Distributed Source Localization under Anchor ...

Chinese Journal of Electronics Vol.23, No.1, Jan. 2014

Distributed Source Localization under Anchor Position Uncertainty∗ LIU Hongqing1 and WU Yuntao2 (1.Department of Electronic and Computer Engineering, National University of Singapore, 12A Kent Ridge Road, Singapore 119223) (2.Key Laboratory of Intelligent Robot in Hubei Province, Wuhan Institute of Technology, Wuhan 430073, China) Abstract — In this work, a distributed source positioning approach is developed based on Alternating direction method of multipliers (ADMM). First, a centralized positioning method is developed under case of the anchor uncertainty. And then, the method is realized in a distributed way using ADMM. Simulation results show that the centralized one is robust to the anchor errors and distributed one has similar performance as the centralized one. Key words — Target positioning, Anchor uncertainty, Distributed Computation.

composition and augmented Lagrangian methods to achieve this simple but powerful formulation, which will be detailed introduced later. The rest of paper is organized as follows. In Section II, the problem formulation is first introduced and then a centralized positioning approach is created. In Section III, the Alternating direction method of multipliers (ADMM) is presented and used in implementing our robust positioning algorithm in a distributed way. In Section IV, simulations are conducted to demonstrate the positioning performance. The conclusions are drawn in Section V.

I. Introduction The source localization problem in wireless sensor network is a important topic in many applications, like military and civil services. In matter of how to collect/computate the data, the positioning approaches are generally categorized into two groups: centralized and distributed algorithms. As the name suggests, in a centralized way, everything including the data collection, computation etc. is done in a central unit[1] . For distributed computation[2] , there are usually two subgroups: the first one is that all computations and data collection are done locally and global estimate is obtained through global averaging algorithm. The second one is each subsystem calculates the local estimates and forward them to a central unit where global estimate is computed. In this work, we focus on the second kind of the distributed computation. In the most work of positioning, one assumes that the perfect knowledge of anchor positions is available. Obviously, the performance degrades as anchor positions are erroneous. To tackle the errors in the anchor position, in this work, we develop a robust positioning algorithm that takes the errors into consideration of the cost function formulation. Therefore, we have the robust centralized positioning algorithm. In order to implement it in a distributed way, we adopt the Alternating direction method of multipliers (ADMM)[3] , which is a well suitable tool to solve distributed convex optimization suppose the original problem is separable. It combines ideas of de-

II. Centralized Positioning 1. Time-of-arrival based localization In this work, we consider simple situation where we localize single source through the Time of arrival (TOA) measurements. In a mathematical form, the measured distance is determined di = x − xi 2 + wi ,

i = 1, 2, · · · , N

T

(1) T

where xi = (xi , y i ) denotes the anchor position, (x, y) represents the source position,  · 2 stands for the 2 -norm and wi is the measurement noise with known variance σi2 . In this work, we only consider Line-of-sight (LOS) propagation. Given the measurements {di }, the goal is to estimate the source position (x, y)T . For the model in Eq.(1), the Maximum likelihood (ML) estimation can be formulated as minimize x

N  1 (di − x − xi 2 )2 2 σ i=1 i

(2)

To represent Eq.(2) in a optimization friendly form, we reexpress it as follows minimize x,{wi }

N  wi2 σi2 i=1

subject to di − wi = x − xi 2 , i = 1, 2, · · · , N

(3)

∗ Manuscript Received Dec. 2011; Accepted Mar. 2012. This work was jointly supported by a research grant from Key Laboratory of Intelligent Robot in Hubei Province, China (Project HBIR 201001) and a grant from National Natural Science Foundation of China (No.61172150).

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However, Eq.(3) is not convex because equality constraints are non-affine. Squaring both side of the constraints in Eq.(3), we have

d2i d2i

d2i − 2wi di + wi2 = xT x + xTi xi − 2xTi x d2i − xTi xi = xT x − 2xTi x + 2w i di − w 2i d2i − xTi xi = xT x − 2xTi x + 2w i di − pi ,

i = 1, 2, · · · , N (4)

where T denotes the transpose and vi = wi2 . Then, Eq.(3) can be rewritten as

x,Y ,p,w

subject to

N  vi 2 σ i=1 i

d2i

−

xTi xi

i = 1, 2, · · · , N

Y = xT x

(5)

where w = [w 1 , w 2 , · · · , w N ]T , v = [v 1 , v 2 , · · · , v N ]T . The above optimization is still not convex since the equality constraints are not affine. To obtain a convex problem, we perform relaxed technique to Eq.(5) to yield the following problem N  vi 2 σ i=1 i

subject to d2i − xTi xi = Y − 2xT xi + 2di w i − v i , vi ≥ wi2 ,

i = 1, 2, · · · , N

T

Y ≥x x

(6)

In the development of Eq.(6), we assume that we have perfect knowledge of the anchor positions, namely, xi is known to us. However, in practice, the anchor positions are usually estimated or provided by Global position system (GPS), which are subject to estimation errors. Therefore, a robust source positioning will be more useful under case of anchor uncertainties. In next subsection, we assume that anchor positions are corrupted by known Gaussian noise and develop a robust positioning approach to tackle this situation. 2. Approach I under anchor position uncertainty We assume that known anchor positions are provided ˜ i = xi + ui , by true ones plus Gaussian noise, namely, x i = 1, 2, · · · N , where ui denotes the uncertainty level. In case of anchor positions error, the signal model becomes[4] ˜ i 2 + wi , di = x − x di = x − (xi + ui )2 + w i ,

i = 1, 2, · · · , N

(7)

Having taken the uncertainty into account, we have a robust positioning optimization problem as

{wi },{u i },x

+ 2xTi ui + uTi ui d2i

N N  wi2  T −1 + ui Ψ i ui 2 σi i=1 i=1

˜ i 2 , subject to di − wi = x − x

i = 1, 2, · · · , N

−

xTi xi

=xT x − 2xT xi + 2xT ui + 2di w i − w 2i + 2xTi ui + uTi ui ,

i = 1, 2 · · · , N

(8)

(9)

T

Then, we define Y = X X, where X = [u1 , u2 , · · · , uN , x]T . According to trace property, we have uTi Ψ −1 i ui =

N 

Tr(Ψ −1 i Ξ i ),

i = 1, 2 · · · , N

(10)

i=1

where Tr(·) denotes the trace operation and Ξ i = ui uTi . Therefore, Eq.(8) can be rewritten as  N   vi −1 + Tr(Ψ Ξ ) minimize i i X ,Y ,v,w σi2 i=1 subject to d2i − xTi xi = y N+1,N+1 − 2xT xi + 2yi,N+1 + 2di wi − vi + 2xTi ui + y i,i , i = 1, 2 · · · , N vi = wi2 ,

i = 1, 2 · · · , N

Ξ i = ui uTi ,

i = 1, 2 · · · , N + 1

Tr(Ξ i ) = yi,i ,

i = 1, 2 · · · , N + 1

Y = XT X

i = 1, 2 · · · , N

minimize

− 2di wi + wi2 =xT x − 2xT xi + 2xT ui + xTi xi

i=1

T

= Y − 2x xi + 2di w i − v i ,

vi = wi2 ,

x,Y ,v,w

− 2di wi + wi2 =xT x − 2xT (xi + ui ) + (xi + ui )T (xi + ui )

N 

i = 1, 2 · · · , N

minimize

where Φ i is the noise variance matrix of the ui . Following the same procedures, the constraints can be linearized by squaring both sides (di − wi )2 =x − (xi + ui )22

(di − wi )2 = x − xi 22

minimize

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(11)

We apply the same relaxation technique on Eq.(11) to obtain the following convex positioning optimization problem  N   vi + Tr(Ψ −1 minimize i Ξ i) 2 X ,Y ,v,w σi i=1 subject to d2i − xTi xi = y N+1,N+1 − 2xT xi + 2yi,N+1 + 2di wi − vi + 2xTi ui + y i,i , i = 1, 2 · · · , N vi ≥

wi2 ,

i = 1, 2 · · · , N

i = 1, 2 · · · , N + 1 Tr(Ξ i ) = yi,i ,   Ξ i ui  03 , i = 1, 2 · · · , N + 1 uTi 1   Y XT  0N+3 X I2

(12)

where 0n is a zero matrix with size n × n. 3. Approach II under anchor position uncertainty In the development of Eq.(12), we assume that the anchor uncertainty Ψ i is known to us. However, in reality, that information is not easy to obtain. Therefore, we will propose another robust positioning approach without needing that information. We assume that the anchor errors fall in a circle ˜ i 2 < i . With this informawith radius of i , namely, xi − x tion, we have the following positioning approach minimize

x,x 1 ,···,x N ,v,w

N  vi 2 σ i=1 i

Distributed Source Localization under Anchor Position Uncertainty 

subject to x − xi 2 ≤ di − w i , i = 1, 2, · · · , N ˜ i 2 < i , xi − x

i = 1, 2, · · · , N

vi ≥ wi2 ,

i = 1, 2 · · · , N

 (13)

4. Relationship between Eqs.(12) and (13) Assuming that we know the anchor error variance in Eq.(13), we set the i equal to σi . By squaring both side ˜ i  < i , we have uTi ui ≤ σi2 , which is the same of xi − x T 2 2 with ui Ψ −1 i ui provided that Ψ i = diag{σ1 , · · · , σN }. This establishes the equivalence between Eqs.(12) and (13).

III. Distributed Positioning In the last section, we consider the scenario where all data have been collected in a central unit, in which we can perform all the computation in a centralized way. However, for the setup of a sensor network where all the sensor are scattered on the filed, a distributed computation is a more suitable choice. Plus considering nowadays sensors have capacity to perform certain level of computations, distributed process would not be much a challenge for them. In this work, we consider a distributed computation based on global consensus form, which can be generally expressed as follows[3] minimize

N 

Yi Xi

ui 1

 03 ,  X Ti  04 I2

(16)

where X i = [ui , x]T , Y i = X Ti X i and ylast,last denotes the last component of the Y i . In the next section, the simulations are provided to evaluate the performance of the proposed approach. For the second distributed computation based on its corresponding centralized version of Eq.(13), following the same produces developed in Eq.(15), we have the similar distributed algorithm like in Eq.(16). The difference is the fi (xi ), which can be expressed as follows vi minimize x,x i ,v i ,w i σ 2 i subject to x − xi 2 ≤ di − w i , ˜ i 2 < i , x i − x vi ≥ wi2 ,

(17)

Note that all the formulations are Semidefinite programming (SDP), which can be solved efficiently. In this work, we use CVX[5] in Matlab as a solver.

IV. Simulation Results fi (xi )

i=1

subject to xi − z = 0,

i = 1, 2, · · · , N

(14)

where xi is the local variable, z is the global variable and writing fi means that the original problem is separable. The distributed computation based on Alternating direction method of multipliers (ADMM) can be expressed as xk+1 :=argmin{fi (xi ) + qikT (xi − z k ) + (ρ/2)||xi − z k ||22 } i z k+1 :=

Ξi uTi

95



N 1  k+1 (x + (1/ρ)qik ) N i=1 i

− z k+1 ) qik+1 :=qik + ρ(xk+1 i

(15)

As we can see that x-update and q-update are local computations, so they can run in parallel. And also, z-update sometimes is called fusion center, which collects all the xi and qi from ith subsystem. In this particular work, each subsystem would be the anchors. First, each anchor calculates the target position by itself using the local measurement based on x-update and q-update in Eq.(15). And then, the global estimation is carried out by z-update in Eq.(15). And note that each subfunction fi (xi ) in Eq.(15) is the constrained optimization in Eq.(12) for each sensor, namely, each sensor solves the following optimization problem   vi −1 + Tr(Ψ Ξ ) minimize i i X i ,Y i ,v i ,w i σi2

Computer simulations have been conducted to evaluate the performance of the proposed source localization approach. The ten BS are randomly deployed in a field of square 10m × 10m and a single target is placed in the same area randomly. All the results are based on averages of 500 independent runs, and Mean square error (MSE) for position is employed as the performance measure. In this work, we assume all anchors have the same error covariance matrices, namely Φi = ε2 I 2 with ε2 = 1. 1. Centralized approaches In the first test, we evaluate the performance of the centralized algorithm for localization problem under anchor uncertainty. Fig.1 shows the MSEs of the position estimates versus measurement noise level σ 2 . The i is set to be 0.1 in Eq.(13). On the one hand, if we do not take the uncertainty into account, the portioning performance degrades significantly. Clearly, this is not the scheme recommended in this case. On the other hand, we can see two proposed methods

subject to d2i − xTi xi = ylast,last − 2xT xi + 2yi,last + 2di wi − vi + 2xTi ui + y i,i , vi ≥ wi2 , Tr(Ξ i ) = yi,i ,

Fig. 1. MSEs versus measurement noise level

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have similar performance and approach the CRLB[6] . Even though, the approach based on Eq.(13) has small performance loss (1dB tops) compared to Eq.(12), it does not require the anchor error information. Therefore, in this case, scheme of Eq.(13) is recommended here. 2. Distributed approaches In the second test, we evaluate the performance of the distributed algorithm for the same problem. In Fig.2, the MSEs of the position estimates are provided. As we can see that, the distributed computation is only 2dB away from CRLB at the worst case. That demonstrates that distributed computation is a promising approach here to provide satisfied estimation and scalable computations.

Fig. 2. MSEs versus measurement noise level

V. Conclusions In this work, we develop robust positioning approaches when anchors have uncertainties. For a distributed sensor network, the corresponding distributed computations are implemented as well. Simulation results demonstrate that the performance of the proposed methods approach the corresponding CRLB. The authors would like to thank Dr. Kenneth Wing Kin Lui for providing a relevant centralized positioning code. References [1] P. Biswas, T.C. Liang, T.C. Wang and Y. Ye, “Semidefinite programming based algorithms for sensor network localization”, ACM Trans. Sensor Networks, Vol.2, No.2, pp.188–220, 2006.

2014

[2] N. Patwari, J.N. Ash, S. Kyperountas, A.O. III Hero, R.L. Moses, N.S. Correal, “Locating the nodes: Cooperative localization in wireless sensor networks”, IEEE Signal Processing Magazine, Vol.22, No.4, pp.54–69,2005. [3] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers”, To appear in Foundations and Trends in Machine Learning, Michael Jordan, Editor in Chief, Vol.3, No.1, pp.1–124, 2011. [4] K.W.K. Lui, W.K. Ma, H.C. So and F.K.W. Chan, “Semidefinite programming algorithms for sensor network node localization with uncertainties in anchor positions and/or propagation speed”, IEEE Transactions on Signal Processing, Vol.57, No.2, pp.752–763, 2009. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press 2004. [6] Y. Shang, H. Shi and A.A. Ahmed, “Performance study of localization methods for ad-hoc sensor networks”, Proc. IEEE International Conference on Mobile Ad-hoc and Sensor Systems, pp.184–193, 2004. Fort Lauderdale, Florida, USA LIU Hongqing was born in Heilongjiang Province, and received his B.S. and M.S. degrees, from Xidian University, Xi’an Shaanxi, China, in 2003 and 2006, respectively, and Ph.D. degree from City University of Hong Kong, Hong Kong, China, in 2009, all in electronic engineering. In 2009, after graduation, he joined Acoustic Research Laboratory (ARL), National University of Singapore (NUS), as a research fellow. His research interests lie in the areas of statistical signal processing and convex optimization, including compressed sensing, localization/tracking, parameter estimation and underwater imaging. WU Yuntao was born in Hubei Province, and received the Ph.D. degree in information and communication engineering from the National Key Lab. for Radar Signal Processing at Xidian University in December 2003. From April 2004 to September 2006, He was a post-doctoral researcher at Institute of Acoustics, Chinese Academy of Sciences, Beijing, China. From Oct. 2006 to Feb. 2008, He worked as a senior research fellow at the City University of Hong Kong. Currently, He is a full-time professor and the vice-dean in the School of Computer Science and Engineering at Wuhan Institute of Technology, and the deputy director of the Key Lab. of Intelligent Robot in Hubei Province. He has published over 50 journal and conference papers. His research interests include signal detection, parameter estimation in array signal processing and source localization for wireless sensor networks, biomedicine signal analysis, etc.