# Motion Planning in UAV-Aided Data Collection with Dynamic Jamming
## 3 Global Replanning Based on CFC
### Global Path Planning Based on SCA  
For the nonconvex problem \(P0\), we perform the following for the nonconvex constraints (26i), constraints (26i) and constraints (26k). We first introduce the auxiliary variables \(\kappa[n]\) for constraints (26i) and constraints (26i) in the following equivalent form: For the non-convex problem P0, we perform the following steps for the non-convex constraints (26i) and constraints (26i). First, we introduce the auxiliary variables \(\kappa[n]\) for the constraints (26i) and (26i) with the following equivalent form:  
\[\sum_{n=1}^{N}\kappa_{k}[n]\cdot r_{k}[n]\cdot\delta\geqslant Q_{th},\forall k \in\mathcal{K}, \tag{28}\]  
\[\kappa_{k}[n]\cdot r_{k}[n]\geqslant\kappa_{k}[n]\cdot r_{th},\forall k\in \mathcal{K}, \tag{29}\]  
\[\kappa_{k}[n]\in\{0,1\},\forall k\in\mathcal{K}. \tag{30}\]  
**Proof. ** Constraint (28) shows that \(\kappa_{k}[n]\) can only be \(1\) when \(r_{k}[n]\geqslant r_{th}\). Otherwise, it is zero. The minimum amount of data collected constantly must be higher than \(Q_{th}\), and the constraint (28) is only valid when some \(\kappa_{k}[n]\) equal to \(1\). Hence, the constraints (28) and (29) can make the UAV \"pass through\" the \"surrounding\" of each SN during its flight while remaining at a low velocity for a period of time based on the minimum amount of collected data. However, constraints (28) and (29) are still non-convex. For \(r_{k}[n]\), the auxiliary variables \(S_{k}[n]\) and \(I_{k}[n]\) are introduced to obtain the following relation:  
\[\sum_{n=1}^{N}\kappa_{k}[n]\cdot r_{k}[n]\cdot\delta\geqslant Q_{th}, \tag{31}\]  
where \(r_{k}[n]\geqslant r_{k}[n]=B\cdot\log_{2}(1+\frac{S_{k}[n]^{-1}}{I_{k}[n]})\). The auxiliary variables \(S_{k}[n]\) and \(I_{k}[n]\) must meet the following conditions:  
\[S_{k}[n]^{-1}\leqslant\beta_{0}p_{0}d_{k}[n]^{-\kappa},S_{k}[n]>0, \tag{32}\]  
Figure 2: Block diagram of presented autonomous system. \[I_{k}[n]\geqslant\sum_{j}\beta_{j}p_{j}d_{j}[n]^{-\alpha}+\sigma^{2}. \tag{33}\]  
For constraint (32), we can rewrite the convex constraint as follows:  
\[d_{k}[n]^{\alpha}\text{ - }\beta_{0}p_{0}S_{k}[n]\leqslant 0,S_{k}[n]>0. \tag{34}\]  
However, constraint (33) is still non-convex. We perform a first-order Taylor expansion:  
\[ \begin{split}& d_{j}^{-\alpha}[n]=\left[(p_{x}-p_{jx})^{2}+(p_{y}-p_{ jy})^{2}+(p_{z}-p_{jz})^{2}\right]^{-\alpha/2}\\ &\geqslant L_{j}^{r}[n]^{-\alpha/2}-\alpha L_{j}^{r}[n]^{-\alpha /2-1}\left[(p^{r}-p_{j})^{T}\cdot(p-p^{r})\right],\end{split}  \tag{35}\]  
where \(L_{j}[n]=(p_{x}-p_{jx})^{2}+(p_{y}-p_{jy})^{2}+(p_{z}-p_{jz})^{2}\), and \(L_{j}^{r}[n]\), \(p^{r}\) are the initial values of the \(r\)-th iteration. This results in a new convex constraint (36), which is a relaxation of constraint (33):  
\[I_{k}[n]\geqslant\sum_{j}\beta_{j}p_{j}\Big{(}L_{j}^{r}[n]^{-\alpha/2}-A_{j}^{ r}[n]\Big{)}+\sigma^{2}, \tag{36}\]  
where \(A_{j}^{r}[n]=\alpha L_{j}^{r}[n]^{-\alpha/2-1}\Big{[}(p^{r}-p_{j})^{T}\cdot(p- p^{r})\Big{]}\). For constraint (31), we further introduce an auxiliary variable \(\varepsilon[n]\), that is less than the first-order Taylor expansion of \(\bar{r}_{k}[n]\). \[ \begin{split}&\varepsilon_{k}[n]\leqslant U_{k}^{r}[n]+V_{k}^{r}[n] (S_{k}[n]-S_{k}^{r}[n])\\ &+W_{k}^{r}[n](I_{k}[n]-I_{k}^{r}[n]),\end{split}  \tag{37}\]  
where \(U_{k}^{r}[n]=B\cdot\log_{2}(1+\frac{1}{S_{k}^{r}[n]\bar{I}[n]})\), \(V_{k}^{r}[n]=\frac{-B\cdot\log_{2}e}{S_{k}^{r}[n]+S_{k}^{r}[n]^{2}\bar{I}[n]}\), and \(W_{k}^{r}[n]=\frac{-B\cdot\log_{2}e}{I_{k}^{r}[n]+I_{k}^{r}[n]^{2}S_{k}^{r}[n]}\). Therefore, a further relaxed form of the constraint (31) can be obtained  
\[\sum_{n=1}^{N}\kappa_{k}[n]\cdot\varepsilon_{k}[n]\cdot\delta\geqslant Q_{th}, \tag{38}\]  
Constraint (38) is still a non-convex constraint owing to the product of the indicator function variable \(\kappa_{k}[n]\) and variable \(\varepsilon_{k}[n]\).