# A Weighted and Distributed Algorithm for Range-Based Multi-Hop Localization Using a Newton Method
## 4 A Robust and Distributed Localization Algorithm Based on the Newton Method
For instance, the remarked arrows, left side, imply that the combination R = 35 m with \(nfe=0.1\) will be applied to all distance measurements on an entire network. Using this combination (R = 35 m and \(nfe=0.1\)), without outliers (\(\Theta=0\%\)), all iterative algorithms are tested using the 10 independent networks. For each one of the 10 networks, the algorithm is run until one of the stopping criteria is satisfied (i.e., 100 iterations or \(\left\|\mathbf{p}_{i}-\mathbf{\tilde{p}}_{i}\right\|\leq\tau\)). Then, the number of iterations in which the algorithm stopped is registered. Finally, Equation (41) is used to calculate the average RMSE (\(RMSE\)) of the 10 networks and also the average of the numbers of iterations (\(Iter\_avg\)) is obtained. The next step consists of testing the same combination (R = 35 m and \(nfe=0.1\)), but now adding 10% of outliers to noisy distances with (38). This process is repeated until reaching 50% of outliers in noisy distances. Therefore, for each outlier step there will be two parameters (i.e., \(RMSE\) and \(Iter\_avg\)) for each evaluated algorithm. The last process is repeated for R = 35 m with \(nfe=0.3\)

R = 45 m with \(nfe=0.1\) and finally for R = 45 m with \(nfe=0.3\). Table 1 summarizes the parameters used in the experimental simulations. ### Range-Based Multi-Hop Localization over Randomly and Uniformly Distributed Sensor Networks

In this section, we test the proposed RWNM algorithm under isotropic WSN topologies. Here, sensor nodes are uniformly and randomly distributed over an area of \(200\times 200\) m\({}^{2}\). Ten multi-hop networks similar to that one presented in Figure 1b are created. The ratio between the number of unknown sensors and anchors is 0.95. Each network has 100 sensor nodes. First, it is assumed that each sensor node has a partial set of neighboring sensors according to Equation (7). Furthermore, anchors and unknown sensors are assumed with the same circular radio range R. Each pairwise distance is affected with a noise factor of \(nfe=0.1\) using Equation (35). This benchmark network, with \(nfe=0.1\) and \(\Theta=0\), is used as a basis to create other five networks, affected with different levels of outliers \(\Theta=10\%,\Theta=20\%,\Theta=30\%,\Theta=40\%\), and \(\Theta=50\%\). Finally, each one of the six networks is run in every iterative algorithm to evaluate its accuracy and rate of convergence (or iterations) at different levels of outliers. It must be remarked that DV-hop [13] and RELM [35] algorithms modify distances between anchors and unknown sensors, where \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Description\textbackslash{}Values** \\ \hline Standard-deviation of the distance error (nfe) & 0.1, 0.3 \\ Outlier levels (\(\Theta\)) & 0\%, 10\%, 20\%, 30\%, 40\%, 50\% \\ Radio Range & 35 m, 45 m \\ Number of Networks & 10 \\ Network Topologies & isotropic: square-shape \\  & anisotropic: circular-shape \\ Sensor Nodes Distribution & uniformly random distributed \\ Anchor Nodes & 5 \\ Unknown Sensors & 95 \\ Deployment Area & 200 \(\times\) 200 m\({}^{2}\) \\ Performance Metrics & RMSE, Iterations \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of the experiments. Figure 2: Methodology followed to test \(RMSEavg\) and \(Iter\_avg\) spent by algorithms using a setup of 10 networks for both circular and uniform node distributions. such distances are unaffected by noise or by outliers as they depend on hop counts between them. Initial estimates average of the 10 networks is around 86.33 m.

It is known that there is at least one path between any pair of sensor nodes in each network. Initial position estimates for the iterative algorithms are simply calculated as the average positions of anchor nodes. The refinement stage is run in a distributed way as second localization process in each unknown sensor to improve position estimates. Figure 3 shows the accuracy iteration behavior of the evaluated algorithms considering \(nfe=0.1\) and R = 35 m at different levels of outliers.