquer method. However divided Newton’s algorithm is a nonlinear function. Based on previous research of uncooperative linearized algorithms, we further use three algorithms, Taylor-series expansion (TS) algorithm [35] , distance-augmented (DA) algorithm [46] and hyperbolic-canceled (HC) algorithm [22] to perform divided-and-conquer method. One of the most important problems is the source of errors, including non-line-of-sight (NLOS) [16, 17] and multipath propagation [29, 37]. Then, tracking [20, 21] a moving unknown position is another important issue in sensor network localization. While the mobile is moving, the main concern is to estimate its trajectory. Kalman filter[18, 19] has been widely applied in trajectory estimation of a moving object. However, we know that the variance of the estimate is bounded by the Cramer-Rao Lower Bound (CRLB) [25]. It reveals the full Fisher Information Matrix (FIM). In cooperative localization, the full cooperative FIM is too complex to see the benefits of cooperation. Therefore, [31] proposed an eigenvalue view of Equivalent FIM (EFIM) to provide some insight in cooperative localization information for only two unknown positions. If there are more than two unknown positions, it still can not see the effect of cooperative. Therefore, we propose a recursive block matrix inversion based on eigenvalue view to derive the Approximation of EFIM (AEFIM). We can see that the more cooperative positions, the better localization accuracy. Then, we further utilize the result to derive the Approximation of Cooperative CRLB (AC-CRLB) and we find that some parameters, variance of measurement error, 2.

(14) numbers of known positions, numbers of unknown positions and cooperative angles can influence the localization accuracy. This thesis is organized as follows. We introduce basic localization system in Chapter 2 and proposed two low-cost cooperative localization methods, joint Taylor-series expansion algorithm (joint TS) and divided-and-conquer method (divided algorithms) based on TOA in Chapter 3. In Chapter 4, we derive AC-CRLB and theoretical converged Mean-Square-Error (MSE) of divided TS for two unknown positions. Computer simulations will evaluate the computation cost and MSE for proposed cooperative algorithms and compare the cooperative CRLB from full cooperative FIM, AEFIM and AC-CRLB in Chapter 5. Finally, we give a conclusion of our work in Chapter 6.. 3.

(15) Chapter 2 Basic Localization System Figure 1 shows a basic localization system. There are N known positions of sensors and M unknown positions of mobiles. Mobiles transmit information to all sensors by wireless networks.. ( xi , yi ). and. (x , y ) j. j. are coordinates of mobile i and. sensor j, respectively.. Mobile_1 ( x1, y1 ) Mobile_2 ( x2 , y 2 ). Sensor_1 ( x1 , y1 ). Sensor_N ( xN , yN ). Mobile_M. ( xM , yM ). Sensor_2 ( x2 , y2 ). …. Sensor_3 ( x3 , y3 ). Figure 2.1 A basic localization system in wireless sensor network.. In order to estimate positions of mobiles, we have to use the measurements between the sensors and mobiles. However, the positioning accuracy is degraded in the NLOS environment when large NLOS error is imposed on the TOA, TDOA, RSS or AOA measurement. There are some algorithms to estimate the positions of mobiles 4.

(16) based on these measurement. Mathematically, assuming that the range of measurement errors are Gaussian distributed, the maximum likelihood (ML) methods for localization correspond to the nonlinear least squares problem [30], but the ML approach cannot guarantee global convergence. In order to ensure a global solution, semi-definite programming (SDP) [1, 32] and classical multidimensional scaling (MDS) [33, 34] have been proposed. These unknown positions of mobiles will be estimated by four common measurements. Then, these measurements are introduced in Section 2.1. Then, the basic localization algorithm based on TOA measurement is discussed in Section 2.2.. 2.1 Measurement Characterization In a localization system, we utilize the measurements between sensors and mobile to estimate the position of mobile. The major measurements are time of arrival (TOA), time different of arrival (TDOA), angle of arrival (AOA) and received signal strength (RSS). In the following, we recommend these four type measurements and model it in Figure 2.1.. 1. TOA [45, 46]: Measuring propagation time from mobile to sensors, the delay time. ti j between transmission at mobile i and sensor j. Thus, the distance di j between mobile i and sensor j can be calculated by multiplying the propagation time of the signal propagation speed. The cornerstone of time-based techniques is the receiver’s ability to accurately estimate the arrival time of the lone-of-sight (LOS) signal. This estimation is likely to suffer both additive noise and multipath signals. The model of TOA is. 5.

(17) ti j =. ri j vc. + wi j. (2.1). where ri j is real distance between mobile i and sensor j and wi j is measurement. (. ). noise modeled as AWGN (additive white Gaussian noise), denoted as N 0, δ i2j . Then we have the measurement distance by multiplying the propagation speed vc ,. di j = ri j + ni j. (. where ni j ~ N 0, σ i2j. ). (2.2). is AWGN as well. The position of mobile i is hided in real. distance because of ri j =. (x − x ) +( y − y ) 2. i. j. i. 2. j. . In this thesis, we only consider the. TOA measurement.. 2. TDOA [5, 6] : Measuring propagation time difference from different sensors, then we can calculate the measurement distance difference between different sensors to the same mobile. With the cross-correlation of different sensors, the unknown time can be differentiated. The model is given by. (t. ij. ). (r. − rik. )+. − wik. ). (2.3). − dik = ri j − rik + ni j − nik. ). (2.4). − tik =. ij. vc. (w. ij. The measurement difference distance is. (d. ij. ) (. ) (. The position of mobile is hided on real distance ri j and rik for mobile i.. 3. RSS [8, 9] : The power on transmitter (sensors or mobile) are known on the system. Measuring the power difference between sensors and mobile to estimate the distance between them. It is can easy to perform by cheap equipment. A model that solely depends on the relative is the so-called Okumura-Hata model [14], 6.

(18) ( ). Ploss = K − 10α log ri j + wi j. (2.5). But in the literature, it shows that RSS is not accurate enough because multi-path, noise, humidity, temperature can affect the RSS measurement [36]. As TOA measurement, the position of mobile is hided on ri j .. 4. AOA [7] : The use of directionally sensitive and complex antenna array to estimate the angle of arrival from mobile to sensors. But AOA is disturbed by many factors. For instance, multipath [40], NLOS and so on. The model is. α i j = βi j + wi j. (2.6). where α i j is measurement angle from mobile i to sensor j, βi j is real angle and wi j is AWGN as well. Every sensors extend the angle to form a intersection which is. ( xi , yi ) . TOA is a good candidate in terms of accuracy, and then we utilize TOA to discuss the localization algorithm in following section.. 2.2 TOA Localization Algorithm Figure 2.2 shows an uncooperative localization system with TOA measurements. We can estimate mobiles i and j locations by individual TOA measurements from all sensors. Without loss of generality, we focus on the position of mobile i. ri j and di j are real distance and measurement distance between mobile i and sensor j.. 7.

(19) Figure 2.2 A basic localization scenario with TOA measurement.. The TOA from sensors j to mobile i can be modeled as follows, di j = xi − x j + ni j = ri j + ni j , j = 1, 2,...M. where xi = [ xi. yi ]. T. (2.7). is the coordinate vector of mobile i, x j = ⎡⎣ x. (. j. y j ⎤⎦. T. is. ). coordinate vector of sensor j and ni j is Gaussian noise N 0, σ i2j . We want to utilize these measurements to estimate the position of mobile i. In previous section, there are different algorithms can be solve the problem. We recommend a nonlinear Maximum Likelihood (ML) estimator in Section 2.2.1 and other linear estimators are discussed in Section 2.2.2.. 2.2.1 ML Estimator According to the model of range error, the probability density function (PDF) of the measurement distance of mobile i is. 8.

(20) (. ). p di j | xi =. (. 1 2πσ i2j. ). ⎛ d − x −x i ij j exp ⎜ − 2 ⎜ 2σ i j ⎜ ⎝. 2. ⎞ ⎟ ⎟ ⎟ ⎠. (2.8). Assume the range errors are independent between sensors. The uncooperative likelihood function [25] can be denoted as N. (. ). p ( di | xi ) = ∏ p di j | xi = j =1. (. 1 2π. N. N. ∏σ j =1. ij. ⎛ N d − x −x i i j exp ⎜ −∑ 2 ⎜⎜ j =1 2σ i j ⎝. ). 2. ⎞ ⎟ ⎟⎟ ⎠. (2.9). where d i = ⎡⎣ di 1 , di 2 " diN ⎤⎦ is the measurement set from mobile i. The ML criterion searches a xˆ i which maximizes the likelihood function (2.9), ⎧ ⎛ N d − x −x ⎪ i j ij 1 ⎪ max ⎨ exp ⎜ −∑ N ⎜⎜ j =1 xi N 2σ i j ⎪ 2π ∏ σ i j ⎝ j =1 ⎩⎪. (. ). 2. ⎫ ⎞⎪ ⎟⎪ ⎟⎟ ⎬ ⎪ ⎠⎪ ⎭. (2.10). In (2.10), it is equivalent to minimization of the summation term. The solution can be rewritten as follows,. (. ⎧ N d − x −x i j ij ⎪ min ⎨∑ 2 xi 2σ i j ⎪ j =1 ⎩. ). 2. ⎫ ⎪ ⎬ ⎪ ⎭. (2.11). (2.11) is an optimal solution of ML estimator [25], it is also a weighted least-squares (WLS) solution [38]. Ignoring the weight of variance σ i2j on (2.11), it can be written as follows ⎧N min ⎨∑ d i j − xi − x j xi ⎩ j =1. (. ⎫. )⎬ 2. ⎭. (2.12). In (2.12), it is simpler because we ignore the statistical characteristic of range error; it is least-squares estimation (LSE) [39]. The above algorithm we mention is a nonlinear functions, it can be solved by iterated Nonlinear Least Square Solution [40], but it has to afford high computation cost, then, three linearization algorithms will be introduced in Section 2.2.2. 9.

(21) 2.2.2 Linearization of Least-Squares Estimator There are three common linearization methods, Taylor-series expansion algorithm (TS), distance-augmented algorithm (DA) and hyperbolic-canceled algorithm (HC). We will introduce these algorithms in Section 2.2.2a, Section 2.2.2b and Section 2.2.2c respectively. In Section 2.2.2d, we compare the localization accuracy of three linearization methods.. 2.2.2a Taylor-series Expansion Our aim is to linearize the nonlinear term, real distance function in (2.7), f j ( xi , yi ) = xi − x j =. (x − x ) +( y − y ) 2. 2. .. (2.13). f j ( xi , yi ) = f j ( xi 0 , yi 0 ) + ⎡⎣∇T f j ( xi 0 , yi 0 ) ⎤⎦ Δ + nT _ i j. (2.14). i. j. i. j. Applying Taylor-Series expansion [35] to (2.13) gives. where. ( xi 0 , yi 0 ). is the reference point of mobile i and the gradient vector. ⎡ ∂f ( xi 0 , yi 0 ) ∇T f j ( xi 0 , yi 0 ) = ⎢ j ∂xi ⎣. ∂f j ( xi 0 , yi 0 ) ⎤ ⎡ xi 0 − x j ⎥=⎢ ∂yi ⎦ ⎣⎢ f j ( xi 0 , yi 0 ). yi 0 − y j ⎤ ⎥ , f j ( xi 0 , yi 0 ) ⎦⎥. ⎡x − x ⎤ Δ = ⎢ i i0 ⎥ ⎣ yi − yi 0 ⎦. and nT _ i j denotes the higher order truncation error of the Taylor approximation for the distance ri j . Rewriting (2.14) as f j ( xi , yi ) = ri 0, j +. xi 0 − x j ri 0, j. ( xi − xi 0 ) +. yi 0 − y j ri 0, j. ( yi − yi 0 ) + nT _ i j. (2.15). where ri 0, j = f j ( xi 0 , yi 0 ) is the distance between reference point i and sensor j. The measurement model in (2.7) can be written as follows 10.

(22) di j = ri 0, j +. xi 0 − x j ri 0, j. ( xi − xi 0 ) +. yi 0 − y j ri 0, j. ( yi − yi 0 ) + nT _ i j + ni j. (2.16). which is a linear function and can be made into a matrix form as H i _ TS xi = bi _ TS + ni _ TS. (2.17). where. H i _ TS. ⎡ xi 0 − x1 ⎢ r ⎢ i 0,1 ⎢ xi 0 − x2 ⎢ = ⎢ ri 0,2 ⎢ # ⎢ ⎢ xi 0 − xN ⎢ r ⎣ i 0, N. yi 0 − y1 ⎤ ri 0,1 ⎥ ⎥ ⎡ cos θi 0,1 ⎥ yi 0 − y2 ⎢ ⎥ ⎢ cos θ i 0,2 ri 0,2 ⎥ = ⎢ ⎥ ⎢ # # ⎥ ⎢cos θ i 0, N yi 0 − yN ⎥ ⎣ ri 0, N ⎥⎦. sin θi 0,1 ⎤ ⎥ sin θi 0,2 ⎥ , # ⎥ ⎥ sin θi 0, N ⎥⎦. bi _ TS. ⎡ di 1 − ri 0,1 ⎤ ⎢ ⎥ ⎢ di 2 − ri 0,2 ⎥ =⎢ ⎥, # ⎢ ⎥ ⎢⎣ diN − ri 0, N ⎥⎦. (2.19). n i _ TS. ⎡ nT _ i 1 + ni 1 ⎤ ⎢ ⎥ ⎢ nT _ i 2 + ni 2 ⎥ =⎢ ⎥ # ⎢ ⎥ ⎢⎣ nT _ iN + niN ⎥⎦. (2.20). (2.18). and. ⎛ xi 0 − x j ri 0, j = ri 0, j − ⎜ ⎜ r ⎝ i 0, j. ⎞ ⎛ yi 0 − y j ⎟⎟ xi 0 − ⎜⎜ ⎠ ⎝ ri 0, j. ⎞ ⎟⎟ yi 0 ⎠. H i _ TS is an angle matrix and ni _ TS is a TS error vector which includes measurement error and higher-order error term due to inaccurate reference position. We can apply weighted least-squares(WLS) solution [38] to get the uncooperative TS estimator xˆ i _ WTS. 11.

(23) xˆ i _ WTS = ( H i _ TS T Wi _ TS H i _ TS ) H i _ TS T Wi _ TS bi _ TS −1. (2.21). where the uncooperative TS weighting is covariance inverse of the TS error vector. (. Wi _ TS = E ⎡⎣n i _ TS n i _ TS T ⎤⎦. The element of E ⎡⎣ni _ TS ni _ TS T ⎤⎦. ). −1. (2.22). is a diagonal matrix. On-diagonal: E ⎡⎣ni _ TS ni _ TS T ⎤⎦ = σ T2 _ ip + σ ip2 pp Off-diagonal: E ⎣⎡ni _ TS ni _ TS T ⎦⎤ = 0 pq. (2.23) (2.24). The resulting covariance matrix of uncooperative TS estimator ei _ WTS = xˆ i _ WTS − x is cov ( ei _ WTS ) = ( H i _ TS T Wi _ TS H i _ TS ). −1. (2.25). The Mean-Square-Error (MSE) of the estimator is −1 σ 2i _ WTS = trace ⎡⎢( H i _ TS T Wi _ TS H i _ TS ) ⎤⎥. ⎣. ⎦. (2.26). As before if we ignore the statistics Wi _ TS , the solution in (2.21) can be further simplified as xˆ i _ TS = ( H i _ TS T H i _ TS ) H i _ TS T bi _ TS −1. (2.27). and the MSE becomes as −1 −1 σ 2i _ TS = trace ⎡⎢( H i _ TS T H i _ TS ) H i _ TS T Wi _ TS −1H i _ TS ( H i _ TS T H i _ TS ) ⎤⎥. ⎣. ⎦. (2.28). TS algorithm can calculate a quite accurate solution with a very good reference point and avoid the high cost of nonlinear iteration. In fact, we can get the better reference point by updating the reference point in (2.21). xˆ i _ WTS (k + 1) = ( H i _ TS T (k ) Wi _ TS (k )H i _ TS (k ) ) H i _ TS T (k ) Wi _ TS ( k )bi _ TS ( k ) −1. (2.29). where. 12.

(24) ⎡ xi 0 (k ) − x1 ⎢ r (k ) ⎢ i 0,1 ⎢ xi 0 (k ) − x2 ⎢ H i _ TS (k ) = ⎢ ri 0,2 (k ) ⎢ # ⎢ ⎢ xi 0 (k ) − xN ⎢ r (k ) ⎣ i 0, N. 1 ⎡ ⎢ σ 2 (k ) + σ 2 i1 ⎢ T _i1 ⎢ 0 ⎢ Wi _ TS (k ) = ⎢ ⎢ # ⎢ ⎢ 0 ⎢ ⎣. yi 0 (k ) − y1 ⎤ ri 0,1 (k ) ⎥ ⎥ ⎡ cosi 0,1 (k ) sin i 0,1 (k ) ⎤ yi 0 (k ) − y2 ⎥ ⎢ ⎥ ⎥ ⎢ cosi 0,2 (k ) sin i 0,2 (k ) ⎥ ri 0,2 (k ) ⎥ = ⎢ ⎥, # # ⎥ ⎢ ⎥ # ⎥ ⎢ cos (k ) sin (k ) ⎥ i 0, N ⎦ yi 0 (k ) − yN ⎥ ⎣ i 0, N ⎥ ri 0, N (k ) ⎦. (2.30). ⎤ ⎥ ⎥ ⎥ # 0 ⎥ ⎥, ⎥ % 0 ⎥ 1 ⎥ 0 2 2 ⎥ σ T _ iN (k ) + σ iN ⎦. (2.31). 0. σ. 2 T _i2. 1 (k ) + σ i22 0 ". ". ⎡ di 1 − ri 0,1 (k ) ⎤ ⎢ ⎥ di 2 − ri 0,2 (k ) ⎥ ⎢ , bi _ TS (k ) = ⎢ ⎥ # ⎢ ⎥ ⎢⎣ diN − ri 0, N (k ) ⎥⎦. 0. (2.32). and k is the iteration index. The reference point is replaced by the TS estimator at global index k,. ( x ( k ) , y ( k ) ) = xˆ i0. i0. i _ WTS. (k ). . In (2.26) and (2.28), we know the angle matrix H i _ TS. and uncooperative TS weighting matrix Wi _ TS can affect the localization accuracy. We will compare it with other two linearized algorithms in Section 2.2.2d. However, we know this method has sensitive reference point. If we have a good point, the localization is quite accurate. Then, we can use some very simple method to find a not-too-bad point based on measurement distances. For example, in a 20m x 20m room, we have four known sensor locations on the four corners. According to four measurement distances, we can use proportion to find a simple reference point.. 13.

(25) 2.2.2b Distance-Canceled The other linear algorithm is distance-canceled (DA) algorithm [46], we summarize it as follows. First squaring up (2.7) gives di j 2 = xi − x j. 2. + 2 xi − x j ni j + ni j 2. (2.33). (2.13) is a vector form, we use a scalar form to indicate di j 2 = ( xi − x j ) + ( yi − y j ) + 2ri j ni j + ni j 2 2. 2. (2.34). Expansion of (2.34) gives di j 2 = xi 2 + yi 2 + x j 2 + y j 2 − 2 xi x j − 2 yi y j + 2ri j ni j + ni j 2. (2.35). we know the nonlinear term is xi 2 + yi 2 . We augment a squared distance variable Ri = xi 2 + yi 2. (2.36). then (2.35) can be rewritten as follows 2 xi x j + 2 yi y j − Ri = g j − di j 2 + 2ri j ni j + ni j 2 .. (2.37). Rewrite (2.37) in a matrix form as follows H DA x i = bi _ DA + ni _ DA. (2.38). where. H DA. ⎡ 2 x1 ⎢ 2x =⎢ 2 ⎢ # ⎢ ⎣ 2 xN. 2 y1 2 y2 # 2 yN. −1⎤ −1⎥⎥ , #⎥ ⎥ −1⎦. ⎡ xi ⎤ ⎡ xiT ⎤ ⎢ ⎥ x i = ⎢ yi ⎥ = ⎢ ⎥ , R ⎢⎣ Ri ⎥⎦ ⎣ i ⎦. (2.39). (2.40). 14.

(26) bi _ DA. ⎡ g1 − di21 ⎤ ⎢ ⎥ g 2 − di22 ⎥ ⎢ = , ⎢ ⎥ # ⎢ 2 ⎥ ⎣⎢ g N − diN ⎦⎥. (2.41). and. ni _ DA. ⎡ 2ri 1 ni 1 + ni 1 2 ⎤ ⎢ ⎥ 2ri 2 ni 2 + ni 2 2 ⎥ ⎢ = . ⎢ ⎥ # ⎢ 2⎥ ⎢⎣ 2riN niN + niN ⎥⎦. (2.42). H DA is a coordinate matrix and ni _ DA is a DA error vector. Here, the solution vector x i (2.34) is different from other two linearized method’s xi because we augment a variable Ri . In (2.38) we do not know the DA error vector, the matrix function we meet H DA x i ≈ bi _ DA. (2.43). As before, we can apply WLS solution to get uncooperative DA estimator −1 xˆ i _ WDA = ( H DAT Wi _ DA H DA ) H DAT Wi _ DAbi _ DA. (2.44). where the uncooperative DA weighting Wi _ DA is the covariance inverse of the DA error vector. (. Wi _ DA = E ⎡⎣n i _ DAn i _ DAT ⎤⎦. ). −1. (2.45). and E ⎡⎣ni _ DAni _ DAT ⎤⎦ is a diagonal matrix as well On-diagonal: E ⎡⎣ni _ DAni _ DAT ⎤⎦ = 4rip2σ ip2 + 3σ ip2 pp Off-diagonal: E ⎡⎣ni _ DAni _ DAT ⎤⎦ = 0 pq. (2.46). (2.47). and the resulting covariance matrix of uncooperative DA estimator error ei _ WDA = xˆ i _ WDA − x is 15.

(27) cov ( ei _ WDA ) = ( H DAT Wi _ DA H DA ). −1. (2.48). The size of covariance matrix is 3x3 and it is different from other two linearized methods. Then, the MSE of the estimator is −1 σ 2i _ WDA = trace ⎡⎢( H DAT Wi _ DA H DA ) ⎤⎥. ⎣. ⎦ 2×2. (2.49). As before, if we ignore the statistics Wi _ DA , the solution in (2.44) can be further simplified as −1 xˆ i _ DA = ( H DAT H DA ) H DAT bi _ DA. (2.50). and the MSE becomes as −1 −1 σ 2i _ DA = trace ⎡⎢( H DAT H DA ) H DAT Wi _ DA −1H DA ( H DAT H DA ) ⎤⎥. ⎣. ⎦ 2×2. (2.51). From (2.49) and (2.51), we know that coordinate matrix H DA and DA weighting matrix Wi _ DA can affect the localization accuracy. DA algorithm is very easy to operate but it also suffers from new variable Ri . However, Ri is not independent on the variable of position mobile i in WLS solution. The accuracy might be the worst .. 2.2.2c Hyperbolic-Canceled Now, we introduce the HC algorithm [22]. In (2.35) we know that xi 2 + yi 2 is nonlinear term. Then, we choice the sensor k as a reference sensor. The reference equation is dik 2 = xi 2 + yi 2 + xk 2 + yk 2 − 2 xi xk − 2 yi yk + 2rik nik + nik 2. (2.52). In order to cancel the nonlinear term, subtraction of (2.35) from (2.52) gives 2 xi ( x j − xk ) + 2 yi ( y j − yk ) = dik 2 − di j 2 + g j − g k + 2ri j ni j − 2rik nik + ni j 2 − nik 2 , j ≠ k (2.53). 16.

(28) where g k = xk 2 + yk 2 , g j = x j 2 + y j 2 Without loss of generality, let k=1. (2.53) is a linear function, we can rewrite (2.53) as a matrix form H HC xi = bi _ HC + ni _ HC. (2.54). where. H HC. ⎡ ( x2 − x1 ) ⎢ (x − x ) =⎢ 3 1 ⎢ # ⎢ ⎢⎣( xN − x1 ). ( y2 − y1 ) ⎤ ( y3 − y1 ) ⎥⎥. ,. (2.55). bi _ HC. ⎡ di21 − di22 + g 2 − g1 ⎤ ⎢ 2 ⎥ 2 1 ⎢ di 1 − di 3 + g3 − g1 ⎥ , = ⎥ 2⎢ # ⎢ 2 ⎥ 2 ⎣⎢ di 1 − diN + g N − g1 ⎦⎥. (2.56). ⎥ ⎥ ( yN − y1 )⎥⎦ #. and. ni − HC. 1 2 ⎡ 2 ⎤ ⎢ ri 2 ni 2 − ri 1 ni 1 + 2 ( ni 2 + ni 1 ) ⎥ ⎢ ⎥ ⎢ r n − r n + 1 ( n2 + n2 ) ⎥ = ⎢ i3 i3 i1 i1 2 i3 i1 ⎥ . ⎢ ⎥ # ⎢ ⎥ ⎢ 1 2 2 ⎥ ⎢ riN niN − ri 1 ni 1 + ( niN + ni 1 ) ⎥ 2 ⎣ ⎦. (2.57). H HC is a coordinate-difference matrix and ni _ HC is a HC error vector for estimated mobile i. In fact, we do not know the HC error vector in (2.54). The matrix equation we meet H HC xi ≈ bi _ HC. (2.58). We can apply WLS solution to get the uncooperative HC estimator xˆ i _ WHC = ( H HC T Wi _ HC H HC ) H HC T Wi _ HC bi _ HC −1. (2.59). 17.

(29) where the uncooperative HC weighting matrix Wi _ HC is the covariance inverse of the HC error vector. (. Wi _ HC = E ⎡⎣ni _ HC ni _ HC T ⎤⎦. ). −1. (2.60). Then, the covariance of E ⎡⎣ni _ HC ni _ HC T ⎤⎦ is. 3 On-diagonal: E ⎡⎣ni _ HC ni _ HC T ⎤⎦ = rip2σ ip2 + ri 21σ i21 + (σ ip2 − σ i21 ) pp 4 3 Off-diagonal: E ⎡⎣ni _ HC ni _ HC T ⎤⎦ = ri 21σ i21 + σ i21 pq 4. (2.61). (2.62). It is not diagonal matrix anymore. The resulting covariance matrix of the uncooperative HC estimator error ei _ WHC = xˆ i _ WHC − x is cov ( ei _ WHC ) = ( H HC T Wi _ HC H HC ). −1. (2.63). We can further get the MSE of the estimator −1 σ 2i _ WHC = trace ⎡⎢( H HC T Wi _ HC H HC ) ⎤⎥. ⎣. ⎦. (2.64). If we ignore the statistic Wi _ HC , the solution in (2.59) can be further simplified as xˆ i _ HC = ( H HC T H HC ) H HC T bi _ HC −1. (2.65). and the MSE becomes as −1 −1 σ 2i _ HC = trace ⎡⎢( H HC T H HC ) H HC T Wi _ HC −1H HC ( H HC T H HC ) ⎤⎥. ⎣. ⎦. (2.66). From (2.51) and (2.53), we know that coordinate-difference matrix H HC and uncooperative HC weighting matrix Wi _ HC can affect the localization accuracy. Then, we will compare H algorithm and Walgorithm of three linearized algorithms in Section 2.2.2d.. 18.

(30) 2.2.2d Summary of Three Linearization Algorithms We know three linearization algorithms have similar matrix equation as Hx = b + n . Now, we compare H, x, and n. Now, we can see the H for TS is angle. matrix in (2.18). H i _ TS. ⎡ xi 0 − x1 ⎢ r ⎢ i 0,1 ⎢ xi 0 − x2 ⎢ = ⎢ ri 0,2 ⎢ # ⎢ ⎢ xi 0 − xN ⎢ r ⎣ i 0, N. yi 0 − y1 ⎤ ri 0,1 ⎥ ⎥ ⎡ cos θi 0,1 yi 0 − y2 ⎥ ⎢ cos θi 0,2 ⎥ ri 0,2 ⎥ = ⎢ ⎢ ⎥ ⎢ # # ⎥ ⎢cos θ i 0, N yi 0 − yN ⎥ ⎣ ri 0, N ⎥⎦. sin θi 0,1 ⎤ ⎥ sin θi 0,2 ⎥ # ⎥ ⎥ sin θi 0, N ⎥⎦. (2.18). which consists of angles from sensors. Then, H for DA is coordinate matrix in (2.39). H DA. ⎡ 2 x1 ⎢ 2x =⎢ 2 ⎢ # ⎢ ⎣ 2 xN. 2 y1 2 y2 # 2 yN. −1⎤ −1⎥⎥ #⎥ ⎥ −1⎦. (2.39). It is made up of coordinates for all sensors. The last H of HC is coordinate-difference matrix in (2.55). H HC. ⎡ ( x2 − x1 ) ⎢ (x − x ) =⎢ 3 1 ⎢ # ⎢ ⎣⎢( xN − x1 ). ( y2 − y1 ) ⎤ ( y3 − y1 ) ⎥⎥. ⎥ ⎥ ( yN − y1 )⎦⎥ #. ,. (2.55). It is made up of differences of coordinates based on reference mobile 1. However, only the variable x = [ x x = [x. y. R ] of DA is different from other linearized methods’. y ] . Finally, we discuss n from three parts, noise source, algorithm weighting. and covariance matrix of estimator error as follows. (1) Noise Source: Noise source can affect the theoretical MSE. In HC algorithm, after linearization 19.

(31) operation, the HC error term is ri j ni j − ri 1 ni 1 +. (. 1 ni j 2 − ni 1 2 2. ). which is involved in the. effect of real distance ri j and ri 1 . The effect amplifies the measurement error and destroys the theoretical MSE. However, in TS algorithm, after linearization operation, the TS error term is nT _ i j + ni j . If the reference position is perfectly ideal, TS error term only involves the measurement error ni j . Then, in DA algorithm, after linearization operation, the DA error term is 2ri j ni j + ni j 2 which is also involved in real distance effect. In addition, it has a new variable Ri . In fact, the new variable is not independent on position of mobile in WLS solution (2.44). Therefore, it has the worst MSE performance. After the comparison of noise source, we will compare the weighting matrix for these linearized algorithms.. (2) Weighting Matrix: First, We assume variances of measurement errors are the same as σ 2 . We know the weighting matrix is gotten from noise source covariance. Then, the uncooperative HC weighting matrix is. Wi _ Hyper. 3 ⎡ 2 2 ⎢ ri 2 + ri 1 + 2 ⎢ ⎢ r2 + 3 1 ⎢ i1 4 = 2⎢ σ ⎢ # ⎢ ⎢ 2 3 ⎢ ri 1 + 4 ⎣. 3 r + 4 ri 23 + ri 21 + 3 r + 4 2 i1. ". 3 2. ri 21 +. −1. 2 i1. 3 4. %. ri 21 +. ⎤ ⎥ ⎥ ⎥ # ⎥ 3 ⎥⎥ ri 21 + 4 ⎥ 3⎥ riN2 + ri 21 + ⎥ 2⎦ 3 r + 4. ". 2 i1. 3 4. (2.67). In (2.53), uncooperative HC weighting matrix compensates the HC error term with real distance effect. In fact, it is related with the reference point mobile i. Next, in TS algorithm, we assume the reference position is very good nT _ i j ≈ 0 .. 20.

(32) Therefore, the uncooperative TS weighting matrix is Wi _ TS ≈. 1. σ2. I N ×N. (2.68). In (2.70), TS weighting compensates Taylor error term ni 1 which is also measurement error. Finally, in DA algorithm, the uncooperative DA weighting can be rewritten as follow as. Wi _ DA. ⎡ 1 ⎢ 4r 2 + 3 ⎢ i1 ⎢ 1 ⎢ 0 = 2⎢ σ ⎢ # ⎢ ⎢ 0 ⎢ ⎣. ⎤ ⎥ ⎥ ⎥ 1 0 # ⎥ 2 4ri 2 + 3 ⎥ 0 0 ⎥ % ⎥ 1 ⎥ 0 " 4riN2 + 3 ⎥⎦ 0. ". 0. (2.69). In (2.71), DA weighting compensates the error term with real distance effect. However, before the compensation of weighting matrix, the best MSE performance is uncooperative TS estimator and the best is the DA estimator. After the compensation of weighted, their have similar theoretical MSE. The computer simulation result for theoretical MSE of three linearized algorithms will show in Figures 5.2 and 5.3.. (3) Covariance Matrix of Estimator Error The covariance matrix includes matrix H algorithm and weighting matrix Walgorithm because of cov ( ealgorithmeTalgorithm ) = ( H algorithmT Walgorithm H algorithm ) . Then, the HC −1. algorithm’s is difficult to calculate because it’s covariance of HC error vector is not a diagonal matrix in (2.61) and (2.62). Therefore, we discuss TS algorithm and DA algorithm. However, in TS algorithm, the covariance matrix of estimator error in (2.26) included angle matrix H i _ TS and its weighting matrix Wi _ TS is given by. 21.

(33) ⎧ ⎪ ⎪⎪ 1 cov ( ei _ WTS ) = ⎨ 2 ⎪σ ⎪ ⎪⎩ If the reference point. 2 ⎡ xi 0 − x j ) ( ⎢ N ⎢ ri 20, j ⎢ ∑ j =1 ⎢ x − x ( i 0 j )( yi 0 − y j ) ⎢ ri 20, j ⎢⎣. ( xi 0 , yi 0 ). ( xi 0 − x j )( yi 0 − y j ) ⎤⎥ ⎫⎪ ⎥ ⎪⎪ ri 20, j ⎥⎬ 2 ⎥ − y y ( i0 j ) ⎥ ⎪ ⎪ ri 20, j ⎥⎦ ⎭⎪. is very close to true point. −1. (2.70). ( xi , yi ) , (2.70) can be. approximated as follows 2 ⎧ ⎡ xi − x j ) ( ⎪ ⎢ ⎪N ⎢ ri 2j 2 ⎪ cov ( ei _ WTS ) ≈ σ ⎨∑ ⎢ ⎪ j =1 ⎢ ( xi − x j )( yi − y j ) ⎪ ⎢ ri 2j ⎪⎩ ⎢⎣. ( xi − x j )( yi − y j ) ⎥⎤ ⎪⎫ ⎥ ⎪⎪ ri 2j ⎥⎬ 2 y y − ( i j ) ⎥⎥ ⎪ ⎪ ri 2j ⎥⎦ ⎪⎭. −1. (2.71). In fact, the right hand side of (2.71) is the full uncooperative Fisher Information Matrix (FIM) (4.4) inverse and is proportion to the noise variance σ 2 . Therefore, the theoretical MSE of TS estimator is very close to uncooperative CRLB. As before, the covariance of estimator error for DA algorithm can be formed as ⎧ ⎡ 4 x j2 4x j y j 2x j ⎤⎫ − ⎪ ⎢ 2 ⎥⎪ 4ri 2j + 3 4ri 2j + 3 ⎥ ⎪ ⎪ ⎢ 4ri j + 3 ⎪N ⎢ ⎪ 4x j y j 4 y 2j 2 y j ⎥⎥ ⎪ 2 ⎪ ⎢ cov ( ei _ DA ) = σ ⎨∑ − 2 ⎬ 2 4ri 2j + 3 4ri j + 3 ⎥ ⎪ ⎪ j =1 ⎢⎢ 4ri j + 3 ⎥⎪ ⎪ ⎢ 2x j 2yj ⎥⎪ 1 ⎪ ⎢− 2 − 2 2 4ri j + 3 4ri j + 3 ⎥⎦ ⎪ ⎪ ⎣ 4ri j + 3 ⎩ ⎭. −1. (2.72). The covariance matrix is proportion to the noise variance σ 2 as well. If rji2 3 , (2.53) can be simplified as follows. 22.

(34) ⎧ ⎡ x j2 ⎪ ⎢ 2 ⎪ ⎢ ri j ⎪M ⎢ x y 2 ⎪ cov ( ei _ WDA ) = σ ⎨∑ ⎢ j 2 j ⎪ i =1 ⎢⎢ ri j ⎪ ⎢ x ⎪ ⎢ − j2 ⎪ ⎣ 2ri j ⎩. xj yj ri 2j y 2j ri 2j −. yj 2ri 2j. x ⎤⎫ − j2 ⎥ ⎪ 2ri j ⎥ ⎪ ⎪ y j ⎥⎥ ⎪ − 2 ⎬ 2ri j ⎥ ⎪ ⎥ 1 ⎥ ⎪⎪ 4ri 2j ⎥⎦ ⎪ ⎭. −1. (2.73). There are negative numbers in (2.73), it can reduce the covariance matrix. Then, the theoretical MSE of DA estimator is larger than TS estimator’s. Finally, the noise source of HC algorithm is similar to DA algorithm’s expect the new augmented variable. Therefore, the two methods have similar theoretical MSE. Simulation results demonstrate the effectiveness of these algorithms in Section 5.1.. 23.

(35) Chapter 3 Cooperative Localization Algorithm In cooperative system, cooperative relay or Ad Hoc short-rang communication among the terminals will be supported [41, 42]. They consider the short-range has lower measurement error interference, therefore it can enhance the location estimation accuracy. Then, they investigate the data fusion of large-scale and small-scale. [26] proposed Cooperative Localization with Optimum Quality of Estimate (CLOQ) which takes advantage of the behavior of the channel to provide accurate indoor positioning. This algorithm uses the quality of ranging and positioning estimates to provide practical and accurate results. More importantly, it reduces error propagation substantially. If Non-Line-Of-Sight (NLOS) exists, cooperative group localization (CGL) scheme is proposed in [13] based on rigid graph theory. However, cooperative localization is not a well solved problem because the distance measurement between any pairs of unknown positions (mobiles) are utilized to assist in the location estimation. This is much more challenging than the traditional localization where only distance measurements between unknown position (one mobile) and known positions (sensors) are employed for localization. Therefore, [12] devised subspace approach to solve that problem and which can outperform the classical MDS algorithm. The other chooses is cooperative ML estimator which can be solved by joint Newton’s algorithm. But the computation cost is quite high because it is nonlinear iterated solution. Base on the linearized algorithm of uncooperative localization, we propose joint Taylor-series expansion algorithm to reduce the cost. But other linear operation can not linearize the cooperative TOA measurement to form joint algorithms. Therefore, we devise divided-and-conquer method to overcome 24.

(36) that problem and it can also reduce computation cost from joint Newton’s algorithm. In divided-and-conquer method, we need uncertain virtual sensors to achieve the method. [15] proposed an error propagation aware algorithm to track the extent of the uncertain virtual position error. It sets threshold to decide which virtual can be involve in localization. By recursive position estimation, the more estimated locations are selected as virtual sensor locations until all estimated positions are virtual sensor locations. In fact, some bad virtual sensor positions still provide the information to localization and the selection of threshold has trade-off between computation cost and localization accuracy. Therefore, our proposed algorithm consider all virtual sensors to help by global iteration and we also discuss the computation cost. Figure 3.1 indicates the cooperative localization system. As before, there are N known positions of sensors and M unknown positions of mobiles. The two individual localization system for mobile i and mobile j are cooperated by cooperative TOA measurement, di j . Then, ri j is real cooperative distance from mobiles i and j.. Figure 3.1 Cooperative localization system with cooperative TOA measurement. 25.

(37) The dot line distance denotes cooperative distance that connects two mobiles. All measurements of cooperative localization are illustrated as follows The uncooperative measurement between mobile i and sensor j is denoted as di j = ri j + ni j , i = 1 ~ M , j = 1 ~ N. (3.1). and the cooperative measurement between mobile i and mobile j is denoted as dij = rij + nij , i < j , i, j = 1 ~ M. (3.2). where ni j and ni j is uncooperative and cooperative measurement error belong to. (. AWGN ni j ~ N 0, σ i2j. ). and ni j ~ N ( 0, σ i2j ). There are M positions of mobile will be estimated, the cooperative likelihood function (ML) [42] is similar to uncooperative likelihood function in (2,9) and it is written as follows p ( dUncoop , d Coop | x ) =. (. ). (. ). ⎛ d − x −x 2 ⎞ M ⎛ d − x −x 2 ⎞ i j ij i j i j 1 ⎟⋅ ⎟ exp ⎜ − exp ⎜ − ∏∏ ∏ 2 2 2 2 ⎜ ⎟ ⎜ ⎟⎟ σ σ 2 2 i =1 j =1 2σ i j ij ⎜ ⎟ ij,>j i=1 2σ ij ⎜ ij ⎝ ⎠ ⎝ . . ⎠ M. N. 1. Uncooperation. Cooperation. (3.3) where cooperative observation set di _ Uncoop is like d i in (2.9), cooperative observation set dCoop = ⎡⎣ d12 , d13 ," , d1M , d 23 , d 24 ," d 2 M ," d M −1, M ⎤⎦ and all positions of mobiles x = [ x1. x2 " x M ] . T. The cooperative ML criterion searches a xˆ which maximizes likelihood function (3.3), ⎧ ⎫ ⎪ ⎪ M 2 2⎪ 1 ⎪M N 1 min ⎨∑∑ 2 di j − xi − x j + ∑ 2 dij − xi − x j ⎬ xˆ i =1 j =1 2σ i j i =1 2σ ij ⎪ ⎪ . j >i . ⎪ ⎪ Noncooperation Cooperation ⎩ ⎭. (. ). (. ). (3.4). 26.

(38) where xˆ = [ xˆ 1. xˆ 2 " xˆ M ] is the cooperative ML estimator. T. We will introduce a common joint Newton’s method to solve it. However, joint Newton’s method is nonlinear function and it computation cost is quite high. Therefore we propose two new methods to reduce it, joint Taylor-series expansion algorithm (joint TS) and divide-and-conquer method (divided algorithms). The structure of the rest of this section is as follows. Section 3.1 discusses joint cooperative algorithms. The divide-and-conquer method is proposed in Section 3.2. In Section 3.3, discusses the issue of divide-and-conquer method, compensation of uncertain virtual sensor.. 3.1 Joint Cooperative Algorithm According to cooperative algorithms, we will introduce a common nonlinear joint Newton’s method to solved (3.4) in 3.1.1 and we propose a joint TS in Section 3.1.2. In 3.1.3, we will illustrate why the other linearized methods can’t form joint algorithms.. 3.1.1 Newton’s Method In order to minimize the object function in (3.4), we can set its gradient function to zero and to get the estimated positions. Let the object function be denoted as M. N. G (x) = ∑∑ i =1 j =1. 1 2σ i2j. (d. ij. − xi − x j. ) + ∑ 2σ1 ( d 2. M. i =1 j >i. 2 ij. ij − x i − x j. ). 2. (3.5). and ⎡ ∂G (x) ∇Tx G (x) = ⎢ ⎣ ∂x1. ∂G (x) ∂y1. ∂G ( x) ∂x2. ∂G (x) ∂G (x) ⎤ " ⎥=0 ∂y2 ∂yM ⎦. (3.6). We can use Newton’s method [26] to solve (3.6) and it is donated as follows 27.

(39) xˆ Joint_Newton ( k + 1) = xˆ Joint_Newton ( k ) − ⎡⎣∇ x∇Tx G ( xˆ Joint_Newton ( k ) ) ⎤⎦. −1. 2 M ×2 M. ∇Tx G ( xˆ Joint_Newton ( k ) ). (3.7) where k is the iteration index, ⎡ ∂ 2G ( xˆ ) ⎢ ⎢ ∂x1∂x1 ⎢ ∂ 2G ( xˆ ) ⎢ ⎢ ∂y1∂x1 ⎢ 2 ⎢ ∂ G ( xˆ ) T ∇ x∇ x G (xˆ ) = ⎢ ∂x2 ∂x1 ⎢ 2 ⎢ ∂ G ( xˆ ) ⎢ ∂y ∂x ⎢ 2 1 ⎢ # ⎢ 2 ⎢ ∂ G ( xˆ ) ⎢ ∂y ∂x ⎣ M 1. ∂ 2G ( xˆ ) ∂x1∂y1. ∂ 2G ( xˆ ) ∂x1∂x2. ∂ 2G ( xˆ ) " ∂x1∂y2. ∂ 2G ( xˆ ) ∂y1∂y1. ∂ 2G ( xˆ ) ∂y1∂x2. ∂ 2G ( xˆ ) ∂y1∂y2. ∂ 2G ( xˆ ). %. ∂x2 ∂y1 ∂ 2G ( xˆ ) ∂y2 ∂y1. %. # ∂ G ( xˆ ) ∂yM ∂y1 2. ". ∂ 2G ( xˆ ) ⎤ ⎥ ∂x1∂yM ⎥ ∂ 2G ( xˆ ) ⎥ ⎥ " ∂y1∂yM ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ # ⎥ ⎥ ⎥ % ⎥ 2 ∂ G ( xˆ ) ⎥ ∂yM ∂yM ⎥⎦. (3.8). is the Jacobian matrix [23] whose element is. (. ). (. ). 2 2 ⎡ d pj − rpj ⎤ ∂ 2G (xˆ ) N 1 ⎢ d pj − rpj ( x p − x j ) ( x p − x j ) ⎥ =∑ + − ⎥ rpj3 rpj2 rpj ∂x p ∂x p j =1 σ pj2 ⎢ ⎣ ⎦ 2 2 ⎡ ( d − r )( x − x ) ( x − x ) ( d − r ) ⎤ p −1 1 ip ip i p i p ip ip ⎥ + ∑ 2 ⎢+ + − 3 2 ⎥ rip rip rip i =1 σ ip ⎢ ⎣ ⎦ i< p 2 2 ⎡ ⎤ 1 ⎢ ( d pi − rip )( x p − xi ) ( x p − xi ) ( d pi − rpi ) ⎥ +∑ 2 + + − ⎥ rpi rpi3 rpi2 i = p +1 σ ip ⎢ ⎣ ⎦ i> p M. (. ). ∂ 2G (xˆ ) N 1 ⎡ d pj − rpj ( x p − x j )( y p − y j ) ( x p − x j )( y p − y j ) ⎤ ⎢ ⎥ =∑ + rpj3 rpj2 ∂x p ∂y p j =1 σ pj2 ⎢ ⎥⎦ ⎣ p −1 1 ⎡ ( dip − rip )( xi − x p )( yi − y p ) ( xi − x p )( yi − y p ) ⎤ + ∑ 2 ⎢+ + ⎥ rip3 rip2 i =1 σ ip ⎢ ⎥⎦ ⎣ i< p +. 1 ⎡ ( d pi − rpi )( x p − xi )( y p − yi ) ( x p − xi )( y p − yi ) ⎤ + ⎥ ∑ 2 ⎢+ rpi3 rpi2 i = p +1 σ pi ⎢ ⎥⎦ ⎣ i> p M. 2 2 ⎡ ⎤ 1 ⎢ ( d pq − rpq )( x p − xq ) ( x p − xq ) ( d pq − rpq ) ⎥ ∂ 2G (xˆ ) =− 2 + − ⎥ rpq3 rpq2 rpq σ pq ⎢ ∂x p ∂xq ⎣ ⎦. 28.

(40) ∂ 2G (xˆ ) 1 ⎡ ( d pq − rpq )( x p − xq )( y p − yq ) ( x p − xq )( y p − yq ) ⎤ =− 2 ⎢ + ⎥ ∂x p ∂yq σ pq ⎢⎣ rpq3 rpq2 ⎥⎦. (. ). (. 2 2 ⎡ d pj − rpj ∂ 2G (xˆ ) N 1 ⎢ d pj − rpj ( y p − y j ) ( y p − y j ) =∑ 2 + − 3 2 rpj rpj rpj ∂y p ∂y p j =1 σ pj ⎢ ⎣ 2 2 ⎡ ⎤ p −1 1 ⎢ ( dip − rip )( yi − y p ) ( yi − y p ) ( dip − rip ) ⎥ +∑ 2 + − ⎥ rip3 rip2 rip i =1 σ ip ⎢ ⎣ ⎦ i< p. ) ⎤⎥ ⎥ ⎦. 2 2 ⎡ ⎤ 1 ⎢ ( d pi − rip )( y p − yi ) ( y p − yi ) ( d pi − rpi ) ⎥ + ∑ 2 + − ⎥ rpi rpi3 rpi2 i = p +1 σ ip ⎢ ⎣ ⎦ i> p M. 2 2 ⎡ ⎤ 1 ⎢ ( d pq − rpq )( y p − yq ) ( y p − yq ) ( d pq − rpq ) ⎥ ∂ 2G (xˆ ) =− 2 + − ⎥ rpq3 rpq2 rpq σ pq ⎢ ∂y p ∂yq ⎣ ⎦. Actually, the Jacobian matrix inverse part ⎡∇Tx ∇ xG ( xˆ Joint_Newton ( k ) ) ⎤ ˆ ⎣ ⎦ 2 M ×2 M ∇ xG ( x Joint_Newton ( k ) ) in (3.7) can be achieved −1. by ∇Tx ∇ xG ( xˆ Joint_Newton ( k ) ) \ ∇ xG ( xˆ ( k ) ) of Gaussian elimination method in MATLAB function. Even if the Gaussian elimination method replaces that part, but the size of Jacobian matrix is quite large and the elements have to calculate a lot of summations when it exists multitudinous mobiles. We assume one global iteration in (3.7) needs F2 M ×2 M flops. G global iterations are needed to converge in (3.7). Then, the total computation cost is given by Joint Newton computation cost = F2 M ×2 M × G joint flops.. (3.9). Then, we derive linearized method, joint TS to reduce the calculation complexity F2 M ×2 M for Gaussian elimination method of large matrix in following section.. 3.1.2 Taylor-Series Expansion Algorithm Now we use Taylor-series approximation method to linearize cooperative nonlinear function. According to (2.17) in Section 2.2.2a, we have M uncooperative 29.

(41) TS matrix equation as follows H i _ TS xi = bi _ TS + ni _ TS i = 1, 2,..., M. (3.10). Now, the remaining task for us is to linearize cooperative distances rij among mobiles. We know the relationship of real distance from mobile i to mobile j is rij = ( xi − x j ) 2 + ( yi − y j ) 2 = f ij (Δxij , Δyij ). (3.11). Because there are four variables in a cooperative real distance (includes two positions of mobiles), we can regard the difference variable as a new variable, difference-variable Δxij and Δyij . ⎡ xi − x j ⎤ ⎡ Δxij ⎤ ⎢ y − y ⎥ = ⎢ Δy ⎥ j⎦ ⎣ i ⎣ ij ⎦. (3.12). later we will convert these difference-variable Δxij , Δyij back to four original variable xi , yi , x j , y j . Apply Taylor-series expansion to (3.11) as follows. fij (Δxij , Δyij ) = fij (Δxij 0 , Δyij 0 ) + ⎡⎣∇T f ij (Δxij 0 , Δyij 0 ) ⎤⎦ Δ + nT _ ij. (3.13). where (Δxij 0 , Δyij 0 ) is the difference reference point , nT _ ij is the higher order truncation error of the Taylor-series expansion for the distance rij , ⎡ ∂f ij (Δxij 0 , Δyij 0 ) ∇T fij (Δxij 0 , Δyij 0 ) = ⎢ ∂Δxij ⎢⎣. ∂fij (Δxij 0 , Δyij 0 ) ∂Δyij. ⎤ ⎡ Δxij 0 ⎥=⎢ ⎥⎦ ⎢⎣ fij (Δxij 0 , Δyij 0 ). ⎤ ⎥ fij (Δxij 0 , Δyij 0 ) ⎥⎦ Δyij 0. and ⎡ Δxij − Δxij 0 ⎤ Δ=⎢ ⎥. ⎣ Δyij − Δyij 0 ⎦. Now the cooperative measurement model (3.2) becomes 30.

(42) dij = rij + nij = rij 0 +. where. Δxij 0 rij 0. ( Δx. ij. − Δxij 0 ) +. Δyij 0 rij 0. ( Δx. ij. − Δxij 0 ) + nTij +nij. (3.14). rij 0 = f ij (Δxij 0 , Δyij 0 ) .. Knowing that (3.14) is a linear function of difference-variable (Δxij 0 , Δyij 0 ) , we may form a cooperative TS equation (. Δxij 0 rij 0. )Δxij + (. Δyij 0 rij 0. )Δyij = dij − rij 0 + nT _ ij + nij. (3.15). ⎛ Δxij 0 ⎞ ⎛ Δyij 0 ⎞ ⎟⎟ Δxij 0 − ⎜⎜ ⎟⎟ Δyij 0 ⎝ rij 0 ⎠ ⎝ rij 0 ⎠. where rij 0 = rij 0 − ⎜⎜. The cooperative TS equation in (3.15), distance-variable. ( Δx , Δy ). T. ij. ij. is different. form the variables of uncooperative matrixes equation in (2.17) for mobile i and mobile j with original variables xi and x j . Consequently we convert the difference-variable back to original variable. (3.15) can be written as follow ⎡ Δxij 0 ⎡ Δx Δy ⎤ Δy ⎤ ) ( ij 0 ) ⎥ xi − ⎢ ( ij 0 ) ( ij 0 ) ⎥ x j = dij − rij 0 + nT _ ij + nij ⎢( rij 0 ⎦⎥ rij 0 ⎦⎥ ⎣⎢ rij 0 ⎣⎢ rij 0 The cooperative TS equation (3.14) has the original variable. (x , x ). T. i. j. (3.16). which is like. uncooperative TS matrix equation’s for mobile i and mobile j. Therefore, we can combine two equations to form a joint TS matrix equaiton. In case of M=4, the joint TS cooperative matrix equation is given by. 31.

(43) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣. H1_ TS 0. 0 H 2_TS. 0. 0. 0 T h120. 0 T −h120. 0 0 H 3_ TS 0. h. 0. 0 T −h130. h. 0 hT230. 0 −hT230. hT240. 0 hT340. T 130 T 140. 0 0 0. 0. ⎡ b1 ⎤ ⎡ n1_ TS ⎤ ⎤ ⎥ ⎢ b ⎥ ⎢ n ⎥ 2 _ TS 2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ b 3 ⎥ ⎢ n3_ TS ⎥ 0 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ H 4 _ TS ⎥ ⎡ x1 ⎤ ⎢ b 4 ⎥ ⎢ n 4 _ TS ⎥ 0 ⎥ ⎢⎢ x 2 ⎥⎥ ⎢ d12 − r120 ⎥ ⎢ nT _12 + n12 ⎥ ⎥ =⎢ ⎥ ⎥+⎢ 0 ⎥ ⎢ x3 ⎥ ⎢ d13 − r130 ⎥ ⎢ nT _13 + n13 ⎥ T ⎥ ⎢ x ⎥ ⎢ d − r ⎥ ⎢ n +n ⎥ −h140 ⎥ ⎣ 4 ⎦ ⎢ 14 140 ⎥ ⎢ T _14 14 ⎥ 0 ⎥ ⎢ d 23 − r230 ⎥ ⎢ nT _ 23 + n24 ⎥ T ⎥ ⎢ d − r ⎥ ⎢ n +n ⎥ −h 240 ⎥ ⎢ 24 240 ⎥ ⎢ T _ 24 24 ⎥ −hT340 ⎥⎦ ⎢⎣ d34 − r340 ⎥⎦ ⎢⎣ nT _ 34 + n34 ⎥⎦ 0 0. (3.17). Where hij 0. ⎡ Δxij 0 =⎢ ⎣⎢ rij 0. T. Δyij 0 ⎤ ⎥ = ⎡⎣cos θ ij 0 rij 0 ⎦⎥. T. sin θij 0 ⎤⎦ .. is a cooperative angle vector between two reference points. ( xi 0 , yi 0 ). and. (x. j0. , y j0 ) .. Here, there are C ( M , 2 ) = 6 pairs cooperative measurements between mobiles. We note a angle vector in (3.17) for one pair is ⎡⎣hij 0. −hij 0 ⎤⎦ because of the. difference-variable in (3.12). In addition, (3,17) can be simplified as H Joint_TS x = b Joint_TS + n Joint_TS. (3.18). where H Joint_TS is joint TS angle matrix. As before, the joint TS estimator is given by xˆ Joint_TS = ( H Joint_TST WJoint_TS H Joint_TS ) H Joint_TST WJoint_TSb Joint_TS −1. (3.19). −1. where the joint TS weighting matrix, WJoint_TS = E ⎡⎣n Joint_TSnTJoint_TS ⎤⎦ . It is like (2.29). We also can get the better reference point set by updating the reference point set from joint TS estimator in (3.19). xˆ Joint_TS (k + 1) = ( H Joint_TST (k ) WJoint_TS (k )H Joint_TS (k ) ) H Joint_TST (k ) WJoint_TS (k )b Joint_TS (k ) −1. (3.20) 32.

(44) In the same reason, the solution (3.20) can be solved by Gaussian elimination method in MATLAB function, xˆ Joint_TS = WJoint_TS H Joint_TS \ WJoint_TS b Joint_TS . Even if the mobiles more, the size of joint TS angle matrix larger, but it doesn’t do a lot of summations. However, it can reduce the computation cost of joint Newton’s method. Simulations are presented in terms of the MSE, convergence rate and computation cost for two joint algorithms in Section 5.2. Alternatively, we also propose a divide-and-conquer method the reduce the cost of calculation complexly (3.7) in Section 3.2. 3.1.3 Other Joint Linearization Algorithms We will brief illustrate that distance-augmented method and hyperbolic-canceled method can not achieve joint algorithms. We know the new challenge to form joint algorithm is to linearize the cooperative real distance rij = ( xi − x j ) 2 + ( yi − y j ) 2 . However, in DA method, we square the measurement model before linearized operation. Therefore, (3.7) can be operated as follows, dij2 = ( xi − x j ) 2 + ( yi − y j ) 2 + 2rij nij + nij2 = Ri + R j − 2 xi x j − 2 yi y j + 2rij nij + nij2. (3.21). In (3.21) even if we augment the new variable Ri and R j , it still exist the nonlinear term xi x j and yi y j . Therefore, DA method can not achieve joint algorithm. Of course, HC method has the same problem because it has to do square operation. Therefore, we do not have joint DA and joint HC algorithms. However, if. (x , y ) j. j. are known, (3.21) can be linearized. Therefore, we proposed divide-and-conquer to solve that problem.. 33.

(45) 3.2 Divide-and-Conquer Method We know that cooperative localization is involved unknown positions connection. It is more difficult than uncooperative localization. Therefore, we can simplify it by divide-and-conquer method. We illustrate the method Figures 3.2 to 3.4.. Figure 3.2 The joint localization method. If the mobile j location is know as. ( xˆ , yˆ ) , we have a individual localization in j. j. Figure 3.3.. Figure 3.3 A part of divide-and-conquer method. 34.

(46) As before, if mobile j location is known, we have another part of divide-and-conquer method in Figure 3.4.. Figure 3.4 Another part of divide-and-conquer method.. However, divide-and-conquer method can reduce the computation cost in (3.7). the system model is described in follows. At first, every position of mobiles can be estimated from uncooperative ML estimator. Then, we have initial virtual sensors.. ⎧ ⎫ ⎪⎪ N 1 2⎪ ⎪ min ⎨∑ 2 di j − xi − x j ⎬ , i = 1 ~ M xi ,0 j =1 2σ ij ⎪ . ⎪ ⎪⎩ ⎪⎭ Noncooperation. (. ). (3.22). From (3.22), we have M initial virtual sensors xˆ 1,0 , xˆ 2,0 ," , xˆ M ,0 . Next, we divide M virtual ML functions to estimate M positions of mobiles again. Every function can be achieved by other M-1 positions of virtual sensors and it can be updated until their locations are converged, we call global iteration. According to the sequential of virtual sensors location updating, there are two type sequential, Jacobi method and Gauss-Seidel method. The detail of divide-and-conquer method and sequential of virtual sensors updating are discuss in Section 3.2.1. A joint Newton’s algorithm and three joint linearized algorithms perform the proposed method are discussed in 35.

(47) Section 3.2.2.. 3.2.1 Two Category of Update Sequence We will introduce Jacobi method and Gauss-Seidel method for divide-and-conquer method in Section 3.2.1a and Section 3.2.1b.. 3.2.1a Jacobi Method Now, we describe the divide-and-conquer method for Jacobi method [23]. After we have initial virtual sensors from uncooperative algorithm, we start form mobile 1. Then, the positions of virtual sensor 2 to virtual sensor M will be helped to estimate the position of mobile 1 at 1st global iteration. The divided ML estimator for the mobile 1 is given by ⎧N 1 ⎪ min ⎨∑ 2 d1 j − x1 − x j x1,1 ⎪ j =1 2σ 1 j ⎩. (. ) + ∑ 2σ ( d M. 2. l =2. 1. 2 1lˆ. 1l. − x1 − xˆ l ,0. ). 2. ⎫ ⎪ ⎬ ⎪ ⎭. (3.23) It is similar uncooperative ML estimator (2.11) because its only includes one unknown position x1 of mobile 1, while virtual sensors locations xˆ l ,0 , l = 2,3,..., M , are already known. After estimating position of mobile 1, do the same step to get virtual sensor positions xˆ 2,1 , xˆ 3,1 ,..., xˆ M ,1 from mobile 2 to mobile M. This procedure is called a global iteration, as shown in the right-hand-side of Figure 3.5. Next, this global iteration can be repeated to update positions from xˆ 1,n , xˆ 2,n ,..., xˆ M ,n to xˆ 1,n +1 , xˆ 2,n +1 ,..., xˆ M ,n +1 . The divided ML estimator for mobile i at nth is given by ⎧N 1 ⎪ min ⎨∑ 2 di j − xi − x j xi ,n+1 ⎪ j =1 2σ ij ⎩. (. ) + ∑ 2σ ( d 2. M. l =1 l ≠i. 1. 2 ilˆ. il. − xi − xˆ l ,n. ). 2. ⎫ ⎪ ⎬ , i = 1, 2,..., M ⎪ ⎭. (3.24). 36.

(48) where xˆ i ,n denotes position of virtual sensor i at the nth global iteration. This divide-and-conquer method in (3.24) stops until all positions of virtual sensors are converged.. Uncooperative Localization Algorithm. xˆ 1,0 xˆ 2,0 # xˆ M ,0 (Initial positions of virtual sensors). Virtual algorithm 1 by virtual sensors xˆ 2,n , xˆ 3,n ," xˆ M ,n Virtual algorithm 2 by virtual sensors xˆ 1,n , xˆ 3, n , xˆ 4,n "xˆ M ,n. xˆ1,n +1 xˆ 2,n +1 # xˆ M ,n +1. … Virtual algorithm M by virtual sensors xˆ 1,n , xˆ 3,n , xˆ 4,n " xˆ M −1,n. Figure 3.5 The Jacobi method diagram for divide-and-conquer method.. 3.2.1b Gauss-Seidel Method The only difference between Gauss-Seidel method [23] and Jacobi method is that in the former, the most recently update step. First, we start from mobile 1, the positions of initial virtual sensors xˆ 2,0 , xˆ 3,0 ," , xˆ M ,0 and positions of sensors x1 , x2 ," , x N estimate position of mobile 1 to get virtual sensor 1, xˆ 1,1 at 1st global iteration. Next, we estimate mobile 2 by position virtual sensors xˆ 1,1 , xˆ 3,0 ," , xˆ M ,0 . We can see the position of virtual sensor 1, xˆ 1,1 had been updated in virtual sensors locations. In Jacobi method, it uses xˆ 1,0 , xˆ 3,0 ," , xˆ M ,0 to estimate position of mobile 2. The divided ML of Gauss-Seidel method is given by. 37.