# LEACH-MTC: A Network Energy Optimization Algorithm Constraint as Moving Target Prediction
## 3 Method: LEACH-MTC Based on Moving Target State Prediction
### Construction of Elliptical Area Based on State Prediction  
Intuitively, reasonably choosing work nodes close to target can effectively reduce redundant information related to observed target, thus enhancing energy efficiency of WSNs. Therefore, we can pre-awaken sleeping nodes nearby the direction of the moving target to perform monitor work. Specifically, to make sure working nodes (i.e., awakened sleeping nodes) are located in direction of the moving target, the coverage area of the awakened sleeping nodes is defined as an ellipse whose major axis coincides with the direction of target moving. According to the network monitoring performance parameter \(C_{w}\) and the target state prediction, an elliptical coverage area can be constructed. The size of the coverage area can be represented by an elliptical area \(S_{ab}\) as follows:  
\[S_{ab}=\pi ab{=}\frac{C_{w}S_{M}}{m} \tag{15}\]  
where \(a\) and \(b\) represent the long semi-axis and the short semi-axis of the elliptical coverage area, respectively. \(C_{w}\) represents the monitoring performance of network. \(S_{M}\) represents the monitored area, and represents the total number of nodes deployed in the network. The schematic figure based on the motion trajectory of moving target and the elliptical coverage area of the working node is shown in Figure 2.  
In Figure 2, the nodes inside the ellipse are in the working state, while the nodes outside the ellipse are in the sleep state. \(O\big{(}O_{x},O_{y}\big{)}\) is the coordinate of the elliptical center. \(F_{1}\big{(}T_{x},T_{y}\big{)}\) is the back focus of the ellipse, and it represents the current position of the moving target. The long axis of elliptical is in the direction of the target motion, and \(v\) represents the current speed of the moving target, and its direction is the tangential direction of the target moving trajectory. \(F_{2}\big{(}J_{x},J_{y}\big{)}\) is the front focus of the ellipse and it represents the prediction position of the moving target. \(a\) is the long semi-axis of the ellipse, \(b\) is the short semi-axis of the ellipse, and \(c\) is the semi-focal length of the ellipse. If \(P\) is any point on the elliptic curve, it satisfies the sum of the distances to the two focal points equal to the long axis, which is Equation (16):  
\[|PF_{1}|+|PF_{2}|=2a \tag{16}\]  
otherwise, if \(P\) is any point inside the ellipse, it has the Equation (17):  
\[|PF_{1}|+|PF_{2}|<2a \tag{17}\]  
where \(|PF_{1}|\) is the distances between the point \(P\) and the focal point \(F_{1}\), and \(|PF_{2}|\) is the distance between the point \(P\) and the focal point \(F_{2}\). According to Equations (16) and (17), this paper determines whether the network node in the working state or in the sleep state. The distribution of network nodes as shown in Figure 3.  
Figure 3: Distribution of working nodes and sleep nodes. Figure 2: Target motion trajectory and elliptical area. In Figure 3, the elliptical area is the area of the coverage area formed by the working node, where F1 is the current position of the moving target and F2 is another focus about the center \(O\) of ellipse. Based on the Figure 2, at time \(k\), we assume that the coordinate of the elliptical center is \(O\big{(}O_{x}(k),O_{y}(k)\big{)}\), the two focal coordinates of the ellipse are \(F_{1}\big{(}T_{x}(k),T_{y}(k)\big{)}\) and \(F_{2}\big{(}I_{x}(k),I_{y}(k)\big{)}\), respectively. According to the state prediction of the moving target, the predicted position coordinate can be received as \(\big{(}T_{x}(k+\delta_{t}),T_{y}(k+\delta_{t})\big{)}\) at time \(k+\delta_{t}\). Thus, the magnitude of speed \(v(k)\) as follows:  
\[|v(k)|=\frac{\sqrt{\big{(}T_{x}(k+\delta_{t})-T_{x}(k)\big{)}^{2}+\big{(}T_{y} (k+\delta_{t})-T_{y}(k)\big{)}^{2}}}{\delta_{t}} \tag{18}\]  
This paper assumes that the shape of the elliptical coverage area formed is related to the target motion velocity, then the semi-focal length of the ellipse \(c(k)\) is received by the Equation (19).