# Joint Optimization of Relay Communication Rates in Clustered Drones under Interference Conditions
## 3 Problem Description
The movement constraints also need to satisfy the following:  
\[\left[\left(x_{r}^{uav}(1)-x^{init}\right)^{2}+(y_{r}^{uav}(1)-y^{init})^{2} \right]\leq\left(V_{max}^{hor}\delta_{T}\right)^{2}\forall k \tag{27}\]  
\[\left[\left(x^{term}-x_{r}^{uav}(K)\right)^{2}+(y^{term}-y_{r}^{uav}(K))^{2} \right]\leq\left(V_{max}^{hor}\delta_{T}\right)^{2}\ \forall k \tag{28}\]  
\[(z_{r}^{uav}(1)-z^{init})^{2}\leq(V_{max}^{ver}\delta_{T})^{2}\ \forall k \tag{29}\]  
\[0\leq E_{Rauv}^{s}\leq E_{max}^{Rauv} \tag{21}\]0  
Since the auxiliary communication operates on a two-hop model, there is a single time slot delay in relay transmission [24]. Therefore, the transmission power of the cluster dronesat the final time slot and the relay drone at the initial time slot is set to zero, expressed as follows:  
\[0\leq E_{Rauv}^{s}\leq E_{max}^{Rauv} \tag{21}\]1  
\[0\leq E_{Rauv}^{s}\leq E_{max}^{Rauv} \tag{21}\]2  
Here, \(K\) represents the last task slot. To ensure that the data volume returned by the cluster drones is sufficient for analysis and decision making at the ground station, it is required that the total communication rate of the cluster drones exceed a specified threshold \(R_{l}\), expressed as follows:  
\[0\leq E_{Rauv}^{s}\leq E_{max}^{Rauv} \tag{21}\]3  
Because of the relay drone's ability to switch between different modulation modes based on varying channel conditions and according to different modulation orders, the formula for calculating the bit error rate (BER) also varies [25], expressed as follows:  
\[ \begin{array}{c}BER_{\{2\}}\doteq erfc(\sqrt{2 SNR_{R,S}})\ BER_{\{4\}} \doteq 4 erfc(\sqrt{SNR_{R,S}}\\ BER_{\{16\}}\doteq 4 erfc(\sqrt{3 SNR_{R,S}/15}\ BER_{\{64\}}\doteq 4 erfc(\sqrt{3 SNR_{R,S}/63} \end{array}  \tag{34}\]  
\[ \begin{array}{c}erfc(x)=\frac{2}{\sqrt{\pi}}\sum\limits_{x}^{\infty}e^{-t^{2}} dt\end{array} \]  
Therefore, for each communication link formed between the cluster drones and the ground station, the BER must meet the threshold requirement \(BER_{q}\), as follows:  
\[BER_{\{2,4,16,64\}}\leq BER_{q} \tag{35}\]  
Thus, the optimization model for the communication rate of cluster drones under interference conditions is constructed as follows:  
\[C_{1}\]  
\[\max\limits_{\{P^{R},\lambda,U,Q\}}\]  
\[R_{S}^{C}\ \forall k \tag{36}\]  
\[s.t.\ \frac{P^{F}\beta^{o}}{D^{2}(k)P_{T}}\geq D_{n} \tag{37}\]  
\[Y=\{0,1\} \tag{38}\]  
\[0\leq\lambda_{i}(k)\leq 1 \tag{39}\]  
\[U=\{2,4,16,64\}\]  
\[(13)-(26),\ (28)\]  
The optimization model for the communication rate of cluster drones under interference conditions integrates several constraints, as follows: channel selection constraints (37) and (38), physical constraints of relay-assisted communication (20)-(30), and communication service performance constraints (31)-(33), (35), (39), and (40). The problem is formulated as a highly coupled nonlinear mixed-integer optimization model involving both discrete and continuous variables, which presents significant computational challenges due to its complexity. To effectively tackle this, the approach is divided into the following two strategic steps: firstly, employing an intelligent channel selection scheme to choose interference-free channels, and secondly, conducting the joint optimization of the relay's modulation method, transmission power, three-dimensional trajectory, and the cluster drones' power allocation factors. This method aims to maximize the data transmission rate of the relay drone under interference conditions while simplifying the optimization process and enhancing solution efficiency.