# An Anchor-Free Location Algorithm Based on Transition Coordinates
## 5 Simulation Result  
In order to verify the performance of the algorithm of the unanchored location, MATLAB R2016a software is used to simulate the algorithm. The positioning area is set to a two-dimensional plane of 100 m \(\times\) 100 m, the number of nodes is set to \(N\), and the real position of each member is randomly generated by the system in \([-50,50]\), one of which is its horizontal coordinate and the other is its vertical coordinate. In order to better observe the relative position graph formed by each member, we adjusted the position coordinate of the first member to the origin, the position of the second member to the positive half axis of the \(x\)-axis, and the position of the third member to the positive half axis of the \(y\)-axis. After the completion of each iteration update, the ratio of the sum of distance errors between the transition coordinates of all nodes and the real coordinates to the sum of distances between the real coordinates of each node is called error proportion (EP). The value of the EP is that the positioning accuracy needs to be achieved in the positioning process. We give the EP formula as follows:  
\[\text{EP}=\frac{\sum\limits_{ \begin{subarray}{c}i,j=\text{1 to}N\\ \text{and}i\
eq j\end{subarray} }|d_{ij}-d^{\prime}_{ij}|}{\sum\limits_{ \begin{subarray}{c}i,j=\text{1 to}N\\ \text{and}i\
eq j\end{subarray} }d^{\prime}_{ij}} \tag{9}\]  
where \(N\) is the total number of nodes, \(|d_{ij}-d^{\prime}_{ij}|\) represents the absolute value of the distance error between the transition coordinates of the node and the real coordinates, and \(d^{\prime}_{ij}\) represents the distance between the real coordinates of the nodes. The authors of [11] proposed the global energy ratio (GER) index, which considers the distance between nodes and neighbors and non-neighbors from the perspective of the whole network, reflecting not only the side length error but also the overall network structure. Let the real coordinate distance between node \(N_{i}\) and \(N_{j}\) be \(r_{ij}\), and the distance between the coordinates after their positioning is completed be \(\hat{d}_{ij}\), and set \(\hat{e}_{ij}=(\hat{d}_{ij}-r_{ij})/r_{ij}\). If the network size is \(N\), GER is defined as follows:  
\[\text{GER}=\frac{\sqrt{\sum\limits_{ \begin{subarray}{c}i,j,i<j\end{subarray}  }\hat{e}_{ij}^{2}}}{N(N-1)/2} \tag{10}\]  
In this paper, the factors that affect the performance of the algorithm mainly include the topological structure of the initial coordinates, the positioning accuracy, and the number of nodes \(N\). After reading a lot of the literature, the simulation conditions set for our first experiment were that the topology structure of the initial coordinates was square, the positioning accuracy was 1%, and the number of nodes \(N\) = 9. This experiment was taken as a control experiment for the rest of the simulation experiments in this paper, and the simulation effect was shown in Figure 2. \begin{table}
\begin{tabular}{c c} \hline
**Approaches** & **Computation Complexity** \\ \hline Centralized & \(O(N^{3})\) \\ Semi-centralized & \(O(N^{5})\) \\ Distributed & \(O(N^{3})\) \\ The algorithm proposed in this paper & \(O((N-1)L)\) \\ \hline \end{tabular}
\end{table} 
Table 2: The computational complexity of different localization algorithms. As can be seen from Figure 2, the initial position coordinates of 9 nodes are generated by a random function and serve as the starting position of transition coordinates, and the initial error EP is 0.342. Then, by constantly correcting the position increment between transition coordinates and real coordinates through the algorithm proposed in this paper, it can be seen that EP is less than 0.1 when it is equal to 0.089 after 7 iterations.