Robust distributed cooperative RSS-based localization for directed graphs in mixed LoS/NLoS environments

2.1 Problem formulation

Consider¹ a directed graph with N_a anchor nodes and N_u agent nodes, for a total of N=N_a+N_u nodes. In a two-dimensional (2D) scenario, we denote the position of node i by \(\boldsymbol {x}_{i} = [\!x_{i} \ y_{i}]^{\top } \in \mathbb {R}^{2 \times 1}\), where ^⊤ denotes transpose. Between two distinct nodes i and j, the binary variable o_j→i indicates if a measure, onto direction j→i, is observed (o_j→i=1) or not (o_j→i=0). In the case when i=j, since a node does not self-measure, we have o_i→i=0. This allows us to define the observation matrix \(\boldsymbol {\mathcal {O}} \in \mathbb {B}^{N \times N}\) with elements \(o_{i,j} \triangleq o_{i \rightarrow j}\) as above. The aforementioned directed graph has connection matrix \(\boldsymbol {\mathcal {O}}\). It is important to remark that, for a directed graph, \(\boldsymbol {\mathcal {O}}\) is not necessarily symmetric; physically, this models possible channel anisotropies, miss-detections and, more generally, link failures. Let m_j→i be a binary variable, which denotes if the link j→i is LoS (m_j→i=1) or NLoS (m_j→i=0). Due to physical reasons, m_j→i=m_i→j. We define the LoS/NLoS matrix²\(\mathbf {L} \in \mathbb {B}^{N \times N}\) of elements \(l_{i,j} \triangleq m_{i \rightarrow j}\), and we observe that, since m_j→i=m_i→j, the matrix is symmetric, i.e., L^⊤=L. We stress that this symmetry is preserved regardless of \(\mathbf {\mathcal {O}}\), as it derives from physical reasons only. Let Γ(i) be the (open) neighborhood of node i, i.e., the set of all nodes from which node i receives observables (RSS measures), formally: \(\Gamma (i) \triangleq \{ j \neq i : o_{j \rightarrow i}=1 \}\). We define Γ_a(i) as the anchor-neighborhood of node i, i.e., the subset of Γ(i) which contains only anchor nodes as neighbors of node i. We also define Γ_u(i) as the agent-neighborhood of node i, i.e., the subset of Γ(i) which contains only agent nodes as neighbors of node i. In general, Γ(i)=Γ_a(i)∪Γ_u(i).

2.2 Data model

for any k₁,k₂,i₁,i₂,j₁∈Γ(i₁),j₂∈Γ(i₂). The previous equations imply that two different links are always independent, regardless of the considered time instant. In this paper, we call this property link independence. If only one link is considered, i.e., j₂=j₁ and i₂=i₁, then independence is preserved by choosing different time instants, implying that the sequence \(\left \{ w_{j \rightarrow i} \right \}_{k} \triangleq \left \{ w_{j \rightarrow i}(1), w_{j \rightarrow i}(2), \dots \right \}\) is white. The same reasoning applies to the (similarly defined) sequence {v_j→i}_k. As a matter of notation, we denote the unknown positions (indexing the agents before the anchors) by \(\boldsymbol {x} \triangleq \left [\boldsymbol {x}^{\top }_{1} \ \cdots \ \boldsymbol {x}^{\top }_{N_{u}}\right ]^{\top } \in \mathbb {R}^{2 N_{u} \times 1}\) and we define η as the collection of all channel parameters, i.e., \(\boldsymbol {\eta } \triangleq \left [\boldsymbol {\eta }^{\top }_{\text {LOS}} \ \boldsymbol {\eta }^{\top }_{\text {NLOS}}\right ]^{\top }\), with \(\boldsymbol {\eta }_{\text {LOS}} \triangleq \left [p_{0_{\text {LOS}}} \ \alpha _{\text {LOS}} \ \sigma ^{2}_{\text {LOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\), \(\boldsymbol {\eta }_{\text {NLOS}} \triangleq \left [p_{0_{\text {NLOS}}} \ \alpha _{\text {NLOS}} \ \sigma ^{2}_{\text {NLOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\).

It is important to stress that, in a more realistic scenario, channel parameters may vary from link to link and also across time. However, such a generalization would produce an under-determined system of equations, thus giving up uniqueness of the solution and, more generally, analytical tractability of the problem. For the purposes of this paper, the observation model above is sufficiently general to solve the localization task while retaining analytical tractability.

2.3 Time-averaged RSS measures

as our new observables⁴. While it would have been preferable to work with the original data from a theoretical standpoint, several considerations lead to the preference of time-averaged data, most notably: (1) comparison with other algorithms present in the literature, where the data model assumes only one sample per link, i.e., K=1, which is simply a special case in this paper; (2) reduced computational complexity in the subsequent algorithms; (3) if the RSS measures onto a given link needs to be communicated between two nodes, the communication cost is notably reduced, since only one scalar, instead of K samples, needs to be communicated; (4) formal simplicity of the subsequent equations.

where \(\sigma _{j \rightarrow i}^{2}\) is either \(\sigma ^{2}_{\text {LOS}}\) or \(\sigma ^{2}_{\text {NLOS}}\) and the general result follows from link independence.

which represents the information available to the whole network.

2.4 Single-agent robust maximum likelihood (ML)

Empirically, we can intuitively interpret λ_i as the fraction of anchor-agent links in LoS (for node i), while ζ_i as the fraction of agent-agent links in LoS (for node i). As in [21], the Markov chain induced by our model is regular and time-homogeneous. From this, it follows that the Markov chain will converge to a two-component Gaussian mixture, giving a theoretical justification to the proposed approach.

where the maximization is subject to several constraints: λ_i∈(0,1), α_LOS > 0, α_NLOS>0, \(\sigma ^{2}_{\text {LOS}} > 0\), and \(\sigma ^{2}_{\text {NLOS}} > 0\). In general, the previous maximization admits no closed-form solution, so we must resort to numerical procedures.

2.5 Multi-agent robust ML-based scheme

where \(\hat {\Gamma }_{u}(i)\) is the set of all agent neighbors of node i for which estimated positions exist. We can iteratively construct (and update) the set \(\hat {\Gamma }_{u}(i)\), in order to obtain a fully distributed algorithm, as summarized in Algorithm 1.

A few remarks are now in order. First, this algorithm imposes some restrictions on the arbitrariness of the network topology, since the information spreads starting from the agents which were able to self-localize during initialization; in practice, this requires the network to be sufficiently connected. Second, convergence of the algorithm is actually a matter of compatibility: if the network is sufficiently connected (compatible), convergence is guaranteed. Given a directed graph, compatibility can be tested a priori and necessary and sufficient conditions can be found (see Section 4). Third, unlike many algorithms present in the literature, symmetrical links are not necessary, nor do we resort to symmetrization (like NBP): this algorithm naturally takes into account the (possible) asymmetrical links of directed graphs.

2.6 Distributed maximum likelihood (DML)

which, in general, does not admit a closed-form solution, but can be solved numerically. After obtaining \(\hat {\boldsymbol {x}}_{i}\), node i can estimate p₀ and α using (23) and (25).

where (again) an initialization phase is required and the set of estimated agents-neighbors \(\hat {\Gamma }_{u}(i)\) is iteratively updated. The key difference with RDML is that, due to the assumption of the links being all of the same type, the estimates of p₀ and α are broadcasted and a common consensus is reached by averaging. This increases the communication overhead, but lowers the computational complexity, operating a trade-off. The DML algorithm is summarized in Algorithm 2.

Similar remarks as for the RDML can be made for the DML. Again, the network’s topology cannot be completely arbitrary, as the information must spread throughout the network starting from the agents which self-localized, implying that the graph must be sufficiently connected. Necessary and sufficient conditions to answer the compatibility question are the same as RDML. Secondly, the (strong) hypothesis behind the DML derivation (i.e., all links of the same type) allows for a more analytical derivation, up to position estimation, which is a nonlinear least-squares problem. However, it is also its weakness since, as will be shown later, it is not a good choice for mixed LoS/NLoS scenarios.

2.7 Centralized MLE with known nuisance parameters (C-MLE)

where \(\left (p_{0_{j \rightarrow i}}, \alpha _{j \rightarrow i}, \sigma ^{2}_{j \rightarrow i}\right)\) are either LoS or NLoS depending on the considered link. It is important to observe that, if all links are of the same type, the dependence from \(\sigma ^{2}_{j \rightarrow i}\) in (28) disappears. From standard ML theory [45], C-MLE is asymptotically (K→+∞) optimal. The optimization problem (28) is computationally challenging, as it requires a minimization in a 2N_u-dimensional space, but still feasible for small values of N_u.

4.1 Gaussian noise

Here, we validate our novel approach by considering fully connected networks in three different scenarios. In Fig. 4, all links are assumed to be LoS. As can be seen from the ECDFs, the robust approach has similar performances to the D-ML algorithm, which was designed assuming all links to be of the same type. In Fig. 5, all links are assumed to be NLoS and again we see a similar result. The key difference is in Fig. 6, where a mixed LoS/NLoS scenario is considered and the fraction of NLoS links is random (numerically generated as \(\mathcal {U}[0,1]\) and different for each MC trial). Here, the robust approach outperforms D-ML, validating our strategy. Moreover, it has a remarkably close performance to the centralized algorithm C-MLE. As a result, the robust approach is the preferred option in all scenarios.

4.2 Non-Gaussian noise for NLoS

In order to evaluate robustness, we consider a model mismatch on the noise distribution by choosing, for the NLoS links, a Studentt distribution with ν=5 degrees of freedom (as usual, serially i.i.d. and independent from the LoS noise sequence). For brevity, we consider only the case of a fully connected network in a mixed LoS/NLoS scenario. As shown by Fig. 7, our proposed approach is able to cope with non-Gaussianity without a significant performance loss, thanks to the Gaussian mixture model.

4.3 Cooperative gain

The proposed algorithm exhibits the so called “cooperative gain,” i.e., a performance advantage (according to some performance metric, typically localization accuracy) with respect to a non-cooperative approach, where each node tries to localize itself only by self-localization (12). In our case, the cooperative gain is twofold: first, it allows to improve localization accuracy; second, it allows to localize otherwise non-localizable agents. To show the first point, we consider a mixed LoS/NLoS environment (for simplicity, with white Gaussian noise) and a network is generated at each Monte Carlo trial, assuming a communication radius R=70 m, with an ideal probability of detection model [17]. Moreover, the network is generated in a way as to allow all agents to self-localize and, as a consequence, it is RDML-compatible. In Fig. 8, the ECDFs of the localization error are shown. To show the second point, we consider the toy network depicted in Fig. 9. In this example, agent X is not able to self-localize, since no anchors are in its neighborhood, Γ_a(X)=∅. Thus, in a non-cooperative approach, the position of agent X cannot be uniquely determined⁷, while, in a cooperative approach, the position of agent X can actually be obtained by exchanging information, e.g., the estimated positions of its neighbors.

4.4 Variable K and NLoS fraction

We next analyze the RDML algorithm by varying the number of samples per link, K, and the fraction of NLoS links. In both cases, we consider (for brevity) fully connected networks with white Gaussian noise; for variable K, the fraction of NLoS is randomized, while, for variable NLoS fraction, K is fixed. In Fig. 10, we observe that the performance increases as K increases, as expected. In Fig. 11, where each point represents 100 MC trials, the median error is chosen as a performance metric⁸ and RDML shows good performances and is not significantly affected by the actual NLoS fraction, which is evidence for its robustness, while D-ML is clearly inferior and suffers from model mismatch.

4.5 Communication and computational costs

The communication overhead is evaluated by computing the number of elementary messages sent throughout the network, where an elementary message is simply defined as a single scalar. Sending a d-dimensional message, e.g., (x,y) coordinates, is equivalent to sending d elementary messages. In Fig. 12, the communication overhead of RDML and DML is plotted with respect to the number of agents, assuming a complete graph: for N_u agents, 2N_u(N_u−1) scalars, representing estimated 2D positions, are needed for RDML, while 4N_u(N_u−1) are needed for DML, as estimates of (p₀,α) are also necessary. Viewing (2D) position has the fundamental information of a message (d=2), in a complete graph, RDML achieves the theoretical minimum cost in order for all nodes to have complete information, while DML uses auxiliary information. For a general graph, the communication overhead depends on the specific graph topology, but is never greater than the aforementioned value. Thus, for a general graph, the communication cost is upper-bounded by a quadratic function of N_u.

Regarding computational complexity, both RDML and DML scale linearly with N_u and they benefit from parallelization, as the main optimization task can be executed independently for each involved node. As already mentioned, DML operates a trade-off between communication and computational complexity; in fact, the DML optimization problem (27) is easier to solve than the RDML optimization problem (19). Both problems are non-convex and may have local minima/maxima, so care must be taken in the optimization procedure.