# Optimal Position and Target Rate for Covert Communication in UAV-Assisted Uplink RSMA Systems
## 5 Numerical Results  
In this section, we present numerical results to investigate the performance of the proposed covert communication scheme. Unless otherwise stated, we set \(\beta_{0}=-20\) dB, \(\mathbf{q}_{b}=[0,0,10]\) m, \(\mathbf{q}_{w}=[0,100,0]\) m, \(\mathbf{q}_{2}=[50,50,0]\) m, \(z_{1}=25\) m, \(P_{1}^{\max}=10\) W, \(P_{1}^{\min}=1\) W, \(P_{2}=2\) W, \(\sigma_{b}^{2}=\sigma_{w}^{2}=-60\) dBm. In this section, we compared the proposed method with NOMA systems, which can be regarded as a special case of RSMA where all power is allocated to \(\mathbf{s}_{11}\).  
Figure 4 shows the maximum ESR versus the expected minimum DEP with different ECR constraints in RSMA and NOMA systems, where \(\hat{R}_{2}=3\) bpcu. We observe that the curves remain stable initially, but decrease as the expected minimum DEP increases. When the covertness constraint is loose (\(t\geq x_{w}^{2}+t_{w}^{2}\)), the placement of U1 remains unchanged at \(x_{1}=y_{1}=0\). As the covertness constraint increases, U1 moves closer to Willie while moving further from the BS, thereby leading to a decrement in ESR. Additionally, a higher ECR constraint results in a lower ESR, implying that \(\mathbb{O}_{11}^{1}\) is not zero at the maximum ESR. Moreover, the proposed scheme has a higher ESR than NOMA with the same minimum DEP.  
Figure 5 depicts \(\mathbb{O}_{11}^{1}\) of \(\hat{R}_{11}^{\epsilon}\) versus \(\hat{R}_{2}\) for different covertness constraints in RSMA and NOMA systems. In this case, we do not consider the ESR constraint. We find that NOMA has a higher outage probability than RSMA to maximize the ESR. Moreover, as \(\hat{R}_{2}\) increases, the \(\mathbb{O}_{11}^{1}\) of RSMA increases, while that of NOMA decreases. When \(\tau=0\) (i.e., at maximum \(\hat{R}_{2}\)), the \(\mathbb{O}_{11}^{1}\) of both systems are equal. In NOMA systems, fixed power is allocated to \(\mathbf{s}_{11}\) (\(\alpha=1\)). As \(\hat{R}_{2}\) increases, decreasing \(\mathbb{O}_{11}^{1}\) to increase \(\overline{R}_{2}\) will result in a larger ESR. In RSMA  
Figure 4: The maximum ESR versus the expected minimum DEP for different ECR constraints in RSMA and NOMA systems, where \(\hat{R}_{2}=3\) bpcu.  
systems, with the increment of \(\hat{R}_{2}\), more power is allocated to \(\mathbf{s}_{11}\), and more outages can be tolerated to increase the ESR.  
Figure 6 plots the maximum ESR versus \(\hat{R}_{2}\) for different covertness constraints, where \(\epsilon=0.8\hat{R}_{2}\). Similar to Figure 5, we observe that the maximum ESR of RSMA increases while that of NOMA decreases as \(\hat{R}_{2}\) increases. Specifically, when the covertness constraint is looser, RSMA achieves a higher \(\mathbb{O}11^{1}\) and ESR compared to NOMA. This can be attributed to the fact that the channel gain between the BS and U1 is stronger. The stronger channel gain results in \(R_{1}\) playing a more crucial role in determining ESR, thus leading to an increase in \(\mathbb{O}_{11}^{1}\).  
Figure 5: The outage probability of \(\hat{R}_{11}^{4}\) under \(\mathcal{H}_{1}\) for different covertness constraints in RSMA and NOMA systems, where \(\epsilon=0.8\hat{R}_{2}\).  
Figure 6: The maximum ESR versus the target rate of U2 \(\hat{R}_{2}\) for different covertness constraints in RSMA and NOMA systems, where \(\epsilon=0.8\hat{R}_{2}\).