# A Novel Dynamic Transmission Power of Cluster Heads Based Clustering Scheme
## 4 Proposed Approach
### Prediction of Cluster Member Movement
#### 4.5.2 Mobility Model  
In the FANET, the node movement is regular and cannot be described by a completely random entity movement model. The group mobility is also taken into account. We applied the Gauss-Markov model [26] to represent the movement trajectory of nodes, which overcomes the defect of the sudden stop and turn. So this model is more consistent with reality.  
For the convenience of analysis, this paper assumes that the UAV node can obtain its position and speed information through GPS. Since we set the UAVs to fly at the same altitude, so we only concentrate on the \(x\) and \(y\) axes of the coordinate system. Moreover, the CH is regarded as static, and the coordinate system is set with it as the center to analyze the node movement.  
**Theory**: When calculating the new speed and direction, each node has an initial speed and direction, and an average speed and direction. In each time step, we calculated the new speed and direction at the next moment, and repeated the operation.  
**Define**: \(Q\) is a square with sides of length \(S\), which represents the moving region of the node, where \(S\) is no more than the communication diameter of the node. It was assumed that all UAVs have a velocity with the same direction and magnitude at the beginning of the experiment. We removed this velocity and just considered the relative motion of each UAV so that the movement of UAVs was easier to analyze. At the beginning, \(n\) nodes are evenly distributed in the set area. Additionally, at the same time, UAVs will start to move according to the Gauss-Markov model, while being restricted in the area \(Q\) based on the actual situation. Since the UAV is regarded as a point and the scale of FANET is not particularly large, the case of collision between the UAVs is not considered. \(U\)=(\(u_{1}\),\(u_{2}\)...\(u_{n}\)), represents the collection of nodes, \(u_{i}\) is the \(i\)-th node, \(\forall u_{i}\in U\); \((x_{i},y_{i})\in Q\) represents the position of the \(i\)-th node; \(v_{i}\) represents the speed of the \(i\)-th node; and \(\theta_{i}\) represents the angle between velocity reversal and the X-axis. The time interval of HELLO message exchange is discretized, i.e., \(\Delta T=t_{n}-t_{n-1}=k\Delta t\), where \(\Delta t\) is time step, \(k\) is the number of time segments. At time \(t_{n}\) the CH obtains the CM\({}_{i}\)'s velocity \(v_{i}(t_{n})\), direction angle \(\theta_{i}(t_{n})\) and the position \((x_{i}(t_{n}),y_{i}(t_{n}))\) from HELLO message exchange. Hence, the CM\({}_{i}\)'s velocity and direction angle at time \((t_{n}+m\Delta t)\) can be calculated as  
\[\begin{cases}v_{i}(t_{n}+m\Delta t)=\alpha v_{i}(t_{n}+(m-1)\Delta t)+(1-\alpha )\vartheta+\sqrt{1-\alpha^{2}}v_{i}(t_{n}+(m-1)\Delta t)\\ \theta_{i}(t_{n}+m\Delta t)=\alpha\theta_{i}(t_{n}+(m-1)\Delta t)+(1-\alpha) \vartheta+\sqrt{1-\alpha^{2}}\theta_{i}(t_{n}+(m-1)\Delta t),\end{cases} \tag{22}\]  
Figure 4: CM\({}_{5}\) moves from yellow grid to red grid.  
where \(\alpha\) is the relevant memory parameter, denoting the correlation between the current time slot and the previous time slot in speed and direction, and \(0\leq\alpha\leq 1\). The strength of the correlation can be changed by adjusting the value of \(\alpha\); \(1\leq m\leq k\), \(m\) represents the \(m\)-th time slot; and \(\bar{\theta}\) represent the average velocity and the average directional angle over the period. Therefore, according to (22), we can predict the velocity and direction angle of the node at any time in the periodic interval, and the position of CM\({}_{i}\) at \((t_{n}+m\Delta t)\) can be obtained by:  
\[\left\{\begin{aligned} x_{i}(t_{n}+m\Delta t)& =x_{i}(t_{n})+\sum_{l=1}^{m}v_{i}(t_{n}+l\Delta t)\Delta t\cos \theta_{i}(t_{n}+l\Delta t)\\ y_{i}(t_{n}+m\Delta t)&=y_{i}(t_{n})+\sum_{l=1}^{m }v_{i}(t_{n}+l\Delta t)\Delta t\sin\theta_{i}(t_{n}+l\Delta t).\end{aligned}\right. \tag{23}\]