# Three-Dimensional Trajectory Optimization for UAV-Based Post-Disaster Data Collection
## 3 System Architecture
The starting point of the UAV is set in the center of the area as \((250,250,0)\), and the location of the UAV at time \(t\) is assumed as \((X_{t}^{\rm UAV},Y_{t}^{\rm UAV},H_{t}^{\rm UAV})\), while the location of the \(m\)\((m\leq M)\)-UE is assumed as \((X^{m-\rm UE},Y^{m-\rm UE},H^{m-\rm UE})\). Note that for simplicity, the UEs are assumed not to change their location. We adopt the 2.4 GHz band for data transmission, because it can be easily integrated with the hardware of UEs nowadays. For simplicity, the shape of the buildings is ignored. Thus, the data transmission model between the UAV and the UEs is defined with the possibilities of Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) propagation. As shown in Figure 5, LoS propagation occurs only when there is no obstacle between the UAV and the UE. Figure 2: Overview of the system. The LoS and NLoS propagation losses between the UAV and the _m_-UE at time \(t\)\(L_{m,t}^{\text{LoS}}\), \(L_{m,t}^{\text{NLoS}}\) are given below:  
\[L_{m,t}^{\text{LoS}}=20\log_{10}(\frac{4\pi f_{c}}{c}d_{m,t})+\eta_{\text{LoS}} \left[\text{dB}\right] \tag{1}\]  
\[L_{m,t}^{\text{NLoS}}=20\log_{10}(\frac{4\pi f_{c}}{c}d_{m,t})+\eta_{\text{NLoS }}\left[\text{dB}\right] \tag{2}\]  
Figure 4: Sample of UE distribution (\(M=40\).) Figure 3: Process of the UAV-based data collection mission. where \(20\log_{10}(\frac{4\pi f_{c}}{c}d_{m,t})\) is the free space propagation loss (FSPL) [34], \(\eta_{\text{LoS}}\), \(\eta_{\text{NLoS}}\) are the LoS and NLoS excessive loss, respectively, and \(d_{m,t}\) is the Euclidean distance between the UAV and the \(m\)-UE at time \(t\). From these equations, the longer the distance, the higher the propagation loss becomes. According to [25], the probabilities of LoS and NLoS propagation \(P_{m,t}^{\text{LoS}}\), \(P_{m,t}^{\text{NLoS}}\) are given below:  
\[P_{m,t}^{\text{LoS}}=\frac{1}{1+a\exp(-b(\theta_{m,t}-a))} \tag{3}\]  
\[P_{m,t}^{\text{NLoS}}=1-P_{m,t}^{\text{LoS}} \tag{4}\]  
where \(a\), \(b\) are environment S-curve parameters, and \(\theta_{m,t}\) is the elevation angle from the \(m\)-UE to the UAV. Thus, the propagation loss between the UAV and the \(m\)-UE at time \(t\)\(L_{m,t}\) can be calculated by  
\[L_{m,t}=P_{m,t}^{\text{LoS}}L_{m,t}^{\text{LoS}}+P_{m,t}^{\text{NLoS}}L_{m,t} ^{\text{NLoS}}\left[\text{dB}\right] \tag{5}\]  
From the Shannon-Hartley Theorem [35], the maximum data rate between the UAV and the \(m\)-UE at time \(t\)\(C_{m,t}\) can be calculated by  
\[C_{m,t}=\alpha_{m,t}B\log_{2}\left(1+\frac{P_{\text{tx}}\times 10^{\frac{L_{m,t }}{10}}}{\sigma^{2}}\right) \tag{6}\]  
where \(P_{\text{tx}}\) and \(\sigma^{2}\) respectively denote the transmit power of the \(m\)-UE and the noise power at the UAV's receiver, \(B\) is the system bandwidth, and \(\alpha_{m,t}\) is an indicator decided by the connection status between the UAV and \(m\)-UE as depicted in Figure 6, which is defined as below:  
\[\alpha_{m,t}=\left\{ \begin{array}{ll}1&\text{(if $m$-UE at transmission mode)}\\ 0&\text{(if $m$-UE at idle mode)}\end{array} \right. \tag{7}\]  
In other words, the data rate is zero when the \(m\)-UE is in idle mode or sleep mode. Figure 5: LoS and NLoS propagation.