# Interference Avoidance through Periodic UAV Scheduling in RIS-Aided UAV Cluster Communications
## 2 System Models and Problem Formulations  
The UAV clusters are depicted in Figure 1, where the red arrows represent the inter-cluster interference. We focus on the downlink signal transmission, where the RIS equipped with RIS-UAV reflects the signal from the BS to the UAVs within the cluster. The RIS-UAV remains fixed in position relative to the BS and is equipped with \(L\) RIS arrays. Estimating the channel of the RIS is often a complex task due to the large number of RIS elements [27], leading to significant delays in UAV cluster transmission. To mitigate the overhead associated with channel estimation, we can leverage the positional information of the UAVs to simplify the computation of RIS coefficients. Figure 1: The system model of reconfigurable intelligent surface (RIS) aided multiple Unmanned aerial vehicle (UAV) clusters. Given the spatial positions of the RIS and a UAV, as \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\), respectively, the direction vector can be written as \(\mathbf{p_{2}}-\mathbf{p_{1}}=(x,y,z)\). The relative distance, zenith angle, and azimuth angle are provided as follows:  
\[ \begin{split} z&=\|\mathbf{p_{2}}-\mathbf{p_{1}}\|\\ \phi&=\arctan\Bigl{(}\frac{y}{x}\Bigr{)}\\ \theta&=\arctan\Biggl{(}\frac{z}{\sqrt{x^{2}+y^{2}} }\Biggr{)}.\end{split}  \tag{1}\]  
The horizontal and vertical angles of RIS are represented by \(\phi_{R}\) and \(\theta_{R}\), which are determined by the orientation and pitch angles of the RIS-UAV. It is assumed that the channel between the RIS and the other UAV is the LoS channel, and the RIS array is a rectangle array. By denoting \(\phi^{\prime}=\phi_{R}+\phi\), and \(\theta^{\prime}=\theta_{R}+\theta\), the channel can be modeled as [28]  
\[\mathbf{h}_{RU}=\alpha_{RU}\mathbf{d}\bigl{(}\phi^{\prime}\bigr{)}\otimes\mathbf{d}\bigl{(} \theta^{\prime}\bigr{)}, \tag{2}\]  
where \(\alpha_{RU}\) is the channel coefficient. \(\mathbf{d}(\phi^{\prime})\) and \(\mathbf{d}(\theta^{\prime})\) are the steering vectors with the horizontal angle \(\phi^{\prime}\) and vertical angle \(\theta^{\prime}\). These steering vectors can be approximated using discrete Fourier transform (DFT) weights. Assuming that both BS and UAV have a single antenna, and RIS consists of \(N_{R}\) elements, the channel between the BS and RIS is denoted as \(\mathbf{h}_{BR}\). The received signal at the UAV can be expressed as  
\[y_{U}=\mathrm{vec}\{\alpha_{RU}\mathbf{d}\bigl{(}\phi^{\prime}\bigr{)}\otimes\bm {d}\bigl{(}\theta^{\prime}\bigr{)}\}\mathbf{c}_{R}\odot\mathbf{h}_{BR}\hat{s}+\hat{n}, \tag{3}\]  
where \(\otimes\) is the Kronecker product, \(\odot\) is the element-wise product, \(\hat{s}\) is the transmitted symbol and \(\hat{n}\) is the Gaussian noise. \(\mathrm{vec}\{\{}\}\) denotes the vectorization operation, and \(\mathbf{c}_{R}\) are the coefficients of RIS. As the motion of the UAV causes rapid changes in \(\alpha_{RU}\), while \(\mathbf{h}_{BR}\) experiences slower changes, it is possible to set the RIS coefficients to match the cascaded channel without considering the coefficient \(\alpha_{RU}\). Therefore, the RIS coefficients should be chosen to satisfy the condition  
\[\beta\tilde{\varepsilon}_{R}\odot\mathbf{h}_{BR}=\bigl{(}\mathrm{vec}\{\{}\mathbf{d} \bigl{(}\phi^{\prime}\bigr{)}\otimes\mathbf{d}\bigl{(}\theta^{\prime}\bigr{)}\} \bigr{)}^{H}, \tag{4}\]  
where \(\beta\) is a normalization parameter to ensure \(\|\tilde{\varepsilon}_{R}\|^{2}=1\). The coefficients \(\tilde{\varepsilon}_{R}\) for all spacial positions are calculated based on \(\mathbf{h}_{BR}\) at the BS, and transmitted to the RIS-UAV. According to (4), the RIS coefficients enable the RIS to generate a directional beam that points toward a specific UAV. Thus, each UAV is associated with its corresponding beam. Consequently, in this system, the inter-cluster interference is predominantly influenced by the positions of the UAVs and the beam directions, owing to the LoS characteristics of the UAV channels [29]. Therefore, the interference received by a UAV can be estimated based on its location [30].