# Cooperative Localization Using Distance Measurements for Mobile Nodes
## 1 Introduction  
Because of the increased availability of low-cost low-power sensors, smart sensors, and multi-functional sensors, wireless sensor networks are becoming increasingly ubiquitous [1, 2, 3, 4]. Wireless sensor networks are being utilized in a diverse array of tracking and monitoring applications from environmental [5] and health monitoring [6] to traffic [7] and border surveillance [8]. In many of these applications, the nodes in the network are mobile and knowledge of their positions is a prerequisite for completing the task, and crucial for information sharing, data collection, and scheduling [9]. For example, if the locations of the nodes are unknown or significantly incorrect, the data they have collected from surrounding environment, such as wildlife [10] or weather information [11], will be useless, since the positional information is not available. Localization algorithms estimate the locations of unknown nodes in the network using the positions of a known subset of the nodes to provide the required positional information. The most widely used localization techniques in the literature are distance-based localization algorithms, such as trilateration, radio interference positioning system (RIPS) [12], and the Hop-Distance algorithm [13]. These algorithms estimate the inter-node distances and require anchors, which is nodes with known locations, to provide the locations of the remaining nodes. The location of the anchor nodes is accessed via global positioning system (GPS) or a priori information [12, 14, 15, 16, 17, 18]. While, GPS is widely used in locating unknown nodes in many situations, such as indoor, urban, and forest environments, the positions of nodes are difficult to obtain from GPS [19, 20, 21, 22]. In this case, the anchors are absent or the positions of the anchors are not available, and the above algorithms cannot be applied. This is widely regarded as the most significant challenge in the positioning and navigation field [23, 24, 25]. Therefore, there is an increasing need for anchorless localization of sensor networks for use in GPS-denied orcontested environments. Using the movement of nodes and the inter-node communication, cooperative localization can be leveraged in order to solve this problem. This scenario arises in the field of robotics swarms, especially in the navigation and formation control of unmanned aerial vehicles swarms under a GPS-denied environment [20; 25; 26; 27]. In practice, cooperative localization can be achieved by utilizing the inter-node distances [28]. However, relative localization only gives node positions that satisfy the distance constraint, which means that there could be ambiguity problems, i.e., ambiguity due to rotation and/or flip [29]. The multidimensional scaling (MDS) algorithm is a widely used algorithm that is capable of tackling the anchorless localization problem [30; 31; 32; 33; 34; 35]. The aim of MDS is to represent the similarity (or dissimilarity) of high dimensional data in a lower dimensional map which describes the relative distances between pairs of objects (in this case sensor nodes). Like other relative localization methods MDS can also be subject to the ambiguity problem; hence, algorithms have been proposed to attempt to address this problem. In [36] a MDS-based algorithm using moving nodes is presented, which constructs a cost function involving velocities and inter-node distances of all nodes at two consecutive time instants. In [37], a similar algorithm is proposed to solve the anchorless localization problem for nodes that can estimate the position via a nonlinear least square estimator; however, in this case, only one node is moving. In this paper, we present an efficient algorithm for anchorless cooperative localization that is based on MDS. The algorithm mitigates the rotation and flip problems by taking advantage of the movement and inter-node communication of the mobile nodes.