## 4 Problem Definition and Analysis
### Problem Formulation for Perception  
The red team UAVs achieve continuous localization of the anomalous radiation source through cooperation. To perform perception and detection of the anomalous radiation source, certain requirements must be met. These include UAV motion constraints and collision avoidance considerations to ensure safe operation. For localizing the anomalous radiation source, successful perception by three or more red team UAVs is required to achieve localization and detection. Therefore, a three-point localization algorithm is employed.  
Each red team UAV \(i\) perceives the anomalous radiation source through Received Signal Strength Indication (RSSI). The distance can be derived from the received RSSI values [30]:  
\[d_{i}(t)=10^{\frac{RSSI_{i}(t)-P_{0}}{-10\pi}}, \tag{11}\]  
where \(d_{i}(t)\) represents the distance from UAV \(i\) to the radiation source at time \(t\), \(RSSI_{i}(t)\) is the measured signal strength, \(P_{0}\) is the RSSI value at the reference distance of 1 m, and \(n\) is the path loss exponent. As shown in Equation (11), the exponential relationship between RSSI and distance allows us to estimate the UAV's distance from the radiation source. Due to the limited sensing range of UAVs, the measurement is valid only when  
\[d_{i}(t)<d_{max}, \tag{12}\]where \(d_{max}\) is the maximum detection distance. Building upon the distance estimation in Equation (11) and the range constraint in Equation (12), the set of UAVs with valid measurements at time \(t\) is defined as follows:  
\[VM(t)=\{i\mid\text{RSSI}_{t}(t)>\text{RSSI}_{t_{th}}\}, \tag{13}\]  
where \(\text{RSSI}_{th}\) is the threshold value for valid RSSI measurements. The three-point localization method requires at least three spatially distributed measurement points to uniquely determine the position of a radiation source through geometric triangulation. Therefore, successful localization is only possible when \(|VM(t)|\geq 3\), meaning three or more UAVs have obtained valid RSSI measurements simultaneously. To simplify the model, each UAV is equipped with a directional sensor that can be oriented in eight discrete directions. The sensor heading angle \(\psi_{i}(t)\) of UAV \(i\) at time \(t\) can only take values from a discrete set of angles separated by \(\pi/4\):  
\[\psi_{i}(t)\in\{0,\pi/4,\pi/2,3\pi/4,\pi,5\pi/4,3\pi/2,7\pi/4\}. \tag{14}\]  
Combining all the measurement and sensing constraints defined in Equations (11)-(14), our objective is to maximize the time duration \(I(t)\) when the radiation source can be effectively localized. Therefore, we formulate the following optimization problem to coordinate the UAVs' motion and sensing directions:  
\[\max_{\{\mathbf{\psi}_{i}(t)\mathbf{\phi}_{i}(t),\mathbf{\psi}_{i}(t)\}_{t=1}^ {T}\lor i} \sum_{t=1}^{T}I(t)\] (15) s.t. ( 4)-( 10 ) ( 12 ) ( 14 ) \[I(t)=\begin{cases}1,&\text{if }|VM(t)|\geq 3\\ 0,&\text{otherwise}\end{cases}. \tag{15b}\]  
This optimization problem aims to find the optimal velocity \(\mathbf{\mathbf{v}}_{i}(t)\), heading angle \(\mathbf{\phi}_{i}(t)\), and sensor orientation \(\mathbf{\psi}_{i}(t)\) for each UAV \(i\) over the time horizon \(\mathcal{T}\), subject to the motion and sensing constraints. At time \(t\), the indicator function \(I(t)\) evaluates to \(1\) only when three or more UAVs have valid measurements simultaneously, enabling three-point localization. This condition may not be satisfied at all times as UAVs may temporarily lose detection due to various factors such as distance, orientation, or environmental interference. By maximizing the sum of \(I(t)\), we seek to maximize the duration during which the radiation source can be effectively localized.