# Joint Optimization of Relay Communication Rates in Clustered Drones under Interference Conditions
## 3 Problem Description
 \tag{16}\]  
From this, it is evident that in time slot \(k\), the communication rate between the cluster drone \(i\) and the ground station is as follows:  
\[ \begin{split} R_{i}(k)&=B_{n}^{Raw}log_{2}(1+SNR_{ i,S})\ \forall k\\ &=B_{n}^{Raw}log_{2}(1+\frac{\lambda_{i}(k)P^{F}(k)P^{R}(k)\left( G_{t}G_{r}(\lambda^{\copyright)}^{2})^{2}}{I_{1}^{t}+I_{2}^{t}})\end{split}  \tag{17}\]  
Because of the dynamic characteristics of the relay drone, the channel conditions for its assisted communication change in real time. Therefore, dynamically adjusting the channel modulation method is an important strategy to flexibly respond to real-time channel variations. Hence, this paper introduces Quadrature Amplitude Modulation (QAM) technology into relay drone-assisted communication to maximize high-performance auxiliary communication. Consequently, in time slot \(k\), the communication rate between cluster drone \(i\) and the ground station becomes the following:  
\[ \begin{split} R_{i}(k)&=\eta B_{n}^{Raw}log_{2}(1+ SNR_{i})\ \forall k\\ &=\eta B_{n}^{Raw}log_{2}(1+\frac{\lambda_{i}(k)P^{F}(k)P^{R}(k) \left(G_{t}G_{r}(\lambda^{\copyright)}^{2})^{2}}{I_{1}^{t}+I_{2}^{t}})\\ \eta&=log_{2}(U(k))\end{split}  \tag{18}\]  
In this case, \(U(k)\) represents the modulation order at time \(k\), and the set of selectable modulation orders is given by \(\{2,4,16,64\}\). Therefore, the expression for the total communication rate between the cluster drone and the ground station over the entire mission period is as follows:  
\[ \begin{array}{rcl}R_{S}^{C}&=&\sum\limits_{k=1}^{K}\sum\limits_{i=1}^{M}R_{i}(k) \ \forall k\\ &=&\sum\limits_{k=1}^{K}\sum\limits_{i=1}^{M}log_{2}(U(k))B_{n}^{Rauv}log_{2}(1 +\frac{\lambda_{i}(k)P^{F}(k)P^{R}(k)(G_{t}G_{r}(\lambda^{G})^{2})^{2}}{I_{1}^{ t}+I_{2}^{t}I_{3}^{t}})\end{array}  \tag{19}\]  
From the above formula, it is evident that optimizing the total communication rate achievable by the ground station is dependent on the power allocation factors of the cluster drones, the modulation method, transmission power, and three-dimensional trajectory of the relay drone. However, it is also necessary to consider the physical constraints of the communication equipment and the performance constraints of the communication process. The transmission power of the relay drone must not exceed its maximum transmission power, \(P_{max}^{R}\), and the energy consumption for auxiliary communication must not exceed the maximum energy storage capacity of the relay drone, \(E_{max}^{Rauv}\), expressed as follows:  
\[0\leq P^{R}(k)\leq P_{max}^{R} \tag{20}\]  
\[0\leq E_{Rauv}^{s}\leq E_{max}^{Rauv} \tag{21}\]  
Similarly, similar constraints exist for all drones in the cluster, namely, the following:  
\[0\leq\lambda_{i}(k)P^{F}(k)\leq P_{max}^{F} \tag{22}\]  
\[0\leq E_{Fauv}^{s}\leq E_{max}^{Fauv} \tag{23}\]  
Additionally, in each time slot, the horizontal movement trajectory of the relay drone cannot exceed the trajectory length that would be covered at its maximum horizontal speed, \(V_{max}^{hor}\), expressed as follows:  
\[\left[(x_{r}^{uav}(k+1)-x_{r}^{uav}(k))^{2}+(y_{r}^{uav}(k+1)-y_{r}^{uav}(k))^ {2}\right]\leq\left(V_{max}^{hor}\delta_{T}\right)^{2}\forall k \tag{24}\]  
Similarly, the vertical movement trajectory of the relay drone cannot exceed the trajectory length that would be covered at its maximum vertical speed, \(V_{max}^{ver}\), and the altitude of the relay drone must be within an appropriate range, expressed as follows:  
\[(z_{r}^{uav}(k+1)-z_{r}^{uav}(k))^{2}\leq(V_{max}^{ver}\delta_{T})^{2}\ \forall k \tag{25}\]  
\[H_{l}\leq z_{r}^{uav}(k)\leq H_{h}\ \forall k \tag{26}\]  
In this case, \(H_{l},H_{h}\) represent the flight altitude limit for the relay drone, with the starting point coordinates given by \(UAV_{init}^{R}=(x^{init},y^{init},z^{init})\) and the ending point coordinates by \(UAV_{term}^{R}=(x^{term},y^{term},z^{term})\).