# A Weighted and Distributed Algorithm for Range-Based Multi-Hop Localization Using a Newton Method
## 1 Introduction  
Nowadays, Wireless Sensor Networks (WSNs) have gained relevance in many aspects of our lives because they have the capacity to innovate our environment including body sensor networks, smart houses, automation, transportation, business, security, agriculture to military applications, to mention a few. A WSN is typically composed of a large number of tiny devices called sensor nodes. Due to the characteristics of a sensor node, which contains constrained resources in processing, memory, sensing, and non-renewable with low-power batteries, wireless transmissions must be used efficiently to save energy. Furthermore, even though RF transmissions are around 10 times more energy-expensive than processing, all aspects of energy saving must be carefully taken into account (e.g., leaking energy) to preserve the autonomy of the entire network [1; 2; 3; 4]. Commonly, WSNs are randomly deployed in large geographical areas with the aim of detecting environmental events on sensor nodes. Such events must be reported to other nodes using short-range wireless transmissions until reaching one or more sink nodes for reporting data. Clearly, routing protocols play an important function in multi-hop networks, that is to say, the shortest route from one sensor node to another sensor node allows the entire network to save energy [5; 6]. However, it is imperative for data reporting to know where the detected event comes from. In other words, for a broadcast event the time-location parameter is more important than the sensor identity itself. Thus, positions of sensor nodes should be known a priori before to run a WSN application, and so the background stage in most WSN applications is the localization process. In this stage, the geographic-location must be discovered using RF transmissions with neighboring sensors and specific algorithms [7; 8; 9; 10; 11]. An interesting research dealing with multi-hoplocalization is presented in [12]. It manages an hybrid approach where the DV-Hop scheme is improved by promoting unknown sensor as anchor nodes, where these virtual anchors are one-hop neighboring sensor of the real anchors, and their localization are obtained using the estimated distances from the RSSI technique. The authors claim that this scheme can be employed when the density of anchors in the network is low. In distributed localization algorithms dealing with realistic multi-hop WSN scenarios, estimating among neighboring sensors can suffer from hardware imperfections, changing environments, network topologies, and node densities, which consequently can affect the estimation of the sensor positions [13; 14; 15; 16; 17]. For instance, the work in [18] addresses the multi-hop localization problem in severe multi-path scenarios deriving from changing environments. It integrates the non-line-of-sight (NLOS) path information to maximize a reduced-complexity pseudo-maximum likelihood scheme to estimate the position of unknown sensors. Another interesting approach is proposed in [19] addressing hardware imperfections called wireless ad hoc system for position (WASP). It handles complex radio propagation signals coming from beacons to determine localization of unknown sensors by using novel extensions for ToA measurements. This approach provides high accuracy at minimum hardware and processing cost. It uses a robust least squares algorithm (RLS) for localization that removes iteratively suspicious bad range measurements by recomputing node positions. However, in our work, we pursuit a different approach where the outlierness level of each measurement is controlled by weights that could attenuate the possible influence of atypical observations on the solution of the optimization problem. In [20], the localization problem is addressed using a robust statistic, which reduces the influence of outliers in the estimation process.