# Resilient Formation Reconfiguration for Leader-Follower Multi-UAVs
## 3 Problem Formulation
\tag{5}\]  
Next, model the kinematics of the leader in the leader-follower multi-UAVs as  
\[q_{0}(t)=q_{0}(t_{0})+p_{0}\cdot(t-t_{0}),t\geq t_{0}\geq 0, \tag{6}\]  
where \(q_{0}(t)\) denotes the 3D coordinate of the leader at time instant \(t\) while \(p_{0}\) refers to the constant linear velocity of the leader. **Remark 1**. : _It is well known that UAVs have slower translational mechanics and faster rotational dynamics. Therefore, the flight controller of the UAV adopts a cascade structure, in which the outer-loop controller focuses on position and speed while the inner-loop controller focuses on attitude stability. The article focuses on the formation reconfiguration of multi-UAVs, where UAVs can be regarded as point-mass systems. Therefore, the article only considers the outer-loop controller of the UAV kinematics model and does not consider the dynamic model attitude controller of each UAV._  
The communication topology of the leader-follower multi-UAVs are illustrated in Figure 1, where the leader is denoted by node 0 and the followers are associated with the rest of the nodes, and satisfies the following assumptions. **Assumption 1**. : _The scale of the leader-follower multi-UAVs is no less than five, with one UAV as the leader._  
**Assumption 2**. : _Only the followers numbered one and two can receive the signals from the leader._  
**Assumption 3**. : _Followers with adjacent odd (even) numbers are neighbors to each other._  
**Remark 2**. : _With the expanding scale of the leader-follower multi-UAVs, some problems are emerging gradually. On the one hand, the communication topology becomes increasingly complicated, which can further make the design of a formation controller more difficult. On the other hand, in terms of each follower, the plethora of neighbors can place more of the burden on topology reconfiguration. In light of these, the topological structure displayed in Figure 1 with minimal edges is devised based on the parity of vertex numbers, which can make the communication inside the multi-UAVs more effective and the formation reconfiguration more resilient._  
**Definition 1** (Leader-follower formation problem). : _Under the designed control input \(u_{i},i=1,\ldots,N\), \(\lim\limits_{t\to\infty}q_{0}-q_{i}=\Delta_{i}\) and \(\lim\limits_{t\to\infty}p_{0}-p_{i}=0\) for any initial condition, where \(\Delta_{i}\), \(i=1,\ldots,N\) represents the desired distance of the \(i\)th follower with respect to the leader._  
**Definition 2** (Neighbor). : _In a topology, node \(j\) is a neighbor of node \(i\) (\(i,j=0,\ldots,N\), \(i\
eq j\)) in the sense that node \(i\) can receive and use the information of node \(j\) as a control input. The neighbor set of node \(i\) is denoted by \(\mathcal{N}_{i}\). When node \(i\) and node \(j\) can communicate in both directions, node \(i\) and node \(j\) are neighbors of each other._  
**Remark 3**. : _In practice, each UAV within the swarm is associated with a unique number in the swarm communication topology, which can be changed based on real-time situations. Each UAV stores the numbers of all its neighbors. Furthermore, each UAV only sends its state information to other UAVs with proper numbers in the swarm communication topology and receives state information from UAVs with proper numbers in the swarm communication topology. Updating the number of each UAV in the swarm communication topology and subsequently updating the numbers of each UAV's neighbors are integral parts of the communication topology reconfiguration that the article focuses on._  
The core target of the article is to design formation reconfiguration strategies for leader-follower multi-UAVs in terms of three scenarios, including the disengagement of the leader, the disengagement of the follower and the addition of new members.