# Trajectory Optimization for UAV-Aided IoT Secure Communication Against Multiple Eavesdroppers +
## 2 System Model and Problem Formulation
### System Model  
Without loss of generality, the positions of all eavesdroppers, the communication destination, and the UAV are represented within a three-dimensional (3D) Cartesian coordinate system. As illustrated in Figure 1, in this UAV-aided IoT communication system, we assume that all \(K\) eavesdroppers (Eves) and the communication destination are located on the horizontal ground plane. Specifically, the set of eavesdroppers is defined as \(k\in\mathcal{K}\triangleq\{1,\ldots,K\}\). The fixed horizontal positions of the \(k^{th}\) eavesdropper and the communication destination are denoted by \(\mathbf{W}_{k}=[x_{k},y_{k}]^{T}\) and \(\mathbf{W}_{g}=[x_{g},y_{g}]^{T}\), respectively. The UAV is assumed to maintain a constant flight altitude, denoted by \(H\), throughout the entire flight duration \(T\). The UAV's instantaneous horizontal position at time \(t\) is denoted by \(\mathbf{U}_{q}[t]=[x_{q}(t),\,y_{q}(t)]\), where \(0\leq t\leq T\). To simplify the trajectory design, the continuous flight time \(T\) is uniformly divided into \(N\) time slots of equal duration, each with a length of \(\delta_{t}=T/N\). The number of time slots \(N\) is chosen to be sufficiently large so that the UAV's position can be approximated as constant within each interval. Consequently, the UAV's trajectory is discretized into \(N\) anchor points, each representing its horizontal position at the corresponding time slot. The resulting discrete trajectory is denoted as \(\mathbf{U}_{q}[n]=[x_{q}(n),\,y_{q}(n)]^{T}\), where \(n\in\mathcal{N}\triangleq\{1,\,2,\,\ldots,\,N\}\). The UAV's horizontal positions at the initial and final time instances of the total flight duration \(T\) are denoted by \(\mathbf{U}_{q}[I]=[x_{q}(I),\,y_{q}(I),\,H]^{T}\) and \(\mathbf{U}_{q}[F]=[x_{q}(F),\,y_{q}(F),\,H]^{T}\), respectively. Accordingly, the Euclidean distance between the UAV and the communication destination at time slot \(n\) is given by  
\[d_{ug}(n)=\sqrt{\left\|\mathbf{U}_{q}[n]-\mathbf{W}_{g}\right\|^{2}+H^{2}}, \quad\forall n\in\mathcal{N}, \tag{1}\]  
Here, \(\|\cdot\|\) represents the Euclidean norm. Likewise, the Euclidean distance between the UAV and the \(k^{th}\) eavesdropper at time slot \(n\) is expressed as  
\[d_{uk}(n)=\sqrt{\left\|\mathbf{U}_{q}[n]-\mathbf{W}_{k}\right\|^{2}+H^{2}}, \quad\forall k\in\mathcal{K},\,\,\forall n\in\mathcal{N}. \tag{2}\]  
When the UAV operates at a higher altitude in environments with minimal obstructions, the wireless channels between the UAV and both the communication destination and the eavesdroppers are predominantly LoS, and the widely adopted free-space path loss model is employed. Accordingly, the channel power gain between the UAV and the communication destination at time slot \(n\) can be expressed as  
\[h_{ug}(n)=\rho_{0}d_{ug}^{-2}(n)=\frac{\rho_{0}}{\left\|\mathbf{U}_{q}[n]- \mathbf{W}_{g}\right\|^{2}+H^{2}},\quad\forall n\in\mathcal{N}, \tag{3}\]  
Figure 1: UAV-aided secure communication model against eavesdropping in an IoT scenario. Likewise, the channel power gain between the UAV and the \(k^{th}\) eavesdropper at time slot \(n\) is expressed as  
\[h_{uk}(n)=\rho_{0}d_{uk}^{-2}(n)=\frac{\rho_{0}}{\left\|\mathbf{U}_{q}[n]-\mathbf{ W}_{k}\right\|^{2}+H^{2}},\quad\forall k\in\mathcal{K},\;\forall n\in\mathcal{N}. \tag{4}\]  
where \(\rho_{0}\) denotes the reference channel power gain measured at a distance of 1 m.  
Furthermore, let \(v_{\max}\) (m/s) denote the UAV's maximum allowable speed during any time slot. Accordingly, the maximum permissible flight distance within a single time slot is expressed as \(L_{\max}=\delta_{i}v_{\max}\). Moreover, to regulate velocity transitions between consecutive time slots, the UAV has a maximum allowable acceleration, denoted as \(a_{\max}\).