# An Anchor-Free Location Algorithm Based on Transition Coordinates
## 2 System Model  
In this paper, the distance \(d^{\prime}_{ij},1\leq i,j\leq N,i\
eq j\) between \(N\) nodes is collected in a self-organizing way. According to the arrangement and combination formula, there are a total of \(N_{d}=N(N-1)/2\) distances, and then the distance information is transmitted to the data processing center of the system, and the processing center attempts to use the \(N_{d}\) distances to calculate the relative position coordinates between \(N\) nodes. In the early stage of the algorithm in this paper, only the anchor-free location problem of these nodes in the same plane is studied, and the position of \(N\) nodes is not known. According to geometric knowledge [18], if the rotation and translation of nodes are not considered, the relative position coordinates of \(N\) nodes will be unique. As shown in Figure 1, we assume that the number of nodes \(N=9\), so the number of distances \(N_{d}\) between nodes is 36, that is, the distance \(d^{\prime}_{ij}\) between two nodes is represented by 36 dotted lines in the figure. For the convenience of drawing, \(A_{1}\) is drawn at the origin of coordinates for all graphs that do not consider rotation and translation,\(A_{2}\) is located in the positive half axis of the \(x\)-axis, and \(A_{3}\) is located in the first and second quadrants. The following section gives the specific algorithm, and the meanings of each symbol are shown in Table 1.  
\(N\) nodes are set to be represented by numbered \(A_{i}\) and the real coordinate \(\mathbf{p}_{i},1\leq i\leq N\) of each node is undetermined, which is a problem that needs to be solved by anchor-free location technology. Traditional algorithms solve \(\mathbf{p}_{i}\) by trigonometry or constructing a cost function and minimizing it, or inverse matrix [8; 9; 10; 19], which all have the problem of many iterations and large amount of computation. In order to solve the above problems, reduce the number of iterations, and simplify the amount of computation, the anchor-free location algorithm based on transition coordinates \(\mathbf{p}_{i}^{k},1\leq i\leq N,0\leq k\leq K\) is proposed in this paper, where \(K\) represents the number of transition coordinates of nodes and is also equal to the number of iterations. When \(N\) is less than 10, \(K\) is generally less than 30, which is far lower than the number of iterations--more than several hundred times in the traditional algorithm. The system first uses a random function to generate the initial transition coordinate \(\mathbf{p}_{i}^{0},1\leq i\leq N\) of \(N\) nodes, then iteratively updates the new position vector \(\mathbf{p}_{i}^{k+1}\), \(1\leq i\leq N,0\leq k\leq K-1\) by adding position increment \(\Delta\mathbf{p}_{i}^{k+1}\) to the previous transition coordinate \(\mathbf{p}_{i}^{k},1\leq i\leq N,0\leq k\leq K-1\). After \(K\) iterations, the final convergent transition  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Definition** \\ \hline \(N\) & Number of nodes \\ \(\mathbf{p}_{i}\) & The actual position coordinates of node \(i\) \\ \(\mathbf{p}_{i}^{k}\) & Transition coordinates of node \(i\) at time \(k\) \\ \(\mathbf{p}_{i}^{k+1}\) & Transition coordinates of node \(i\) at time \(k+1\) \\ \(\Delta\mathbf{p}_{i}^{k+1}\) & The position increment of the transition coordinate of node \(i\) from time \(k\) to time \\ \(d^{\prime}_{ij}\) & The distance between the actual position coordinatescoordinate \(\mathbf{p}_{i}^{K},1\leq i\leq N\) will be infinitely close to the real coordinate \(\mathbf{p}_{i},1\leq i\leq N\). The iterative algorithm based on transition coordinates proposed in this paper does not construct cost function, nor does it use matrix inversion and other complex operations. The core innovation is that the part of position increment \(\Delta\mathbf{p}_{i}^{k+1}\) only uses the difference between the transition distance \(d_{ij}^{k+1}\) and the real distance \(d_{ij}^{k}\). By compensating the difference, the transition coordinate \(\mathbf{p}_{i}^{k},1\leq i\leq N,0\leq k\leq K-1\) approximates the real coordinate.