## 4 The Proposed Tracking Anchor Based Clustering Method
### Tracking Anchor Determination
The new position of the anchor is calculated based on the weighted average of the lower approximation and boundary, which is given by Equation (11):  
\[a_{i}^{RF}=\left\{ \begin{array}{cc}\lambda_{1}\times\mathfrak{C}+\lambda_{2 }\times\mathfrak{D},&if\underline{A}(\beta_{i})\
eq\varnothing,\;B(\beta_{i}) \
eq\varnothing\\ \mathfrak{C},&if\underline{A}(\beta_{i})\
eq\varnothing,\;\;B(\beta_{i})= \varnothing\\ \mathfrak{D},&if\underline{A}(\beta_{i})=\varnothing,\;\;B(\beta_{i})\
eq \varnothing\end{array} \right.\;, \tag{11}\]  
\[\mathfrak{C}=\frac{1}{|\underline{A}(\beta_{i})|}\sum_{x_{j}\in\underline{A}( \beta_{i})}s_{j}\;, \tag{12}\]  
\[\mathfrak{D}=\frac{1}{n_{i_{s_{j}}\in B(\beta_{i})}}(\mu_{ij})^{m}s_{j}\;, \tag{13}\]  
where  
\[n_{i}=\sum_{s_{j}\in B(\beta_{i})}(\mu_{ij})^{m}\;, \tag{14}\]  
where \(|\underline{A}(\beta_{i})|\) denotes the number of sensors in the lower approximation region. The position of the anchor depends on the parameter \(\lambda_{1}\) and \(\lambda_{2}\), and the fuzzifier \(m\) rules their relative influence. Since sensors in the lower approximation region definitely belong to a  
Figure 4: Cluster \(\beta\) and its lower approximation region and fuzzy boundary region. certain anchor, they are assigned a higher weight \(\lambda_{1}\) compared to \(\lambda_{2}\) of the sensors in the boundary region. Thus, the values are given by \(0<\lambda_{2}<\lambda_{1}<1\). The optimization process of RFCM to the objective function \(J_{RF}\) is calculated by Equation (11). Firstly, randomly assign \(K\) positions to anchors to form an initial population \(\mathfrak{A}=\{a_{1},\ \ a_{2},\ \dots,\ a_{K}\}\), and the fuzzy membership values of sensors are calculated according to (10). These anchors and sensors constitute the initial population. Then, we sort the membership value of each sensor and obtain the highest value, which is assumed to be \(\mu_{ij}\), and the second-highest value, which is assumed to be \(\mu_{kj}\). If \((\mu_{ij}-\mu_{kj})\leq\delta\), we have \(s_{j}\in B(\beta_{i}),\ \ s_{j}\in B(\beta_{k})\), and the sensor \(s_{j}\) does not belong to the lower approximation region of any anchor. Otherwise, if \((\mu_{ij}-\mu_{kj})>\delta\), then \(s_{j}\in\underline{A}(\beta_{i})\). The performance of RFCM depends on the value of \(\delta\), which can be defined by Equation (15):  
\[\delta=\frac{1}{N}\underset{j=1}{\overset{N}{\sum}}\left(\mu_{ij}-\mu_{kj} \right)\,. \tag{15}\]  
where \(N\) is the total number of sensors, and \(\mu_{ij}\) and \(\mu_{kj}\) are the highest and second-highest membership of the sensor \(s_{j}\), respectively. The value of \(\delta\) represents the average difference between the two highest memberships value of all the nodes in the network, which is dependent on the location of the anchors and the nodes. After determining the region to which the sensor belongs, the anchor's position is updated according to Equation (11). The above steps are repeated until the maximum number of iterations \(I_{max}\) is reached or the anchor position and membership relationship obtained by the two iterations are stable, that is \(\left|\frac{\mu_{ij}(t)-\mu_{ij}(t-1)}{\mu_{ij}(t-1)}\right|<\varepsilon\). Both \(I_{max}\) and \(\varepsilon\) are used as signs to stop the iteration. \(I_{max}\) can be obtained from the empirical convergence iterations of the RFCM algorithm in the experiment. \(\varepsilon\) is the threshold of the change rate of the membership value for two iterations, generally taking a small value, e.g., \(\varepsilon=0.01\). After obtaining the anchor position and the membership relationship of sensors, there are two ways to deploy the anchor: if the environment allows, we can use vehicles, UAVs, etc. to transport additional sensors to the designated position as anchors. Otherwise, we make the sensor closest to the anchor position be the anchor.