## 2 System Model and Problem Formulation
### Problem Formulation  
Considering the NOMA method in each beam, intra-beam superposition coding at the transmitter and SIC at the receiver are performed. In the case of uplink mmWave NOMA, the users begin to transmit uplink signals \(x_{1}\) and \(x_{2}\) at the same time and in the same frequency band. For the proposed multiple-UAV relay network shown in Figure 1, each UAV decodes the mixed signals of the two users in its beam (called, for example, UE\({}_{1}\) and UE\({}_{2}\)) and then transmits the superimposed signal \(\mathbf{s}\) to the BS according to  
\[\mathbf{s}=\sqrt{\mathbf{P_{1}}}\hat{x_{1}}+\sqrt{\mathbf{P_{2}}}\hat{x_{2}}, \tag{10}\]  
where \(\hat{x_{1}}\) and \(\hat{x_{2}}\) are the decoded signals of UE\({}_{1}\) and UE\({}_{2}\), respectively, and \(\mathbb{E}[|\hat{x_{1}}|^{2}]=\mathbb{E}[|\hat{x_{2}}|^{2}]=1\) and \(P_{1}+P_{2}\) cannot exceed the maximum transmission power of the UAV.  
In the conventional NOMA with a single-antenna, usually the information of the user with a lower channel gain is decoded first to maximize the sum rate. In contrast, in mmWave NOMA, the decoding order depends on both channel gain and beamforminggain. Without loss of generality, assuming that UE\({}_{1}\) has a better channel condition with respect to UE\({}_{2}\) in the area covered by UAV\({}_{k}\)[4, 29], the achievable rates are  
\[R_{1}=log_{2}(1+\frac{|\mathbf{h_{1}^{H}U}|^{2}P_{1}}{|\mathbf{h_{2}^{H}U}|^{2}P _{2}+\sigma_{k}^{2}}) \tag{11}\]  
\[R_{2}=log_{2}(1+\frac{|\mathbf{h_{2}^{H}U}|^{2}P_{2}}{\sigma_{k}^{2}}), \tag{12}\]  
and the achievable rate from \(UAV_{k}\) is  
\[R_{UAV_{k}}=log_{2}(1+\frac{|\mathbf{h_{k}^{H}W}|^{2}P_{UAV_{k}}}{\sigma_{w}^{ 2}}), \tag{13}\]  
where \(P_{1}\), \(P_{2}\) and \(P_{UAV_{k}}\) are the transmission power of UE\({}_{1}\), UE\({}_{2}\) and UAV\({}_{k}\) respectively, \(\mathbf{h_{1}}\), \(\mathbf{h_{2}}\) are the channel response vector between the UE\({}_{1}\), UE\({}_{2}\) and UAV\({}_{k}\), \(\mathbf{h_{k}}\) is the channel response vector between the UAV\({}_{k}\) and the BS, \(\mathbf{U}\) and \(\mathbf{W}\) represent jointly the analog and digital precoding matrices at the UAV\({}_{k}\) and at the BS, whereas \(\sigma_{k}^{2}\) and \(\sigma_{w}^{2}\) are the power of the zero-mean additive Gaussian white noise at the UAV\({}_{k}\) and BS, respectively.  
Therefore, the available rate received at the BS considering the UAV\({}_{k}\) acting as a relay is  
\[R_{sum}=min(R_{1}+R_{2},R_{UAV_{k}}). \tag{14}\]  
To maximize the uplink average achievable sum rate of all terminals of the multiple-UAV-aided NOMA mmWave system by dynamically tracking the DoAs of multiple UAVs, the optimization problem can be formulated as  
\[\begin{split}\underset{\theta_{k}}{\text{Maximize}}& \sum_{k=1}^{K}R_{sum}\\ s.t.&\\ R_{1}&\geq\tilde{r}\\ R_{2}&\geq\tilde{r}\\ R_{sum}&\geq\tilde{r}\\ P_{1}+P_{2}&\leq P_{UAV_{k}}\\ P_{UAV_{1}}+P_{UAV_{2}}+\...\ P_{UAV_{k}}\leq P\end{split} \tag{15}\]  
where \(\tilde{r}\) denotes the minimal data rate constraint for each user and the last constraint indicates the transmitted power constraint with P being the maximum total transmitted power.  
This optimization problem is very complicated to be solved directly because the problem is non-convex and may not be converted to a convex problem with simple manipulations. Consequently, in order to validate the effectiveness of the proposed optimization problem and the accuracy of the derived analytical model, we resort to numerical simulations.