## 4 The Proposed Tracking Anchor Based Clustering Method
### Tracking Anchor Determination  
After being deployed to the surveillance area, sensor nodes remain in the sleep state. To activate a set of tracking nodes in advance, a certain number of tracking anchors are selected in the network. Tracking anchors are responsible for sending scheduling instructions to nearby nodes according to the predicted target position. First, we determine the number of required tracking anchors. Considering that the sum of the sensing coverage of all anchors is not less than the surveillance area, we can obtain:  
\[K\pi r_{S}^{2}\geq M\times M\, \tag{5}\]  
where \(K\) denotes the number of tracking anchors. For energy efficiency, we select as few anchors as possible while the coverage condition is satisfied. \(K\) is given by:  
\[K=\left\lceil\frac{M\times M}{\pi r_{S}^{2}}\right\rceil. \tag{6}\]  
After determining the number of anchors, the RFCM algorithm proposed in [29] is used to perform unsupervised clustering of sensor nodes. \(K\) cluster centers obtained after iteration can be designated as the position of tracking anchors in the network. In RFCM, \(N\) nodes are partitioned into \(K\) clusters by minimizing the objective function \(J_{RF}\):  
\[J_{RF}=\left\{ \begin{array}{cl}\omega_{1}\times\mathfrak{A}+\omega_{2}\times \mathfrak{B},&if\ \underline{A}(\beta_{i})\
eq\varnothing,\ B(\beta_{i})\
eq \varnothing\\ \mathfrak{A},&if\ \underline{A}(\beta_{i})\
eq\varnothing,\ B(\beta_{i})= \varnothing\\ \mathfrak{B},&if\ \underline{A}(\beta_{i})=\varnothing,\ B(\beta_{i})\
eq \varnothing\\ \end{array} \right.\, \tag{7}\]  
\[\mathfrak{A}=\sum_{i=1}^{K}\sum_{s_{j}\in\underline{A}(a_{i})}(\mu_{ij})^{m} \|s_{j}-a_{i}\|^{2}\, \tag{8}\]  
\[\mathfrak{B}=\sum_{i=1}^{K}\sum_{s_{j}\in B(a_{i})}(\mu_{ij})^{m}\|s_{j}-a_{i} \|^{2}\, \tag{9}\]  
where \(\beta_{i}\) denotes the cluster with the anchor \(a_{i}\) as the center; \(\underline{A}(\beta_{i})\) and \(B(\beta_{i})\) denote the lower approximation region and boundary region of cluster \(\beta_{i}\), respectively; parameters \(\omega_{1}\) and \(\omega_{2}\) correspond to the relative importance of the lower and boundary regions, and \(\omega_{1}+\omega_{2}=1\); \(m\) is the fuzzifier, where \(1\leq m<\infty\), which is introduced to calculate the relevance of each node to the anchors, rather than assigning it to the nearest anchor. If the value of \(m\) is too large, good clustering cannot be obtained, and if \(m\) is too small, the algorithm is close to the normal hard clustering C-means algorithm; \(\mu_{ij}\) is the probabilistic membership of the sensor \(s_{j}\) to anchor \(a_{i}\), and \(\mu_{ij}\in[0,1]\); \(\|\cdot\|\) denotes the distance norm. In the applied RFCM, each cluster consists of a clear lower approximation region and a fuzzy boundary region, as shown in Figure 4. The tracking anchor is located at the centroid of the cluster. There are two important properties about \(\underline{A}(\beta_{i})\) and \(B(\beta_{i})\):_Property I:_  
If the sensor \(s_{j}\in\underline{A}(\beta_{i})\), then \(x_{j}\
ot\in\underline{A}(\beta_{k}),\forall k\
eq i\). That is, the sensor \(s_{j}\) is contained in \(\beta_{i}\) definitely. _Property II:_  
If the sensor \(s_{j}\in B(\beta_{i})\), then \(s_{j}\) possibly belongs to \(\beta_{i}\) and potentially belongs to another cluster. Sensors located in the lower approximation region and the boundary region have different influences on the location of anchors. In RFCM, the membership value of sensors in the lower approximation is fixed to 1, that is, \(\mu_{ij}=1\) if \(s_{j}\in\underline{A}(\beta_{i})\). Moreover, the membership value of the sensors in the boundary region is calculated according to the relative distance to the anchor:  
\[\mu_{ij}=(\sum_{k=1}^{K}{(\frac{d_{ij}}{d_{kj}})}^{\frac{2}{n-1}})^{-1}\;\;,\; \;\;if\;s_{j}\in B(\beta_{i})\;, \tag{10}\]  
where \(d_{ij}\) and \(d_{kj}\) denote the distance from sensor \(s_{j}\) to anchor \(a_{i}\) and \(a_{k}\), respectively.