## 4 The Proposed Tracking Anchor Based Clustering Method
### Tracking Anchor Determination
There will also be competition between nodes for the same communication channel, resulting in unnecessary energy loss. Therefore, we set the second constraint to ensure that the CMs in the sensing-transmitting state are not redundant nodes according to the center angle of the overlapping coverage area being less than \(2\pi\), that is, \(\phi(x_{j})\leq 2\pi\), so the constraint is given by:

\[x_{j}\cdot\phi(x_{j})\leq 2\pi. \tag{20}\]

Regarding \(\phi(x_{j})\), according to Equation (20), the calculation is meaningful only when the node \(s_{j}\) is in the sensing-transmitting state, that is \(x_{j}=1\). To obtain \(\phi(x_{j})\), we first set the sensing area of node \(s_{j}\) as \(A_{j}\). The neighbors of \(s_{j}\) are defined as other CMs that are in the sensing-transmitting state: \(\mathcal{N}\big{(}x_{j}\big{)}=\{s_{k}\in\mathit{cluster}\ |\ x_{k}=1,\ k\
eq j\}\). If \(s_{j}\) is a redundant node, it needs to satisfy [30]:

\[A_{j}\subseteq\mathop{\cup}\limits_{s_{k}\in\mathcal{N}(s_{j})}\left(A_{k} \cap A_{j}\right). \tag{21}\]The overlap of two nodes' sensing areas can be simplified as a sector, which can be represented by a central angle, as shown in Figure 7. Therefore, calculating the overlapping coverage area of a node can be equivalent to calculating the sum of the central angles of the neighbors covering the node, which is given by:

\[\phi(x_{j})=\mathop{\cup}\limits_{s_{k}\in\mathcal{N}(s_{j})}\theta_{A_{k}\cap A _{j}}\, \tag{22}\]

According to Equations (17)-(20), we can define the linear 0-1 programming problem to be solved as:

\[min\ \ F(x)=\sum\limits_{j=1}^{n}e_{j}x_{j} \tag{23}\]

\[s.t.\left\{ \begin{array}{c}\sum\limits_{j=1}^{n}x_{j}\geq n_{0}\\ x_{j}\cdot\phi(x_{j})\leq 2\pi\\ x_{j}=0\ \ or\ \ 1\end{array} \right.. \tag{24}\]

To solve this node state scheduling problem, we adopt the implicit enumeration method [31]. First, we find a feasible solution tentatively, that is, by selecting \(q\) CMs to be in the sensing-transmitting state, satisfying the constraints. Then we calculate the objective value of this feasible solution. Assuming \(F=\mathcal{Q}\) is calculated, we will add it to the constraints as a filtering constraint, and (24) will become:

\[s.t.\left\{ \begin{array}{c}\sum\limits_{j=1}^{n}x_{j}\geq n_{0}\\ x_{j}\cdot\phi(x_{j})\leq 2\pi\\ \sum\limits_{j=1}^{n}e_{h}x_{h}\leq\mathcal{Q}\\ x_{j}=0\ \ or\ \ 1\end{array} \right.. \tag{25}\]

Figure 7: The overlap of the sensing area of two nodes. In this way, we have four constraints and enumerate all the \(2^{n}\) solutions and filter out infeasible solutions using the constraints in (25). For each solution, we substitute it into each constraint to see if it satisfies the inequalities. If one constraint is not satisfied, the other constraints do not have to be checked. Due to the increased constraint, the number of computations can be reduced. During the calculation, if the objective value of a particular solution is \(\mathcal{U}\), and \(\mathcal{U}<\mathcal{Q}\), then we replace the third constraint in (25) with \(\sum\limits_{h=1}^{q}\,e_{h}x_{h}\leq\mathcal{U}\). This improvement of filter conditions can further reduce the amount of calculation. The optimal solution can be obtained by iterating the above calculation until the value of \(F\) cannot be smaller. The optimal solution denotes the optimal status of the CMs in the cluster. The intra-cluster state scheduling process is described in Algorithm 2. ```
0: Cluster Member Number \(n\), node in the sensing or sensing-transmitting state \(x=0\) or \(1\)
0: Optimal State of Cluster Members
1: Parameter Initialization: Consumed Energy \(e\), Minimum Number of Nodes to Transmit \(n_{0}\)
2: Initial Solution: Randomly selecting \(q\) CMs to be in the sensing-transmitting state, satisfying Equation (24), and calculate the objective function value as \(Q\) using Equation (23). 3: Add \(F\leq Q\) to the constraints, Equation (24) becomes Equation (25)
4: Enumerate \(2^{n}\) solutions.