# An Anchor-Free Location Algorithm Based on Transition Coordinates
## 1 Introduction  
Many application scenarios of wireless sensor networks need to know the node location. Node location has an important reference value for information collection and data processing, so positioning is a hot topic in wireless sensor network research [1; 2]. The traditional positioning methods are usually positioned with the global positioning system (GPS) or anchored points with geographic location information [3; 4; 5]. Anchored points refer to nodes [6; 7] with geographic coordinates obtained in advance, and the rest nodes are called unanchored points. However, the positioning of anchor points in wireless sensor networks has many inconveniences and defects in indoor or suburban scenarios [8; 9]. Anchor-free location technology [10] does not need to master the geographical location information of nodes in advance, and does not use GPS or anchor points, but calculates the location of nodes through the local distance information and connectivity between node pairs, so as to determine the relative coordinates and network topology of the entire network. The earliest anchor-free location algorithm is anchor-free localization (AFL) proposed by [PERSON] et al. [11], which is a completely distributed location algorithm. The location algorithm is divided into two steps: the first step is to obtain the model, obtain the collapsible network model by using the heuristic principle, and determine the rough coordinates of the unknown node by using the centroid location algorithm; the secondpart corrects the error, using the particle spring model to correct and optimize the node positioning error. However, there is no clear calculation method of single jump distance in AFL algorithm. The centralized boundary discovery (CBD) algorithm proposed in [12] for unanchored point location can define the energy spent in the initialization process, but the algorithm cannot be used to calculate the energy consumed by a single node. An assumption-based coordinates (ABC) anchor-free location algorithm was proposed by [PERSON], [PERSON], [PERSON], [PERSON] et al. [13], which is a distributed location algorithm. The coordinates of an unknown node are calculated by one estimate, and then the coordinates of the unknown node are used as the reference coordinates to calculate the coordinate position of the next adjacent unknown node, and so on, and the coordinates of all unknown nodes in the network are computed. However, its positioning error is relatively large, and it is generally used as an auxiliary positioning mechanism. [PERSON], [PERSON] et al. proposed an improved ABC algorithm based on the dynamic Levy mechanism [14], which effectively expanded the search scope and avoided falling into local areas, but it still needs further research and testing in practical engineering applications. The above algorithms need to master the position distance information between nodes, and then establish the cost function based on it, and then find the optimal solution to get the position coordinates of nodes. In addition, there is also a method of matrix inversion to solve the position coordinates. For example, [PERSON] et al. proposed a multidimensional positioning algorithm without anchors based on the principle of multidimensional scaling (MDS), which is a centralized location algorithm [15]. The MDS algorithm makes full use of the local distance information and connectivity between unknown nodes, and can simultaneously locate multiple unknown nodes. However, the algorithm replaces the Euclidean distance with the shortest path distance between node pairs. When the shortest path distance differs greatly from the Euclidean distance, there is a large error. In addition, the accumulation and propagation errors of a centralized location algorithm will also bring some positioning errors. For irregular networks, the semi-centralized MDS method proposed in literature [16] has lower complexity and more robustness and accuracy than the centralized MDS, but it still has the problem of error accumulation.