Distributed and Cooperative Localization Algorithms for WSNs in GPS-less Environments Stefano Tennina, Fabio Graziosi, Fortunato Santucci DEWS, Università degli Studi di L'Aquila, Monteluco di Roio - L'Aquila (Italy):
[email protected],
[email protected],
[email protected] Marco Di Renzo Telecommunications Technological Center of Catalonia, Av. Canal Olimpic Castelldefels, Barcelona, Spain:
[email protected].
Abstract Nowadays, one of the most representative and vibrant examples of networked embedded systems is represented by Wireless Sensor Networks (WSNs). For many WSNs applications, it is well-known that the determination of the node’s location is of prime importance. Moreover, for WSNs the most popular solution for localization, i.e., the GPS-aided (Global Positioning System) one, is unanimously considered as not realistic because WSN nodes are supposed to operate at low-complexity and low-power consumptions. The aim of this paper is to outline some of the advanced solutions to address the problem of WSN’s localization in the so-called GPSless environments.
1 Introduction WSNs are distributed networked embedded systems where each node combines sensing, computing, communication, and storage capabilities [1]. Due to their unprecedented design challenges and potentially large revenues, in recent years WSNs have witnessed a tremendous upsurge in interest and activities in both academia and industry [2]. In particular, they have become increasingly popular in military and civilian sectors, and have been proposed for a wide range of application domains, e.g., control and automation, logistics and transportation, environmental monitoring, healthcare and surveillance. In general, WSNs are required to possess self-organizing capabilities, so that little or no human intervention for network deployment and setup is required. A fundamental component of selforganization is the ability of sensor nodes to “sense” their location in space [3], [4], i.e., determining where a given node is physically located in a network. In particular, node localization is a key enabling capability to support a rich set of geographically aware protocols for distributed and self-organizing WSNs [5], and for achieving contextawareness. Moreover, cooperative localization in WSNs provides a potential for many applications in the commercial, public safety and military sectors. In commercial applications, there is a need for localizing and tracking inventory items in warehouses, materials and equipment in manufacturing floors, elderly in nursing homes, medical equipment in hospitals, and objects in
residential homes. In public safety and military applications, indoor localization systems are needed to track inmates in prisons and navigate policemen, fire fighters and soldiers to complete their missions inside buildings. It is well-known that the GPS can greatly facilitate the task of location estimation by potentially allowing every GPS-equipped receiver to accurately localize itself in any point located on or above the Earth surface [6]. However, GPS-based localization solutions are often considered a non-completely viable and well-suited solution for position estimation in WSNs since sensor nodes are supposed to operate at low-complexity and low-power consumptions [7]. Moreover, GPS-based solutions have the undesirable side-effect that they cannot provide reliable location estimates in indoor environments, and in the presence of dense vegetation [8], [9]. As a consequence of the above, much research has been done in the WSN community to develop new techniques for localization in those environments where GPS-aided positioning is either unfeasible or does not meet the design requirements and paradigms of networked embedded systems, i.e., the so-called GPS-less environments. The result of this intensive research work has been the proposal of many new solutions (alternative to GPS) to address the problem of distributed network location discovery (see, e.g, [10] and references therein). However, in [4], [11] the authors conclude that among the existing algorithms no one seems to perform better than the others, and claim that the definition of location algorithms with accurate positioning capabilities and low communication and computation costs for GPSless environments is still an ongoing area of research at both theoretical and experimental levels. Moreover, effective cooperative localization techniques in GPS-less scenarios, e.g., indoor scenarios, hinge on the ranging technology used to feed the localization algorithm [12]. As a matter of fact, most cooperative and distributed localization techniques for WSNs rely on the knowledge of the distance between pairs of nodes in the network, i.e., ranging information. The more accurate these distances are estimated, the more accurate the estimated final position of the node to be localized is. However, ranging accuracy in typical GPS-less environments is severely affected by the dense
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multipath propagation that often characterizes these environments [13]. As a result, even the most effective cooperative localization algorithms may provide completely unreliable estimates of the node’s position due to large ranging errors. Among the proposed emerging techniques to counteract this problem, Ultra Wide Band (UWB) Time of Arrival (ToA)-based ranging has recently received considerable attention in the WSN research community [14]. In addition to its high data rate communications, UWB has been selected as a viable candidate for precise ranging and localization. This is mainly due to its large system bandwidth, which allows for centimetre accuracies, low-power and lowcost implementations [15]. However, UWB-based ranging estimation in indoor environments is a very challenging task on its own, and several issues have to be still addressed to fully understand the performance of its technology in realistic propagation environments characterized by dense multipath propagation and co-located narrow-band interference. Moreover, UWB has been recently considered as a viable candidate for enabling dynamic spectrum access capabilities [16], which further improves the interest of this physical layer technology in the wireless community, and open the way to further research for the definition of ranging-capable UWB nodes with cognitive radio capabilities [17]. The main goals of this project are: 1) outline some of the distributed and cooperative positioning algorithms and 2) evaluate and compare their performances [18] – [20].
2 System Description Let us consider NA wireless nodes distributed in the region of interest, whose exact locations in the considered scenario are known. These nodes will be denoted in the text as “startup anchors”. We also assume that in the same area NU wireless nodes with unknown location have been deployed. We will refer to these nodes as “unknown nodes”. These wireless nodes have a relatively simple radio interface to communicate among them, which allows not only data exchange but also distance measurements. The main goal of localization algorithms is to use the anchor nodes to somehow estimate the position of the unknown nodes in the specified coordinate system. In particular, position estimation algorithms require a minimum of either three or four anchor nodes in a two– and tree– dimensional coordinate system, respectively [20].
2.1
Recursive Positioning Methods
We will consider a recursive positioning method for network location discovery. In particular, the well– known recursive and hierarchical method proposed in [8], [10] is analyzed for sake of illustration. The basic version of the algorithm in a 3D scenario involves the following steps. Phase 1) Unknown nodes that are
connected (i.e., they are in the neighborhood) to at least four "startup anchors" compute their position. Phase 2) Once an unknown node has estimated its position, it becomes a "converted anchor" and broadcasts its estimated position to other nearby unknown nodes, thus enabling them to estimate their positions. Phase 3) This process is repeated until the positions of all the nodes that eventually can have either four "startup anchors" or "converted anchors’ are estimated. As a consequence, depending on the current step of the algorithm, the four reference nodes with known position may be either "startup anchors" or "converted anchors", where the latter are nodes with unknown position at the beginning of the location discovery procedure, but which have localized themselves during one of the previous steps of the iterative algorithm. Of course, differently from the "startup anchors", the position of the "converted anchors" is affected by a certain error. In what follows, we will denote with "reference nodes" both "startup" and "converted" anchors.
2.2
Position Computation
The recursive positioning method described in Section 2-1 requires a technique to compute the location of an unknown node from the position of four anchor nodes, which may be in part "startup anchors" and in part "converted anchors". In general, the computation of the position of the unknown node involves two basic steps: i) measuring the distances between pairs of sensors, and ii) estimating the node’s position via the optimization of a given cost function obtained from the measured distances. With regard to distance estimation between pairs of nodes, we consider the time–of–flight technique [14]. In this method, the time–of–flight of a RF signal between the unknown node and a reference node is used to estimate the distance between them. In particular, after measuring the time–of–flight through ranging packet exchange, the distance can be obtained using the well–known speed–distance relationship. Accurate ranging estimates can be obtained, e.g., using either Spread Spectrum or Ultra Wide Band technologies [14]. In particular, the measured distances are noisy due to channel impairments, and to errors in distance estimation. Throughout the paper, the ranging error will be modeled as a Gaussian random variable with mean value given by the actual distance and standard deviation denoted by σR. With regard to position computation from range estimates, in the literature two basic algorithms are often considered: i) triangulation, which foresees to estimate the position of the unknown node by finding the intersection of four spheres in a 3D environment, and ii) multilateration, according to which the estimated position is obtained by reducing the difference between the actual measured distances and
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Figure 1 - Reference Scenario and Network Topology
the estimated Euclidean distances between unknown and reference nodes, i.e., via the minimization of an error cost function. The main difference between the two approaches is that multilateration algorithms are more robust to noisy range measurements.
3 Optimization Algorithms for Positioning In this section we briefly summarize the algorithms that have been compared in this contribution by considering the scenario sketched in figure 1. The following notation will be used: i) bold symbols are used to denote vectors and matrices, ii) (·)T denotes transpose operation, iii) ∇(·) and ∇2(·) are the gradient and Hessian operators, respectively, iv) ||·|| is the Euclidean distance and |·| the absolute value, v) ∠(·, ·) is the phase angle between two vectors, vi) (·) −1 denotes matrix inversion, vii) will denote the estimated position of the unknown node U1, viii) will be the trial solution of the optimization algorithm, ix) are the positions of the reference nodes , which are exact when the reference node is a "startup anchor", and noisy when the reference node is a "converted anchor", and x) will denote the estimated (via ranging measurements) distance between reference node and the unknown node U1.
3.1
Triangulation Method
In this method, the position of node U1 is obtained by inferring a geometric triangulation among estimated and actual distances. Accordingly, the unknown position is obtained by finding a solution that simultaneously solve the following set of equations:
The system of equations in (1) can be solved using a Least Squares solution, thus yielding , where matrix A and vector b can be found in [8]. In general, triangulation methods may fail to find a solution for the system in (1) when range and reference position estimates are noisy. Multilateration methods are, in general, preferred in this case. The triangulation method will be denoted as INV method throughout the paper.
3.1
Multilateration Method
In this method, the position of node U1 is obtained by minimizing the error cost function F(·) defined as follows:
such that . The minimization of (2) can be done using a variety of numerical optimization techniques, each one having its advantages and disadvantages in terms of accuracy, robustness, speed, complexity, and storage requirements [23]. Some optimization algorithms will be compared in what follows. Note that since optimization methods are iterative by nature, we will denote with index k the k–th iteration of the algorithm and with F(u1(k)) and the error cost function and the estimated position at the k–th iteration, respectively. 1) Steepest Descent (SD): The Steepest Descent is an iterative line search method that allows to find the (local) minimum of the cost function in (2) at step k+1 as follows [23]:
where is a step length factor, which can be chosen as described in [14, pp. 36, ch. 3], and is the search direction of the algorithm. In particular, when the optimization problem is linear, some expressions exist to compute the optimal step length in order to improve the convergence speed of the algorithm. On the other hand, when the
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optimization problem is non–linear, as considered in this contribution, a fixed and small step value is in general preferred in order to reduce the oscillatory effect when the algorithm approaches the solution. In such a case, we have = 0.5μ [10], where μ is the learning speed. 2) Enhanced Steepest Descent (ESD): The Steepest Descent method provides, in general, a good accuracy in estimating the final solution. However, it may require a large number of iterations, which may result in a too slow convergence speed for mobile ad–hoc wireless networks. In order to improve convergence speed, we propose in this contribution an enhanced version, which we call Enhanced Steepest Descent (ESD). The basic idea behind the Enhanced Steepest Descent algorithm is to adjust the step length value αk as a function of the current and previous search directions and , respectively. In particular, is adjusted as follows:
Differently from Steepest Descent and Conjugate Gradient methods, they use both gradient and Hessian information to find the solution [23]. However, they are more complex than Steepest Descent and Conjugate Gradient methods because matrix factorization and Hessian computation are required. More specifically, we consider the Gauss– Newton method, which allows to estimate the position of the unknown node in a linear search fashion by using (3), where is obtained by solving the equation where is an approximated expression of the Hessian matrix and is the Jacobian operator.
5 Numerical Results and Testbed Platform The numerical results are obtained via MATLAB simulations by considering the network scenario shown in figure 1. Algorithm
where = ∠ , , γ < 1 is a linear increment factor, δ > 1 is a multiplicative decrement factor, and and are two threshold values that control the step length update. By using the four degrees of freedom γ, δ, and , we can simultaneously control the convergence rate of the algorithm and the oscillatory phenomenon when approaching the final solution in a simple way and without appreciably increasing the complexity of the algorithm when compared to the Steepest Descent method. 3) Non–Linear Conjugate Gradient (CG): The Non– Linear Conjugate Gradient methods have been used extensively to solve non–linear optimization problems since they do not require matrix storage and need, in general, a smaller number of iterations than Steepest Descent methods. Despite several specific implementations can be found [20], [23], in general the Conjugate Gradient method allows to find the (local) minimum of the cost function in (2) at step k+1 by iteratively using the two recursive equations as follows [23]:
where at the first (i.e., k = 0) iteration, and and are two degrees of freedom of the algorithm, which can be iteratively updated using a wide variety of techniques [20]. 4) Non–Linear Least Squares Method (NLS): Non– Linear Least Squares methods are a class of algorithms that have been specifically designed for the minimization of non–linear cost functions.
CG1
CG2
SD
ESD
NLS
INV
Computatio n Time (s) 0.0253 (T1) 0.0090 (T5) 0.0060 (T9) 0.0255 (T1) 0.0090 (T5) 0.0058 (T9) 0.2206 (T1) 0.0264 (T5) 0.0115 (T9) 0.0793 (T1) 0.0096 (T5) 0.0058 (T9) 0.2615 (T1) 0.0363 (T5) 0.0202 (T9) 0.0001 (T1) 0.0001 (T5) 0.0001 (T9)
Mean Error (m) 7.47 (T1) 1.93 (T5) 1.21 (T9) 7.44 (T1) 1.93 (T5) 1.21 (T9) 6.79 (T1) 1.93 (T5) 1.23 (T9) 6.79 (T1) 1.93 (T5) 1.23 (T9) 6.72 (T1) 1.92 (T5) 1.23 (T9) 15.67 (T1) 3.50 (T5) 2.26 (T9)
Std. Error (m) 6.28 (T1) 1.17 (T5) 0.56 (T9) 6.23 (T1) 1.18 (T5) 0.56 (T9) 4.12 (T1) 1.06 (T5) 0.59 (T9) 4.12 (T1) 1.06 (T5) 0.59 (T9) 4.12 (T1) 1.03 (T5) 0.58 (T9) 9.96 (T1) 2.19 (T5) 1.36 (T9)
Table 1: Optimization algorithms comparison (σR = 0.8m). CG1 and CG2 are the the Fletcher–Reeves Polak–Ribière and Hestenes–Stiefel algorithms with secant method [20].
In table 1 we observe a comparison of some optimization algorithms in terms of computational time, mean and standard deviation positioning error. We observe that: i) the positioning error increases when moving the unknown node from T1 to T9 due to the network topology, as expected, ii) the triangulation algorithm (INV) provides the worst performance in terms of error accuracy, iii) the ESD algorithm provides the same accuracy as the SD and NLS algorithms, but reaches the final solution faster (this is an important result for, e.g., mobile networks), iv) the ESD performs as well as the CG algorithms in most scenarios, but it outperforms CG algorithms in those network topologies that are prone to ambiguities (e.g., when the unknown node is located in T1–T4 positions).
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Figure 2 shows the performance of the optimization algorithms with respect to the standard deviation of the ranging error σR. This figure confirms the results already shown in table 1 and also illustrates that: i) the mean value of the error position increases when the ranging standard deviation increases, as expected, ii) the standard deviation of the error position is almost insensitive to the value of the ranging standard deviation, and iii) the error position gets large when the unknown node moves from position T9 to position T1.
node sees the reference nodes. For every sensor node's position (Th), 40 independent acquisitions have been taken and shown in the figure. Looking at figure 4, we figure out that the positioning error increases as the network topology is more prone to ambiguities, as well as when sensor nodes are further away from reference nodes. We also observe that in every independent acquisition the ESD algorithm shows a good stability in terms of performance.
Figure 2 - Mean value and standard deviation of the positioning error vs. the position of the unknown node (T1, T5, T9).
In order to analyze the performance of the ESD algorithm when used in a real sensor node platform working in an indoor environment, we have reproduced [21], [22] in the new and recently established Networked Control Systems Laboratory of the European Embedded Control Institute (EECI) at the Center of Excellence DEWS [24], the same network scenario already considered in figure 1, and used for simulative analysis, by using CrossBow’s MICAz sensor nodes [25] and relying on their on board RSSI-based ranging capabilities. The deployed testbed is shown in figure 3. In particular, we have four anchor nodes A1, A2, A3, A4, and a set of unknown nodes Ui located on the ground floor in positions Th with h = 1, 2, . . . , 16 (i.e., T1 is the furthest node from the anchors, while T16 is the nearest one). The sensor node deployment shown in figure 3 has been chosen with the aim to investigate the effect of the network geometry on the performance of the ESD algorithm. In particular, in every Th position an unknown node sees the reference nodes with an increasing angle when moving from T1 to T16: this corresponds to moving from a scenario (T1) with a bad geometry where ambiguities may arise during position estimation, towards a scenario (T16) where the unknown node is surrounded by reference nodes, thus giving an ideally optimal network topology for position estimation regardless of the specific algorithm. In figure 4, we have reported the experimental results obtained with the testbed shown in figure 3. In particular, this figure shows the positioning error with respect to the angle under which the unknown
Figure 3 - Deployed testbed using MICAz sensor nodes for performance analysis.
Figure 4 - Positioning error of the ESD algorithm w.r.t. the angle (figure 3), and the acquisitions (run) taken with the testbed.
6 Conclusion In this paper, we have compared the performance of various optimization algorithms and analyzed the position error distribution of each of them, showing that our recently proposed ESD algorithm may outperform the others in some scenarios. In addition, experimental activities carried out on a testbed
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platform deployed in the Networked Control Systems Laboratory of the European Embedded Control Institute, at the University of L’Aquila, Italy, have well validated the simulative analysis conducted in previous research works, as well as shown that the ESD algorithm can be efficiently implemented in a commercial off–the–shelf sensor node platform with a minimum effort. Both the simulative framework and the experimental testbed have been used to analyze the effect of ranging errors and network topology on the performance of the algorithm.
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