# Relational Positioning Method for 2D and 3D Ad Hoc Sensor Networks in Industry 4.0
## 2 Related Research
After a node calculates its position, it is promoted as anchor node assistant to improve the positioning coverage. Finally, a quasi-Newton method is used to optimize the positioning result. In ref. [15], the authors propose a sequence localization algorithm based on the use of a 3D Voronoi diagram. The Voronoi diagram is used to divide the working space depending on the real beacon nodes (known location nodes) and construct a rank sequence table of virtual beacon nodes (determined after partition). This algorithm uses RSSI estimations between nodes as a reference to correct the measured distances and to fix the location sequence of unknown nodes. Then, the algorithm commands a weighted location estimate with N valid virtual beacon nodes by a normalization process of rank correlation coefficients. In a different approach, the authors of [7] propose a method that is based on the convenient deployment of access points (APs) with known coordinates and on the Manhattan of the network space. A set of algebraic relationships is defined for the node's coordinates based on distances from APs to the node of interest. In addition, a maximum a posteriori probability (MAP) criterion is used to improve estimated distances and to improve the performance of the algebraic method. This method is evaluated in multi-hop scenarios, and the distance between nodes is estimated based on the hop count criterion [16]. The main contribution is a new method for location estimation suitable for 3D network spaces. Table 1 summarizes some characteristics of the 3D localization methods explained above, as well as some evaluation performance and scenarios setup. In ref. [17], the authors present a survey of multidimensional scaling (MDS) based localization techniques. These are range-free localization techniques that use proximity information instead of actual distance measurements. MDS keeps the important information while reduces multidimensional data into a lower-dimensional space. In this paper, the authors present an overview of the different variants of outdoor and indoor MDS-based localization systems. They also present some 3D MDS-based localization methods for WSN and discuss how some proposed methods for 2D are not effective in 3D environments, as complexity increases due to several factors, such as node deployment, increment of anchors needed for location calculation, connectivity, and physical obstructions. As explained before, in this paper, we present a different positioning approach based on distance relationships and on a likelihood function. The relational positioning method formulates a searching equation based on a likelihood function that depends on a weights proportion where the points closest to the node to be located will have greater weight according to an assignment based on the ratio of distance relationships between ANs and the node of interest. For the sake of simplicity, we describe the method both for planar and 3D scenarios in the following sections. Note that in this paper, we present the conceptualization of a new 3D localization method and its evaluation in different network scenarios. However, concerns about the implementation of the method in terms of the network layer perspective (e.g., control messages, message passing, etc.) are not necessary to assess the performance of the proposed algorithm.