## 3 System Model
### UAV Kinematic Model  
This section provides a detailed description of the UAV model's movement, with particular focus on changes in speed and direction. We simultaneously establish a safe distance model to ensure safe UAV operation in various environments, while exploring methods for locating target radiation sources. To simplify, we use a 2D model instead of 3D, and make it discrete to fit a map.  
We begin by defining the position formula, followed by the introduction of speed and direction sets based on real-world scenarios, which are used to update positions. The UAV position is given by the following equations:  
\[p_{i}(t)=(x_{i}(t),y_{i}(t)), \tag{3}\]  
where \(p_{i}(t)\) represents the UAV \(i\) position coordinates at time \(t\). Based on Equation (3), the UAV position update is given by the following equations:  
\[p_{i}(t+1)=p_{i}(t)+v_{i}(t)[\lceil\cos\phi_{i}(t)\rceil,\lceil\sin\phi_{i}(t) \rceil]^{T}, \tag{4}\]  
where \(v_{i}(t)\) is the speed, and \(\phi_{i}(t)\) is the direction angle of UAV \(i\) at time \(t\). This update equation captures the discretized motion of UAVs in the 2D plane. To model realistic UAV capabilities, the speed \(v_{i}(t)\) is defined as follows:  
\[v_{i}(t)\in\{0,1\}, \tag{5}\]  
where 0 represents that the UAV is hovering in place, and 1 indicates moving horizontally. Each UAV has two types of physical constraints. For the onboard radiation detection system, the sensing direction constraint is defined as follows:  
\[|\phi_{i}(t+1)-\phi_{i}(t)|\leq\Delta\phi_{\max}, \tag{6}\]  
where \(\Delta\phi_{\max}\) is the maximum allowable change in angle per step. As shown in Equation (6), this constraint ensures that the radiation detector's orientation change remains within its physical capabilities. For the UAV's movement, the acceleration constraint is defined as follow:  
\[|v_{i}(t+1)-v_{i}(t)|\leq\Delta v_{\max}, \tag{7}\]where \(\Delta v_{\max}\) is the maximum allowable change in speed per step, ensuring smooth transitions in UAV motion control.  
To ensure the safe operation of the red team's UAVs, it is essential to maintain a certain safe distance to protect it from external interference and potential hazards. This approach effectively prevents damage from environmental factors and ensures normal and stable operation. The safety constraints are formulated through three key equations. Based on the position vectors defined in Equation (3), the physical-safe distance constraint is formulated as follows:  
\[|p_{i}(t)-p_{j}(t)|\geq D, \tag{8}\]  
where \(D\) is the minimum physical-safe distance between any two UAVs, and the constraint applies for all pairs of different UAVs (\(i\
eq j\)) at all time steps \(t\). This safety constraint is essential for collision avoidance in multi-UAV systems. Each UAV must remain within the map boundaries:  
\[p_{i}(t)\in M, \tag{9}\]  
where  
\[M=\{(x,y)\ |\ 0\leq x\leq X_{\max},0\leq y\leq Y_{\max}\}. \tag{10}\]  
This boundary constraint requires all UAV positions to stay within the valid region at all times, restricting their movement when approaching these limits. Together, Equations (8)-(10) establish comprehensive safety constraints that ensure both collision avoidance between UAVs and containment within the operational area. This approach establishes an appropriate safety margin for the red team's UAVs, ensuring their optimal operational effectiveness.