Efficient and Accurate Localization in Multihop Networks - IEEE Xplore

Efficient and Accurate Localization in Multihop Networks Stefano Severi†, Giuseppe Abreu, Giuseppe Destino and Davide Dardari†

† Dipartimento di Elettronica Informatica e Sistemistica, Universit`a degli Studi di Bologna Via Venezia 52, Cesena, Italy 47023 Email: {stefano.severi,ddardari}@ieee.org  Centre for Wireless Communications, University of Oulu, Finland P.O.Box 4500 FIN-90014 Email: {destino,giuseppe}@ee.oulu.fi

Abstract—We present evidence that multihop node-to-anchor distance information is sufficient to allow accurate selflocalization in multihop wireless networks (such as ad hoc and sensor networks, as well as future cellular systems based on LTE). To this purpose we have implemented two new distancebased source localization algorithms, which prove highly robust to inaccurate range information characterized by distance estimates exceeding the correct ones. Our contribution is a contrasting alternative to current distributed self-localization algorithms, which are founded on the idea of “diffusing” the known location of a few nodes (anchor) to the entire the network via a typically large number of message exchanges amongst neighbors, resulting in high communications costs, low robustness to mobility, and little (location) privacy to end users. To the best of our knowledge, this work is the first example that the aforementioned disadvantages are not an unavoidable price to be payed for accurate location information in multihop networks.

I. I NTRODUCTION In typical distributed cooperative algorithms currently considered for self localization in autonomic multihop network, nodes estimate their own location by collecting information on the location of and distances to neighboring nodes [1]– [3]. While such a message-passing concept is solidly founded on sequential Bayesian inference, a characteristic problem of those algorithms is that the communication cost or complexity – measured in terms of the number of message exchanges required for convergence [4] – grows geometrically with the number N of nodes in the network. In addition to the resulting low efficiency, when the number nA of nodes with known location (anchors) is small compared to the network size (nA  N ), convergence time easily exceeds typical network coherence times causing the localization algorithm to diverge. Last but not least, an inherent disadvantage of such algorithms is that cooperation comes at the cost of non-anonymity, since nodes must disclose their location to neighbors. In this article we demonstrate – to our knowledge for the first time – that the aforementioned drawbacks are not a necessity of distributed cooperative localization algorithms in order to achieve good localization performances. The multihop localization is not a totally new idea. Savvides [5] exploits the location information over multiple hops to perform, in his Collaborative Multilateration a gradient descend optimization. This approach however has the initial requirement of a considerable number of nodes having knowledge of their aboslute position; otherwise, as shown in subsection II-B, it leads to poor localitazion performances.

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The DV-hops algorithm is a similar approach developed by Niculescu [6]. In this case first is computed the distance between any pair of nodes in terms of number of hops; then the average hop-length is derived and finally localitazion is performed via multilateration. Although is a very simply method, the performances can not be very accurate increasing the size of the network. In our work we describe a self-localization algorithm in which any node seeking to find its own location simply collects the multihop distances between itself and a sufficient number of surrounding anchors, which can be performed efficiently, e.g. via distributed tree-based discovery algorithms [4] or geographic routing [7]. While the scheme is distributed (nodes find their own location) and cooperative (neighboring nodes must relay packets towards/from anchors), fundamental differences with respect to current distributed cooperative self-localization algorithms are: a) nodes do not repeatedly update/exchange location information with neighbors (with clear benefits for the security of the network [8]), but rather forward inquires on the location of anchors to the anchors themselves, via multihop routing; b) nodes need not disclose their location to neighbors, and instead simply add each estimated hop distance to the routes length, and c) the algorithm is not iterative, such that its “convergence” time equals the delay associated with the slowest route to the anchors. In section II the model used in simulations is introduced and it is demonstrated that in such scenario classic algorithms are not able to guarantee sufficient accuracy. Two more robust algorithms, one proposed for the first time within this article and the second only recently developed, are then described in section III. Performance comparison and comments are given in section IV while conclusions follow in section V. II. P RELIMINARIES A. Network scenarios and model A typical example of a multihop network is represented in fig. 1(a). Here, the network is understood as a set of N interconnected nodes randomly deployed. Only the location of a small fraction (nA  N ) of nodes is known exactly a priori. These are referred to as anchors (marked by black dots) and A  [a1 , · · · , anA ] is the node coordinate matrix carrying the column vectors with their corresponding locations. The coordinate matrix of the remaining nodes, X  [x0 , · · · , xN −nA −1 ] is unknown. The i-th node xi can however perform ranging measurements to any neighbor xj , obtaining a noisy estimate

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Target

(a) Randomly deployed and connected network.

(b) BFS-like tree built from the target.

Anchor 1

(c) BFS-like tree built from the anchor node 1.

Anchor 3

Anchor 2

(d) BFS-like tree built from the anchor node 2. (e) BFS-like tree built from the anchor node 3. (f) Network connectivity model used during the simulations. Fig. 1. A typical realization of network connectivity (a), a typical BFS-like tree built by the target (b), typical BFS-like trees built by anchor nodes (c,d,e), and a typical multihop network model (f).

of their mutual distance dij , defined as d˜ij (measurement)  dij  xi − xj  = xi − xj , xi − xj , where  ·  denotes the Euclidean norm and · denotes inner product. Without loss of generality we will focus on the localization of the node x0 , which will be referred to as the target and is depicted by a white square in fig. 1. In the distributed scenario, the target node want to estimate its own location computing the distances between itself and a sufficiently large subset of anchors (nA ≥ 3). Since the target is not directly connected to the anchors, it has to determine those distances through the lengths of the multihop paths to the anchors. For convenience (and no consequence in the analysis), it will be assumed that each target-to-anchor path has nHi hops. The estimated multihop distance a target at x0 to an anchor nHifrom d˜k−1 k , where (x1 , . . . , xnHi −1 ) at ai is therefore p˜0i  k=1 are the nodes involved in the path and d˜nHi −1 nHi is the measured distance between anchor ai and the last node of the path xnHi −1 . Since the connectivity of a wireless network can not be determined a priori, due to traffic load, packet collisions and similar challenges, it will be assumed that such paths are the result of a distributed version of the well known Breadth First Search (BFS) algorithm [4], taking into account packet losses and collisions, which is referred to as BSF-like algorithm. Consequently, as can be observed from the examples in fig. 1(b)-1(e), the branches of the resulting trees are not determined

by the shortest paths between the root and the leaves, but rather by less “straight” paths generally growing from the root towards the leaves. Fig. 1(b) illustrates paths obtained in connection to a distributed self-localization scheme, where the target builds its own BFS-like tree in order to find paths to all anchors. In contrast, figs. 1(c)-1(e) illustrate the paths obtained in connection with a centralized systemic localization scheme, where anchors (assumed to be connected to a central station) build their own independent BFS-like trees. Obviously in both cases multihop target-to-anchor distances will be estimated under similar circumstances, namely, through irregular routes which result typically in p˜0i  d0i . With basis on the latter, we will study the performance of localization algorithms with target-to-anchor distances constructed using a model as illustrated in fig. 1(f), in which the target is located at the center of a circle of unitary1 diameter (d = 1) where the anchors lye (randomly placed). The route from the target to the anchor at ai is built by randomly selecting nHi −1 nodes uniformly distributed inside a cone centered at ai . B. A Typical Localization Algorithm in the Multihop Scenario In this section we briefly review a classical localization algorithm, namely the SMACOF technique [9], known for its 1 This is without loss of generality and amounts to a normalization of all distance quantities.

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topped with a ˆ. Then, we have

SMACOF: Mean Location Error (nA = 4)

2.5

  ˆ (m−1) ) = c + tr Q ˆ (m)T UQ ˆ (m) ˆ (m) , Q T (Q   ˆ (m)T B(m) Q(m−1) , −2 · tr Q

2.25 2 1.75

where c is a constant that can be neglected, the entries of U are given by  nA , i = j, (3) uji = −1 , i = j,

ε¯

1.5 1.25 1 0.75 0.5 0.25 0

0

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6

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Fig. 2. Linearized mean location error of SMACOF algorithm for networks with different nHi . Variance, max and min error are shown for every network size. SMACOF: CDF of Mean Location Error Performances for different nHi

1 0.9

0.7

Pr{ε < ε¯}

while diagonal and above-diagonal elements of the symmetric matrix B(m) are given by ⎧ n A +1 p˜0k ⎪ ⎪ ⎪ , i = 1, j = 1, ⎪ (n−1) ⎪ ⎪ k=2 p ˆ0k ⎪ ⎪ ⎪ ⎪ ⎪ n A +1 p ⎪ ⎪ j−1 k ⎪ , i = j, j = 1, ⎪ ⎪ ⎨ k=1 pˆ(n−1) j−1 k (m) i=j (4) bji = ⎪ ⎪ p ˜ ⎪ 0 i−1 ⎪ − , i ≥ j, j = 1, ⎪ ⎪ ⎪ pˆ(n−1) ⎪ 0 i−1 ⎪ ⎪ ⎪ ⎪ ⎪ pj−1 i−1 ⎪ ⎪ ⎪ ⎩ − pˆ(n−1) , i ≥ j, j = 1. j−1 i−1 ˆ (m) of the maAt the m-th iteration the global minimum Q min ˆ (m) , Q ˆ (m−1) ) is computed via the Guttman jored function T (Q transform [11],

0.8

0.6

ˆ (m) = U† B(m) Q ˆ (m−1) , Q min min

0.5 0.4 0.3 0.2

n Hi = 1 nHi = 4 nHi = 7 nHi = 10

0.1 0

(2)

0

0.25

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0.75

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1.25

ε¯

1.5

1.75

2

2.25

2.5

Fig. 3. Cumulative density functions of the mean location error obtained with SMACOF for network with different number of hops and with nA = 4. From the left to right: nHi = 1, 4, 7, 10.

accuracy and efficiency, and study its performance under the conditions described in subsection II-A. This algorithm attempts to find the minimum of a nonconvex function by tracking the global minimum of the soˆ (m−1) ) which ˆ (m) , Q called majorized convex functions T (Q is successively constructed from the original objective and the previous solution obtained in the iterative procedure [10], and ˆ (m) is the estimate of Q = [x0 ; A] after the m-th where Q iteration. Let pij = ai − aj , (1) and denote all estimated quantities by the corresponding letter

(5)

where † denotes the pseudoinverse. Fig. 2 illustrates the (linearized) mean location error ε¯, its variance, its maximum and its minimum value, for a network with nA = 4, varying the number of hops nHi . As can be seen, ε¯ is directly proportional to nHi . To increase the number of hops means also to increase the variance between the real target-to-anchor distances and the multihop paths. As consequence, there is an higher target-to-anchor distances estimate error. For a network characterized by nHi = 2, ε¯ ≈ 0.5, i.e., the mean location error is half of the size of the considered scenario. The linear dependence of the mean location error from the number of hops can be seen also from fig. 3, which illustrates the Cumulative Density Function (CDF) of ε¯ for networks with nA = 4 but different nHi . The four CDFs, obtained respectively for 1, 4, 7 and 10 hops, are almost equispaced on the horizontal axis, meaning that to equal increase of nHi corresponds equal increase of ε¯. It follows that a classic localization algorithm like SMACOF is not able to work properly with considerable target-to-anchor distances estimate error, which is typical of the multihop scenario. The result is a totally inaccurate estimate of target location. These considerations motivate our decision to use, in the context of the multihop localization, algorithms more resilient and robust to the target-to-anchor distances estimate error.

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III. L OCALIZATION IN M ULTIHOP N ETWORKS Following the conclusion of the previous section, here are introduced and described two algorithms able to guarantee accurate localization performances even in presence of a not negligible target-to-anchor distances estimate error. The first, Constrained SDP (CSDP), is based on the well known theory of the Semidefinite Programming. The second, Distance Contration (DC), is developed on the base of the Least Square localization algorithm [12]. A. Constrained SDP The target-to-anchor squared-norms (or squared-distances) can be written as

   I2    ai 2 ai − x0  = aTi −1 = d20i , (6) I x 2 0 xT0 −1 where I2 is a 2-by-2 Identity matrix. Define the matrix

 I2 x0 Z . xT0 y

(7)

Then, the multihop localization problem can be stated as follows ˆ 0 ∈ R2×1 , y ∈ R find x  ai = d20i , such that (8) aTi −1 Z −1 T ˆ 0, ˆ0x y=x The SDP variation of the network localization problem formulated above is obtained [13] by relaxing the equality ˆ 0 into the positive semidefinite condition ˆ T0 x constraint y = x Z  0. Under such relaxation and with elementary algebra ˆ 0 and y we arrive at introduced to eliminate the variables x maximize 0 subject to Z1:2,1:2 = I2 tr(CA Z) = d20i ,

The DC algorithm is described in detail in a companion article [12], and is designed to minimize the effect of biased measurements in source localization problems under NonLine-Of-Sight (NLOS) conditions. The inherent properties of the algorithm are, however, suitable to the multihop localization problem considered here. In particular, the DC algorithm is a reformulation of the distance-based Least-Square (LS) localization problem in which distances are contracted (subtracted by sufficiently large) by coefficients structured in relation to the null-space of the Grammanian matrix (or equivalently the relative angular matrix) describing the geometry in the problem (see [12]). The effect of the contraction is that the solution is found within a feasibility region I where the LS cost function is ensured to be convex. The limitation of the algorithm, however, is that it can only be applied to targets within the convex-hull defined by the anchors. Here we will utilize a simplified version of the DC algorithm, which can be described as follows. First the feasibility region I is constructed from the intersection of the following inequalities: (12) (ˆ p0i − p˜0i ) ≤ 0, ∀i, then it searches, for every anchor at ai , its nearest point such that a¯i ∈ I. The contracted distances are computed as follows: p¯0i = ¯ ai − x ˆ0 .

(13)

Finally the cost function to minimize becomes: ˆ 0) = f (A, x

nA 

(¯ p0i − pˆ0i )2 .

(14)

i=1

(9)

Z  0, where Z1:2,1:2 is the 2 × 2 first-minor of Z and the constants CA is defined as   ai (10) CA  aTi −1T . −1 As remarked in the previous section, the knowledge of d0i is not available. We have now to transform the equality of equation (9) in a set of two inequalities, using 2 constrains. As lower bound we put simply 0, since we know that two nodes can not occupy the same physical location, while as upper bound we can use the estimated target-to-anchor distances p˜0i . Finally the problem becomes: maximize 0 subject to Z1:2,1:2 = I2 .................. 0 < tr(CA Z) ≤ p˜0i , .................. Z  0.

B. Distance Contraction

(11)

IV. P ERFORMANCES C OMPARISON AND D ISCUSSION The same set of data used to run simulations for SMACOF in section II-B, is now used to evaluate the performances of both CSDP and DC algorithms. Fig. 4 illustrates the mean location error ε¯, for networks with nA = 4 anchors, varying the number of hops nHi . As it has been remarked before, the error on target-to-anchor distances estimate is proportional to nHi . It is evident that DC can perform accurate localizations regardless the variation of the hops, i.e., this algorithm is robust to the error that makes SMACOF useless in this scenario. The degradation of the performance proportional to nHi affects also the CSDP algorithm, even if not like SMACOF. Nevertheless CSDP proves to be efficient especially for a small number of hops (nHi ≤ 3), which is a typical case for wireless multihop networks. Under such conditions it even outperforms DC. Mean location error of both algorithms for network with nHi = 4, varying nA , are presented in fig. 5. It can be noted that the minimum number of anchor nodes (nA = 3) is sufficient to guarantee accurate performances. There is no substantial gain increasing nA : from 3 to 8 anchor nodes the

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Mean Location Error (nA = 4)

0.398

Distance Contraction Constrained SDP 0.348

0.298

ε¯

0.248

0.198

0.148

0.098

0.048

1

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nHi

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9

10

Fig. 4. Mean location error as function of the number of hops nHi for networks with nA = 4 anchor nodes. Mean Location Error (nHi = 4)

0.175

Distance Contraction Constrained SDP 0.156

mean location error is reduced of 10% for DC and only of 5% for CSDP. Finally fig. 6 illustrates the CDF of the mean location error obtained for a network with nA = 4 and varying the number of hops: nHi = 1 in the first case (white markers) and nHi = 9 in the second (black markers). The effect of the number of hops on performances already highlighted in fig. 4, is here even more evident. While for the CSDP the differences between the two CDF curves are substantial, in the case of DC they are almost perfectly matching. After the simulation session, DC has proved to be a really robust and efficient algorithm, but with two main drawbacks. It can localize only targets inside the convex-hull determined by anchor nodes, and furthermore it can localize only one target per time. In particular the latter drawback can be annoying for the centralized systemic localization scheme (figs. 1(c)-1(e)), where the central unit has to run the algorithm as many times as the number of targets to localize. On the contrary, CSDP can localize more than one target per time and even outside the convex hull. This complementary suggests a possible combined approach for the two algorithms. CSDP can be used to obtain a first location estimate. If the target results to be inside the convex-hull, and if the medium number of hops is greater than 3, then DC can be run to refine the estimate.

0.137

ε¯

V. C ONCLUSION

0.118

0.099

0.08

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nA

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Fig. 5. Mean location error as function of the number of anchor nodes nA for nHi = 4 hops networks. CDF of Mean Location Error (nA = 4) Distance Contraction versus Constrained SDP

1 0.9 0.8

Pr{ε < ε¯}

0.7 0.6

We properly employed two localization algorithms to obtain accurate performances in multihop networks only operating with the multihop target-to-anchor distance estimates. The result permits an huge reduction of the complexity respect to the classic algorithms, in which every node needs to collect location and distance estimates from all its neighbors. This is possible because the firsts are robust and resilient to the target-to-anchor distances estimate error, that is proportional to the number of hops, while the latter are instead forced to perform the message-passing phase. The gain in the algorithmic complexity is straightforwardly transformed in a reduction of the network traffic, with evident benefits regarding the energy consumption and the packet collisions. Another advantage of this new approach to multihop localization is the improvement of the network security. It is sufficient that only anchors send their location all over the network. Targets can therefore compute their position estimate without sharing it with their neighbors like in the messagepassing approach.

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R EFERENCES 0.4 0.3 0.2 0.1 0

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Constrained SDP, nHi Constrained SDP, nHi Distance Contraction, Distance Contraction,

=1 =9 nHi = 1 nHi = 9

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Fig. 6. Cumulative density function of the mean location error for networks with different number of hops.

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[1] F. Gustafsson and F. Gunnarsson, “Mobile Positioning using Wireless Networks: Possibilities and Fundamental Limitations based on Available Wireless Network Measurements,” Signal Processing Magazine, IEEE, vol. 22, no. 4, pp. 41–53, July 2005. [2] U. Ferner, H. Wymeersch, and M. Win, “Cooperative anchor-less localization for Large Dynamic Networks,” vol. 2, Sept. 2008, pp. 181–185. [3] S. Y. Wong, J. G. Lim, S. Rao, and W. Seah, “Multihop Localization with Density and Path Length Awareness in Non-Uniform Wireless Sensor Networks,” vol. 4, May-1 June 2005, pp. 2551–2555 Vol. 4.

[4] B. Awerbuch and R. Gallager, “A New Distributed Algorithm to find Breadth First Search Trees,” Information Theory, IEEE Transactions on, vol. 33, no. 3, pp. 315–322, May 1987. [5] A. Savvides, C.-C. Han, and M. B. Strivastava, “Dynamic fine-grained localization in ad-hoc networks of sensors,” in MobiCom ’01: Proceedings of the 7th annual international conference on Mobile computing and networking. New York, NY, USA: ACM Press, 2001, pp. 166– 179. [6] D. Niculescu and B. Nath, “Dv based positioning in ad hoc networks,” Telecommunication Systems, vol. 22, no. 1-4, pp. 267–280, 2003. [7] S. Lee, B. Bhattacharjee, and S. Banerjee, “Efficient Geographic Routing in Multihop Wireless Networks,” in MobiHoc ’05: Proceedings of the 6th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM, 2005, pp. 230–241. [8] A. Perrig, J. Stankovic, and D. Wagner, “Security in Wireless Sensor Networks,” Commun. ACM, vol. 47, no. 6, pp. 53–57, 2004. [9] T. F. Cox and M. A. A. Cox, Multidimensional Scaling, 2nd ed. Chapman & Hall/CRC, 2000. [10] J. de Leeuw and P. Mair, “Multidimensional Scaling Using Majorization: SMACOF in R,” 2008. [Online]. Available: http://repositories.cdlib.org/ uclastat/papers/2008010903 [11] L. Guttman, “A general nonmetric technique for finding the smallest coordinate space for a configuration of points,” Psychometrika, vol. 33, no. 4, pp. 469–506, 1968. [Online]. Available: http: //dx.doi.org/10.1007/BF02290164 [12] G. Destino and G. Abreu, “Reformulating the Least-Square Source Localization Problem with Contracted Distances,” Asilomar Conference on Signals, Systems, and Computers, vol. (submitted), 2009. [13] A. M. So and Y. Ye, “Theory of Semidefinite Programming for Sensor Network Localization,” in Proc. IEEE Sixteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA), Vancouver, British Columbia Canada, January 2005, pp. 405–414.

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