## 2 Problem Formulations
### Traditional Unmanned Aerial Vehicles Swarm Dynamics Model  
In the present investigation, two swarm systems are present \(\varpi\in[sub-swarm,\ parent-swarm]\). In the present investigation, two swarm systems comprising a total of N agents are examined. It is observed that these swarms traverse through a three-dimensional space without any imposed boundary constraints. It is noteworthy that the two swarms under consideration are equivalent and autonomous, with each adhering to its own set of swarm movement regulations. The subsequent double integrator governs the motion of the agent:  
\[\left\{ \begin{array}{l}\dot{\chi}_{i}^{\varpi}=v_{i}^{\varpi}\\ m_{i}^{\varpi}\dot{v}_{i}^{\varpi}=u_{i}^{\varpi}-\xi||v_{i}^{\varpi}||^{2}v_{ i}^{\varpi}\end{array} \right. \tag{5}\]  
where \(\chi_{i}^{\varpi}=\left(x_{i}^{\varpi},y_{i}^{\varpi},h_{i}^{\varpi}\right) \in R^{3}\) is the position of UAV \(i\) in \(\varpi\in[sub-swarm,\ parent-swarm]\), \(v_{i}^{\varpi}\in R^{3}\), indicating the velocity vector, \(u_{i}^{\varpi}\in R^{3}\) denotes the control input acting on; \(m_{i}^{\varpi}\) is the quality, \(-\xi||v_{i}^{\varpi}||^{2}v_{i}^{\varpi}\) is the friction against air, and \(\xi\) is the damping factor of air. The majority of internal interactions between agents adhere to the cohesion alignment [22] and separation rules [21]. A prototypical implementation is outlined as follows:  
\[u_{i}^{\varpi}=u_{i}^{\varpi_{out}}+u_{i}^{\varpi_{gai}}+u_{i}^{\varpi_{pos}}+ \xi||v_{i}^{\varpi}||^{2}v_{i}^{\varpi}+\gamma^{in\varpi}v_{i}^{\varpi} \tag{6}\]where \(u_{i}^{\varpi_{val}}\) is the velocity alignment, \(u_{i}^{\varpi_{val}}\) is the navigation function, \(u_{i}^{\varpi_{val}}\) is navigational force, \(u_{i}^{\varpi_{pos}}\) is the position of cooperation, and \(\gamma^{ine}\) is the inertia coefficient. \(u_{i}^{\varpi_{pos}}\) is defined as follows:  
\[u_{i}^{\varpi_{pos}}=\sum_{j\in\mathbb{N}_{i}^{\varpi}(t)}\Gamma^{pos}||d_{ij}^{ \varpi}||\left(1-\left((\frac{l_{a}^{\varpi}||\chi_{j}^{\varpi}-\chi_{i}^{ \varpi}||}{\chi_{j}^{\varpi}-\chi_{i}^{\varpi})}\right)^{2}\right)exp(\frac{l_{ a}^{\varpi}}{d_{ij}^{\varpi}})\right) \tag{7}\]  
where \(\Gamma^{pos}\) is the coefficient of position cooperation, \(l_{a}^{\varpi}\), \(l_{c}^{\varpi}\) are the agent desire spacing and the motion attenuation factor, \(d_{ij}^{\varpi}\) is the distance between two agents, \(N_{i}^{\varpi}\) is utilized to represent the number of agents that are interacting with agent \(i\) in \(\varpi\), and \(N_{i}^{\varpi}(t)\) is the collection of the neighbors in \(\varpi\) that are interacting with agent \(i\) at time \(t\). Typical definitions of \(u_{i}^{\varpi_{val}}\) is given as follows:  
\[u_{i}^{\varpi_{val}}=\frac{\Gamma^{vel}}{N_{i}^{\varpi}}\sum_{j\in N_{i}^{ \varpi}(t)}\left(\tau_{j}^{\varpi}-\tau_{i}^{\varpi}\right) \tag{8}\]  
where \(\Gamma^{vel}\) is the parameter of \(u_{i}^{\varpi_{val}}\). Equation (8) represents the widely recognized \"velocity consensus\" algorithm, which allows the swarm to quickly reach a common speed and sustain a homogeneous state [18]. The present study aimed to investigate the ability of a swarm using Equation (8) to perform self-organized fission movements when only a few agents in the swarm detect the presence of an obstacle. However, researcher findings revealed that the swarm was unable to execute self-organized fission movements in such a scenario [34]. To address this limitation, [PERSON] et al. [17] introduced an intermittent selective mechanism that allows swarms encountering static obstacles to engage in fission-fusion motion. However, when confronted with dynamic obstacles possessing tracking capabilities or uncertain interference directions, the UAV swarm often struggles to complete its flight with optimal resource utilization. Furthermore, such situations may severely disrupt the normal movement of the swarm.