A practical approach for outdoors distributed target localization in wireless sensor networks

The deployment of a large number of scattered sensors in a certain area constitutes a very powerful tool for sensing and retrieving information from the environment (e.g., temperature, humidity, motion). The main features of wireless sensor networks (WSN) are that of a large number of low-cost nodes with limited computational and power resources. WSNs must also be scalable and robust against changes in topology (i.e., node failure or addition of new nodes), as well as energy efficient. These are the major design issues in WSNs that make the development of simple and efficient algorithms a challenging problem. These limitations also make centralized approaches not very suitable for being used in WSNs. Localization is a key task (often mandatory) in many applications [1] and therefore, distributed localization algorithms are of high practical importance.

There exist different measurement sources that can be fused in order to get an estimate of the target's position [1, 2], like time-of-arrival (TOA), time-difference of arrival (TDOA), angle of arrival (AOA) or received signal strength indicator (RSSI). In this article, we focus on single antenna nodes without tight synchronization abilities, which leads us to the use of RSSI measurements for the localization task. One of the main challenges when using RSSI measurements is that the mapping between the measurement and target's position is nonlinear and hence, finding a suitable solution becomes more challenging. Some approaches to deal with non-linearities are based on particle filtering principles [3]. In the context of WSN particle filtering approaches have also been proposed for localization and tracking using RSSI measurements [4–8]. In general, particle filtering approaches have shown very good performance when dealing with RSSI measurements but they are centralized and suffer from a high computational cost and hence, their applicability in a real scenario is questionable.

A recent approach based on convex optimization concepts has been proposed in [9, 10] for the node localization problem. In [9] a semidefinite relaxation approach is used to cast the localization problem into a semidefinite program (SDP) that can be solved efficiently via interior point methods, see [11] and references therein. The obtained position estimate through SDP is then further refined via an iterative algorithm. Although the proposed method provide near optimal results (i.e., close to the Cramer-Rao bound) they are centralized so their application to WSN may be limited. The problem of source localization using energy measurements has also been treated in [12], where a distributed algorithm based on projections onto convex sets is presented. The algorithm is shown to asymptotically approach the maximum likelihood (ML) estimate as the number of nodes increases when the target lies in the convex hull defined by the node's coordinates. In [13], an alternative approach is presented that can handle variations in the path-loss exponent. However, in both approaches no restrictions are imposed in the communication among nodes. In real applications this will cause rapid battery depletion if far away nodes are to communicate. Further, in both approaches the estimation is performed only by a subset of nodes that are selected according to their received signal to noise ratio. The main drawback is that such subset must be known to every node in the network. In a real scenario, the required signaling and routing overhead necessary for node coordination may limit their application.

In this contribution, we propose a distributed algorithm for localization in WSN's by fusing RSSI measurements. We approach the ML estimation problem by solving a simplified and more tractable problem which allows the use of convex optimization tools for its distributed solution. More precisely, we use an augmented Lagrangian approach with a primal-dual decomposition [14, 15]. The developed approach offers an advantage over centralized approaches as it is scalable, robust against changes in network's topology and requires only local communication among neighboring nodes. These are key properties very desirable in the context of WSN's.

The article is organized as follows: Section 2 introduces the localization problem and the underlying propagation model. In Section 3, we present the localization approach based on RSSI readings and its distributed implementation is presented in Section 4. Simulations are provided in Section 6, while some comments and concluding remarks are given in Section 7.

Consider a wireless sensor network, as the one depicted in Figure 1, consisting of M nodes randomly deployed on a certain area (in the same x-y plane). Nodes are static and able to communicate with adjacent nodes that lie within a given range for communications. Nodes are aware of their own location but not aware of the location of any other element in the network. Assume the presence of a target node that emits beacon frames that can be heard by all nodes in the network. The goal is to determine the location of the target node in the x-y plane.

where p _m is the received power (in dB) at reference distance d₀, α _m is the path-loss exponent, d _m is the true distance between the target and the m th node and $n_m~\mathcal{N}\left(0,{\sigma }_m^2\right)$ is a Gaussian random variable of zero mean and variance ${\sigma }_m^2$. The received power r _m will be used to get an estimate of the true target position.

In this section, we present different localization strategies using the received power (in dB) at the nodes. We first consider the (centralized) ML estimate and then we propose to use a suboptimal strategy based on local distance estimate at each node. We show that the proposed localization strategy can be implemented in a fully distributed way by only local communication among neighboring nodes.

3.1 ML estimation

Consider the presence of a centralized unit that gathers all measurements coming from the nodes. Let r = [r₁, . . . , r _M ]^T be a vector whose components are the different measurements taken by each sensor and denote x = [x _t y _t ]^T ∈ ℝ² as the target's position. The true distance between the target and the m th sensor can be then expressed as

$d_m=\left|\right|\mathbf{x}-{\mathbf{c}}_m\left|\right|,$

(2)

where c _m = [x _m y _m ]^⊤ ∈ ℝ² are the coordinates of node m with m = 1, . . . , M. The vector of measurements r can now be written as

$\mathbf{r}=\left[\begin{array}{c}\hfill p_1-10{\alpha }_1\mathsf{\text{lo}}{\mathsf{\text{g}}}_{10}\left(\frac{\left|\right|\mathbf{x}-{\mathbf{c}}_1\left|\right|}{d_0}\right)\hfill \\ \hfill ⋮\hfill \\ \hfill p_M-10{\alpha }_M\mathsf{\text{lo}}{\mathsf{\text{g}}}_{10}\left(\frac{\left|\right|\mathbf{x}-{\mathbf{c}}_M\left|\right|}{d_0}\right)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill n_1\hfill \\ \hfill ⋮\hfill \\ \hfill n_M\hfill \end{array}\right]=\mathit{\mu }\left(\mathbf{x}\right)+\mathbf{n}$

(3)

where the vector $\mathbf{n}~\mathcal{N}\left(\mathbf{0},\mathbf{\sum }\right)$ is jointly Gaussian with zero mean and covariance ∑. It is easy to see that r will follow a Gaussian distribution with mean μ(x) and covariance ∑, that is

$p\left(\mathbf{r};\mathbf{x}\right)=\frac{1}{{\left(2\pi \right)}^{M/2}\sqrt{det\mathbf{\sum }}}exp\left[-1/2{\left(\mathbf{r}-\mathit{\mu }\left(\mathbf{x}\right)\right)}^{\mathsf{\text{T}}}{\mathbf{\sum }}^{-1}\left(\mathbf{r}-\mathit{\mu }\left(\mathbf{x}\right)\right)\right]$

(4)

where p(r ; x) is the probability density function of r with parameter x. The ML estimate of the target position is then

${\widehat{\mathbf{x}}}_{\mathsf{\text{ML}}}=\underset{\mathbf{x}}{arg\; max}p\left(\mathbf{r};\mathbf{x}\right),$

(5)

which is equivalent to maximizing the log of p(r ; x). Neglecting all terms that do not depend on x it is easy to see that

${\widehat{\mathbf{x}}}_{\mathsf{\text{ML}}}=\underset{\mathbf{x}}{arg\; min}\mathit{\mu }{\left(\mathbf{x}\right)}^{\mathsf{\text{T}}}{\mathbf{\sum }}^{-1}\mathit{\mu }\left(x\right)-2{\mathbf{r}}^{\mathsf{\text{T}}}{\mathbf{\sum }}^{-1}\mathit{\mu }\left(\mathbf{x}\right).$

(6)

The objective to be minimized in (6) is not convex and therefore, finding the global optimum is not an easy task. In Figure 2 (left), we have an illustration of how the objective in (6) looks like for a network of 20 nodes over a normalized square area. It is clear that the function is not convex and that several local minima and saddle point may exist. Instead of dealing directly with the ML estimate we propose to use a suboptimal approach that offers a reasonable good performance and that allows for its distributed computation.

One of the main advantages that offers the considered approach is that it allows for a distributed implementation. We assume that nodes communicate with their one-hop neighbors as dictated by the communication graph G(V, E), where V is the set of vertices and E is the set of edges of the graph. By only local communication exchange, nodes can agree to compute some desired (global) quantity using consensus algorithms [19].

where ${\mathbf{M}}_m=\mathbf{I}-2\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill {\mathbf{c}}_m^{\mathsf{\text{T}}}\hfill & \hfill 0\hfill \end{array}\right]$ and I is the identity matrix. With the above problem definition we have the following equivalence:

Lemma 1. The two problems (14) and (18) are equivalent. Further, if we denote Z* as the optimal solution to (18), then the optimal solution x* of (14) is given by x* = [Z*(3, 1), Z*(3, 2)]^T.

Proof. See the Appendix. □

Now that we have established the equivalence between problems (14) and (18) through Lemma 1 we show how to solve it in a distributed way. For that purpose we use the optimization framework for consensus-networked systems as proposed in [15] that uses an augmented Lagrangian approach [14]. The framework in [15] generalizes the previous work of [20] as it can also handle convex but not necessarily strictly convex objective functions. The augmented Lagrangian method adds a quadratic penalty term to the objective function that is zero at the optimal solution. The resulting problem is then equivalent to the original problem as both of them end up with the same solution. Augmented Lagrangian methods are also attractive because they offer better convergence properties than standard primal-dual decomposition methods. A detailed treatment about augmented Lagrangian methods and their properties can be found in [14].

with ${\mathbf{\Psi }}_m^{\left(0\right)}=\mathbf{0}$ and m = 1, . . . , M.

The network will then operate as follows: At the beginning of the k th iteration, each node locally solves (31). Then, nodes broadcast the computed estimates ${\mathbf{Z}}_m^{\left(k+1\right)}$ to their neighbors. With the local estimates of the corresponding neighbors at hand, each node updates its multipliers as in (32). The process is repeated until all nodes converge to the same solution which, in turn would be the same as in the centralized case.

So far, we have shown how to solve (11) by formulating the relaxed problem (14). We have also provided conditions under which the solutions to (11) and (14) coincide. Further, the solution can be computed in a distributed fashion using convex optimization tools. However, the performance of the followed approach in (11) is below that of the ML estimate (6). In order to come closer to the ML solution we could perform an additional local search that improves the obtained estimate through the solution of (14). The idea is to run a distributed optimization routine, taking the solution of (14) as the starting point, to solve for (6). If the previously computed estimate by solving (14) is close to the ML estimate we may converge to it by optimizing in the neighborhood of the solution of (14), otherwise we will converge to a local optima but still improving performance.

Algorithm 1 Gauss-Newton method

1: x⁽⁰⁾← x₀k = 0 {Initialization}

2: while ! found & k < k_maxdo

3:   h_gn← - (J^(k)TJ^(k))^{- 1}J^(k)Tf_ML (x^(k)) {Descent direction}

4:   if ∄ h_gnthen

5:     found = true

6:   end if

7:   x^(k+1)← x^(k)+ h_gn {Update}

8:   k ← k + 1

9: end while

and therefore, the above quantities can be computed in a distributed fashion by means of average consensus [19]. Once we have computed the products (36) and (37), it is straightforward to compute the descent search direction h_gn.

Based on these observations we propose here a fully distributed algorithm, shown as Algorithm 2, which asymptotically approaches the same result as in the centralized case using only local information and the exchange of low-volume intermediate results within each node's 1-hop neighborhood. We immediately note that the Steps 3-6 and 11-12 can all be performed locally by each node. The only communication occurs in the Steps 8 and 9 via standard average consensus Algorithms [19]. Seeing how ${\mathbf{\Delta }}_n^{\left(k\right)}\in {\mathbf{R}}^{2\times 2}$ and is symmetric, and ${\gamma }_n^{\left(k\right)}\in {ℝ}^2$, we conclude that each consensus round requires a broadcast of only five real values.

Algorithm 2 Distributed Gauss-Newton localization

1: ${\widehat{\mathbf{x}}}^{\left(0\right)}$← same initial value ∀m ∈ M

2: for k = 0 to K - 1 do

3:   ${\mathbf{J}}_m^{\left(k\right)}\leftarrow \frac{-10{\alpha }_m}{{\sigma }_m{∥{\widehat{\mathbf{x}}}^{\left(k\right)}-{\mathbf{c}}_n∥}^2}{\left({\widehat{\mathbf{x}}}^{\left(k\right)}-{\mathbf{c}}_n\right)}^{\mathsf{\text{T}}}$

4:   $f_m^{\mathsf{\text{ML}}}\left({\widehat{\mathbf{x}}}^{\left(k\right)}\right)\leftarrow {\sigma }_m^{-1}\left(r_m-p_m+5{\alpha }_m{log}_{10}{∥{\widehat{\mathbf{x}}}^{\left(k\right)}-{\mathbf{c}}_n∥}^2+10{log}_{10}{d}_0\right)$

5:   ${\mathbf{\Delta }}_m^{\left(k\right)}\leftarrow {\mathbf{J}}_m^{\left(k\right)\mathsf{\text{T}}}{\mathbf{J}}_m^{\left(k\right)}$

6:   ${\gamma }_m^{\left(k\right)}\leftarrow {\mathbf{J}}_m^{\left(k\right)\mathsf{\text{T}}}{f}_m^{\mathsf{\text{ML}}}\left({\widehat{\mathbf{x}}}^{\left(k\right)}\right)$

7:   begin consensus

8:     ${\mathbf{\Delta }}_{*}^{\left(k\right)}\leftarrow \frac{1}{M}{\sum }_{m=1}^M{\mathbf{\Delta }}_m^{\left(k\right)}=\frac{1}{M}{\mathbf{J}}^{\left(k\right)\mathsf{\text{T}}}{\mathbf{J}}^{\left(k\right)}$

9:     ${\gamma }_{*}^{\left(k\right)}\leftarrow \frac{1}{M}{\sum }_{m=1}^M{\gamma }_m^{\left(k\right)}=\frac{1}{M}{\mathbf{J}}^{\left(k\right)\mathsf{\text{T}}}{\mathbf{f}}_{\mathsf{\text{ML}}}\left({\widehat{\mathbf{x}}}^{\left(k\right)}\right)$

10:   end consensus

11:   ${\mathbf{h}}^{\left(k\right)}\leftarrow {\mathbf{\Delta }}_{*}^{{\left(k\right)}^{-1}}{\gamma }_{*}^{\left(k\right)}={\left({\mathbf{J}}^{\left(k\right)\mathsf{\text{T}}}{\mathbf{J}}^{\left(k\right)}\right)}^{-1}{\mathbf{J}}^{\left(k\right)\mathsf{\text{T}}}{\mathbf{f}}_{\mathsf{\text{ML}}}\left({\widehat{\mathbf{x}}}^{\left(k\right)}\right)$

12:   ${\widehat{\mathbf{x}}}^{\left(k+1\right)}\leftarrow {\widehat{\mathbf{x}}}^{\left(k\right)}+{\mathbf{h}}^{\left(k\right)}$

13: end for

In this section, we provide several numerical examples in order to evaluate the performance of the proposed approach. For the simulations we consider a network of randomly deployed nodes over an area of 100 × 100 squared meters. We have used the same propagation model for all the nodes with reference power p _m = - 40 dB at reference distance d₀ = 1 m and path-loss exponent α _m = 2 for m = 1, . . . , M. We further assume that the noise processes are independent and identically distributed with $n_m~\mathcal{N}\left(0,{\sigma }_{\mathsf{\text{dB}}}^2\right)$ for all m.

In Figure 3, we have simulated the distributed localization task using a network of 50 nodes (see Figure 1) with a randomly located target. We have performed distributed estimation of the target's position by the alternating between (31) and (32) with penalty parameter c = 0.05. We have plotted in Figure 3 the error between each node's local estimate and the centralized solution of the problem. As it can be appreciated, the distributed algorithm converges to the optimal centralized solution as the number of iterations increases.

We have also simulated the localization task over the same network of Figure 1. For the propagation model the measurement noise variance has been set to ${\sigma }_{\mathsf{\text{dB}}}^2=9$. We have evaluated the performance of the proposed approach with and without weights (labeled SDP and wSDP, respectively) over 1000 random target locations (test points). We compare our approach with Multilateration localization approach [2]. As shown in [2], if one node is chosen as a reference, then the localization problem can be linearized. Therefore, it allows its distributed implementation by means of consensus since it can be cast into a least-squares problem where each node locally contributes to the global cost function. Note, however that nodes must first agree on a common reference. For the simulations, we have chosen the node with the closest distance estimate to be the reference. We also provide the performance of the ML estimate for comparison purposes. The empirical cumulative distribution function (CDF) of the localization error is represented in Figure 4. As it can be observed in Figure 4, the performance of the proposed scheme outperforms that of Multilateration. Further, we observe that the use of weights improves the localization accuracy of the algorithm so that we come closer to the ML estimate.

The performance of the algorithm has also been tested for different values of the measurement noise variance. The results are displayed in Figure 5 where the average error over 1000 locations is depicted as a function of the measurement noise standard deviation. Again, we can appreciate the performance improvement of the proposed approach compared to multilateration as can be seen in Figure 5. We have also displayed the results when combining the proposed distributed localization approach with a local search (wSDP+local). The local search is performed in a distributed fashion following the steps in Algorithm 2. As it can be observed the results of such combination provide close to ML performance. This implies that our method is capable of providing good estimates that could be used to run a local solver in order to come close to the ML estimate. Although not guaranteed to converge to the ML solution, the local search can only provide better estimates (in the ML sense).

We have presented a distributed localization approach over sensor networks using consensus and convex optimization. An alternative problem to the ML position estimation problem has been proposed based on local ML distance estimates at each node. In order to circumvent the non-convexity of the problem, semidefinite relaxation technique has been employed and conditions that guarantee zero gap between the relaxed and the original problem have been given. A distributed algorithm based on an augmented Lagrangian approach using primal-dual decompositions have been proposed and it has been shown to converge to the centralized solution. The approach is suitable for its real implementation in WSN as it is scalable, robust against changes in topology and energy efficient by the use of only local broadcast-type communication among nodes. Another interesting property of the proposed algorithm is that it allows the introduction of additional convex constraints to the localization problem in a straightforward manner.

The proposed algorithm is intended to be usable in real networks and its suitability in terms of accuracy would be determined by the application at hand. However, if higher accuracy is required, we could run an additional optimization step around the found solution. We have verified by means of simulations that the combination of our suboptimal method with a local search provides a localization error close to the ML estimate.

It is worth to mention that the proposed approach has a direct application to distributed tracking in WSN's as well. The tracking procedure would be based on the jointly estimated target's position. As all nodes share the same estimate, they could use that estimate to locally run a tracking filter in order to follow the movement of the target.

Therefore, Ψ will be equal to 0 only if (47) has a solution where x = z and Tr(X) = t. Additionally, we have that for the solution to be a feasible point it must be satisfied that Tr(X - xx^T) ≥ 0 which implies that Tr(X) ≥ ||x||² or, equivalently ||z||²≤ t. This implies that if problem (17) is infeasible, then Ψ ≠ 0 and hence, X = xx^T so that the solution of the relaxed problem (14) coincides with that of the original problem (12), which proves Proposition 1.

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By rearranging terms and the previous conditions on Z we end up with problem (18) and the equivalence is established. The equivalence between the two solutions follows directly from the definition of Z.

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