# Motion Planning in UAV-Aided Data Collection with Dynamic Jamming
## 2 Problem Statement and Formulation
### Problem Formulation  
The optimization goal is to minimize the energy consumption and flight time of the UAVs when performed data collection missions. The optimization goal is to keep the flight time and energy consumption of the UAV minimal while collecting data. The energy consumption of a UAV typically comprises two major components: the propulsion energy and communication-related energy [18]. The latter consists of the energy utilized for communication circuits, signal radiation/reception, signal processing, and so on. For the purpose of this study, we consider the communication-related energy constant. To maintain the UAV at high altitude and to support its motion, propulsion energy is consumed. Typically, the propulsion energy depends on the flight speed, acceleration, jerk of the UAV, and so on. We take interest in the effort or smoothness of the trajectory, i.e., the square \(L^{2}\)-norm of the control input \(j[n]\), \(a[n]\) and \(v[n]\), which represent the energy cost in a dynamic system. Hence, we express the energy optimization function as below:  
\[ \begin{split}& f_{1}(\mathbf{J})=\omega_{1}\mathbf{V}^{T} \mathbf{V}+\omega_{2}\mathbf{A}^{T}\mathbf{A}+\omega_{3}\mathbf{J}^{T}\mathbf{ J}\\ &=\mathbf{J}^{T}\Big{(}\omega_{1}\mathbf{T}_{v}^{T}\mathbf{T}_{v}+ \omega_{2}\mathbf{T}_{a}^{T}\mathbf{T}_{a}+\omega_{3}\mathbf{I}\Big{)}\mathbf{ J}+\\ & 2\Big{(}\omega_{1}\mathbf{B}_{v}^{T}\mathbf{T}_{v}+\omega_{2} \mathbf{B}_{a}^{T}\mathbf{T}_{a}\Big{)}\mathbf{J}+constant.\end{split}  \tag{24}\]  
where \(f_{1}(\mathbf{J})\) represents the smoothness and energy consumption of the UAV trajectory. \(\mathbf{I}\) is the identity matrix; \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\) are the weighting factors. The \(constant=\omega_{1}\mathbf{B}_{v}^{T}\mathbf{B}_{v}+\omega_{2}\mathbf{B}_{a}^{T} \mathbf{B}_{a}\) does not need to be considered in the optimization. Therefore, we formulate the optimization problem as follows:  
\[ \begin{split}\mathrm{P0}:\quad\min_{J,N}&\mathbf{J}^{T} \Big{(}\omega_{1}\mathbf{T}_{v}^{T}\mathbf{T}_{v}+\omega_{2}\mathbf{T}_{a}^{T} \mathbf{T}_{a}+\omega_{3}\mathbf{I}\Big{)}\mathbf{J}+\\ & 2\cdot\Big{(}\omega_{1}\mathbf{B}_{v}^{T}\mathbf{T}_{v}+\omega_{2} \mathbf{B}_{a}^{T}\mathbf{B}_{a}\Big{)}\mathbf{J}\end{split}  \tag{25}\]\[st: \mathbf{P}=\mathbf{T}_{p}\mathbf{J}+\mathbf{B}_{p} \tag{26a}\] \[\mathbf{V}=\mathbf{T}_{v}\mathbf{J}+\mathbf{B}_{v}\] (26b) \[\mathbf{A}=\mathbf{T}_{a}\mathbf{J}+\mathbf{B}_{a}\] (26c) \[-\mathbf{V}_{\max}-\mathbf{B}_{v}\leqslant\mathbf{T}_{v}\mathbf{ J}\leqslant\mathbf{V}_{\max}-\mathbf{B}_{v}\] (26d) \[-\mathbf{A}_{\max}-\mathbf{B}_{a}\leqslant\mathbf{T}_{a}\mathbf{ J}\leqslant\mathbf{A}_{\max}-\mathbf{B}_{a}\] (26e) \[\mathbf{H}_{\min}-\mathbf{B}_{p_{z}}\leqslant\mathbf{T}_{p}\mathbf{ J}_{z}\leqslant\mathbf{H}_{\max}-\mathbf{B}_{p_{z}}\] (26f) \[p[0]=p_{0},\,v[0]=v_{0},\,a[0]=a_{0}\] (26g) \[p[N]=p_{N},\,v[N]=v_{N},\,a[N]=a_{N}\] (26h) \[\sum_{n=1}^{N}\xi_{n}^{k}r_{k}[n]\delta\geqslant Q_{th}^{k}, \forall k\in\mathcal{K}\] (26i) \[\xi_{n}^{k}=\mathcal{I}\{\max(r_{k}[n]-r_{th},0)\},\forall n\in \mathcal{N}\] (26j) \[\mathbf{V}=\mathbf{T}_{v}\mathbf{J}+\mathbf{B}_{v}\]0  
and  
\[\mathcal{I}(x)= \begin{cases}1,x\geqslant 0\\ 0,x\leqslant 0\end{cases}  \tag{27}\]  
where (26a), (26b) and (26c) are the UAV motion primitives based on model prediction. (26d) and (26e) are the motion constraints of the UAV, which limit the range of the maximum velocity and maximum acceleration: and \(\mathbf{V}_{\max}=v_{\max}\cdot\mathbf{I}_{N\times 1}\) and \(\mathbf{A}_{\max}=a_{\max}\cdot\mathbf{I}_{N\times 1}\). (26f) is the height constraint of the UAV, and (26g) and (26h) are the start and end state constraints. (26i) represents the collected data amount of the \(k\)-th sensor, where \(\zeta_{n}^{k}\) is the effective collection indication, and \(Q_{th}^{k}\) is the minimum collection requirement. (26i) is the minimum communication rate of the sensor given according to the actual communication system.