## 2 Preliminaries and Problem Statement
### Path Planning  
In this study, the UAVs are modeled as curvature-bounded vehicles and their kinematics are given by:  
\[\left\{\begin{array}{l}\dot{x}_{i}=V\cos\varphi_{i}\\ \dot{y}_{i}=V\sin\varphi_{i}\,\quad i\in\mathcal{N}\\ \dot{\varphi}_{i}\big{|}\leq\mu_{\max}\end{array}\right. \tag{13}\]  
where \(x_{i}\) and \(y_{i}\) represent the coordinates of UAV \(i\); \(V\) represents the flying speed of the UAVs; \(\varphi_{i}\) represents the heading angle of UAV \(i\); \(\mu_{\max}\) represents the maximum curvature of the heading angle.  
To formulate the path planning problem, paths should be described correctly. Suppose that \(T\mathbb{R}^{2}\) is the tangent bundle of \(\mathbb{R}^{2}\) and \(\mathbf{a}=(A,\overset{\rightarrow}{a})\), \(\mathbf{b}=(B,\overset{\rightarrow}{b})\) are the two elements in \(T\mathbb{R}^{2}\). Thus \(A\),\(B\) are two points in \(\mathbb{R}^{2}\) and \(\overset{\rightarrow}{a},\overset{\rightarrow}{b}\) are the corresponding velocity vectors, respectively. Given \(\mathbf{a},\mathbf{b}\in T\mathbb{R}^{2}\), the curve \(p(z):[0,Z]\rightarrow\mathbb{R}^{2}\) is defined as a path that connects \(\mathbf{a}\) and \(\mathbf{b}\) for a UAV, and if the following conditions hold:  
1. \(\forall z\in[0,Z]\), \(\|p^{\prime}(z)\|=1\), \(\|p^{\prime\prime}(z)\|\leq\mu_{\max}\).
2. \(p(0)=A,p^{\prime}(0)=\overset{\rightarrow}{a},p(Z)=B,p^{\prime}(Z)=\overset{ \rightarrow}{b}\).
3. \(p(z)\) is a continuous curve and it has a continuous first derivative.  
As mentioned above, path planning is performed based on the recovery position series and the expected path length is generated in the recovery scheduling process. Given \(S\in S\) that is obtained in the scheduling layer, there exist  
\[S\overset{S}{\longrightarrow}\mathbb{R}^{+}, \tag{14}\]  
\[P\overset{\mathcal{L}}{\longrightarrow}\mathbb{R}^{+} \tag{15}\]  
where \(P\) indicates the set of flyable paths for a UAV; \(S\) represents the mapping from the recovery sequence to the lengths of the recovery paths; the function \(\mathcal{L}\) represents the mapping from the paths in \(P\) to their lengths.  
Based on the above definitions, the path planning process of a UAV focuses on searching for a curvature-bounded path with a minimum deviation from the expected path length when satisfying the boundary point constraints. Thus, the path planning problem can be formulated as  
\[\min D(\mathcal{L}(p)) \tag{16}\]  
subject to  
\[(p(0),p^{\prime}(0))=\mathbf{a}, \tag{17}\]  
\[(p(\mathcal{L}(p  
This module aims to find the optimal recovery sequence and generate the recovery position series and the expected recovery path lengths, and passes the solution to the next layer. The second layer, which is another major concern of this study, is the path planning module. The path planning module designs the desired paths with an expected length based on the recovery position series and the path lengths are obtained in the previous layer. Moreover, it generates and passes the control parameters to the control module of each UAV.  
It is impractical to connect the scheduling layer and the path planning layer with a simple architecture because the two layers are coupled. The performance of the scheduling layer is affected by the path planning layer since the potential path estimation results can influence the results of the recovery time window calculation. On the contrary, the frequent path planning based on the results of the scheduling layer costs a considerable amount of time. To overcome these difficulties, the recovery planning framework is proposed, and it is shown in Figure 2.