Centralized techniques

1) Anchor-based methods

The classical techniques involve the resolution of a single unknown position of a sensor node at a time by means of RSSI values coming from a fixed number of surrounding anchor nodes (or landmarks) denoted by M (find a comparative made by [99] or [71]). We denote the un-

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Figure 4.8. Classification of the existing classical methods on localization.

known node position by z = (x, y) (two dimensions p = 2) and any anchor node position by Ai = (ai, bi). Since the sensor node only uses the information from known positions, z can be

expressed in absolute coordinates, i.e. anchor positions in GPS-coordinates. In anchor-based methods, the solution (the unknown sensor node position) is issued to M T measurements where T denotes the number of RSSI values collected from each of the M anchor nodes. Note that, when dealing with more than one unknown position, e.g. a network of N sensor nodes, the problem may be performed sequentially, i.e. solving one position at each time. We distinguish between two ways to address the problem:

Geometrical point of view

The following methods were first studied for aerospace or aeronautical systems and con- sider the unknown position as being the intersection point of a fixed number of curves M , one for each landmark. Although they are originally based on time measurements4

(see [137]), the related system of equations can be finally described by the set of land- marks positions and the distances between the landmarks and the unknown position. If the LNSM is considered for the RSSI signal, distances can be estimated as described in Section 4.2.1. If distances are perfectly known, the system of equations is described from different geometrical curves such: circles (called also trilateration [110]), hyperbo- las [137], quadrilaterals (also called min-max [141]) or triangles (known as multilateration [79]). Figure4.9illustrates instances of these approaches.

Note that, unless for the quadrilaterals’ case, a landmark is designed as the reference of

(a) Trilateration (M = 3).

(b) Intersection of three hyperbolas (M = 4). (c) Intersection of three quadrilaterals (M = 3).

(d) Multilateration: cosine rule applied on 5 triangles which forms a linear system of 5 equations following the expres- sion on the right.

Figure 4.9. Classical methods: geometrical point of view.

the coordinate system. As shown in Figure4.9 (a), trilateration considers explicitly one reference landmark and applies a linear transformation (translation t and rotation R eas- ily computed) to the whole system to solve the problem. In Figures4.9 (b) and 4.9 (d)

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Note that, multilateration (geometrically based on the cosine’s theorem) is an extension of trilateration considering M intersecting circles. Indeed, from M equations of circles, subtracting the equation of the reference landmark to the M − 1 remaining equations and adding the square distance between the reference landmark and each M − 1 remaining landmarks, it results in the same system of M − 1 equations of the multilateration tech- nique. When considering a noisy scenario in which distances between the unknown sen- sor node and the landmarks are estimated by means of RSSI values following the LNSM, several works coupled the latter methods with a least squares problem. The most rele- vant works are those from [118], [140] and [141] considering multi-hop communications between the sensor nodes. Any pair of nodes within the network which are not directly connected are able to communicate with the help of one or more intermediate/relay nodes. The main drawback of these methods is the dependence with the anchor’s positions since the solutions must lie inside the convex-hull formed by the set of known positions. In addition, the solution loses in robustness since it may be affected by the choice of the reference landmark, e.g. a landmark with an error in its known position or placed too far from the sensor node.

Statistical point of view

This methods focus on the statistical distribution of the received RSSI measurements com- ing from the M landmarks. The goal is to consider a parametric model for the received signal and to apply maximum likelihood estimator (MLE). Thus, the LNSM in (4.2) is the parametric model assumed for the observed RSSI and the unknown parameter to estimate is the unknown position of the sensor node z. Set (Pm(1), . . . , Pm(T )) the RSSI values

collected from any landmark m. Then, the MLE of z (in two dimensions) is:

ˆ z = arg min z=(x,y) M X m=1 T X t=1  Pm(t) + PL0+ 10η log10 p (x − am)2+ (y − bm)2 2 (4.11)

Since (4.11) is not a non-convex problem, the solution is affected by the existence of local minimums. Iterative algorithms need in general a suitable initialization, e.g. a noisy version of the target position. Indeed, the solution of (4.11) lies on the intersection of the M circles of the form:

(x − am)2+ (y − bm)2 = r ∀m = 1, . . . , M

where r is set as the square distance (4.10). If the dimensions of the area issued to the problem are previously known, one can set a grid of points, evaluate the function (4.11) on each center point and set ˆz as the point where the minimum value is achieved (see for instance [86], [47] or [54]). Note that, the accuracy depends on the choice of the grid’s resolution and so the required computational cost. As an alternative, iterative algorithms can be used to solve this non-linear optimization problem such: the conjugate gradient used in [128], the Nelder-Mead method used in [42] and more recently the Levenberg- Marquardt algorithm in [7].

It is worth noting that the LNSM may not be the most suitable model for more complex indoor scenarios (see [11]). Other works proposes the MLE on different statistical models. In [159] the environment is considered to be dynamic and time-varying. They proposed a modified version of the LNSM by adapting progressively the estimation of the noise vari- ance σ2. In [53] the modified LNSM considers different environment parameters (PL0, η

and σ2) for each landmark and includes a bias factor to (4.2) to model the possibly out- lier/multipath effects. More recently, [55] assumes the RSSI as a Gaussian Mixture of two classes and the problem (4.11) is solved through an Expectation-Maximization algorithm.

2) Anchor-free techniques

When dealing with distributed sensor networks, e.g. a WSN of N nodes, anchors may not be present, too far away or GPS signal is not available, e.g. indoor scenario. Nevertheless, sensor nodes can still benefit of the RSSI measurements obtained from their neighbors whose positions are unknown. The configuration of the network can be recovered on a relative coordinate sys- tem instead of the GPS absolute coordinate system. One should therefore rely on anchor-free methods.

When distances between nodes are view as similarity metrics, the positioning problem is referred to multidimensional scaling (MDS). Yet, the structure of distance-like data is related to the underlying geometric configuration of the network, i.e. find an embedding from the N nodes such that distances are preserved. For instance, in classical MDS [27, Chapter 12] positions are obtained by principal component analysis (PCA) of the input N × N matrix constructed from the Euclidean distances. If distances are issued to some noise, e.g. estimated from RSSI measurements as (4.8), [143] propose a MDS-MAP algorithm based on the classical MDS prob- lem. Indeed, the WSN localization problem is solved by enable each sensor node to infer all the estimated pairwise distances. It is worth noting that MDS problem for WSN localization is related to the rigid graph theory. Indeed, rigid graph theory explores the property of a given network configuration to be an embedding of a graph in an Euclidean space, i.e. the globally rigid property. This property, also known as fold-freedom (see [77] or [44]), suffers from two problems: non-uniqueness and NP-hard complexity. To overcome this issues, [130] proposed a two-phase algorithm based on fold-freedom initial configuration joint with a mass-spring model to refine the estimated positions.

Alternative approaches within the localization context are based on optimization techniques. In metric or modern MDS, positions are obtained by the stress majorization algorithm called SMACOF (see [27, Chapter 8] and [101]). The minimization problem is performed by using an auxiliary quadratic function that majorizes the stress function. Semidefinite programming (SDP) with convex constraints can be found in [57] and [24]. Recently the works of [41], [85] give further error bound analysis of the SDP approach. In general, even if these centralized techniques achieve high accuracy and solve the N unknown positions at once, they require high computational cost and may be especially complex to be implemented in a real wireless sensor network. Yet, classical MDS and SDP require aboutO(N3) of computational complexity.

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