## 5 Distributed Heterogeneous Cluster Task Planning Based on Nash Equilibrium  
### Construction of a Task Planning Model for Heterogeneous Clusters  
The system consists of several small detection satellites deployed in different orbits. These satellites perform collaborative detection tasks in a dynamic space environment. When facing multi-target tracking or sudden detection demands, the system needs rapid response and task re-planning capabilities. In this task planning process, orbit maneuver optimization and multi-target allocation are two core problems. The goal is to ensure detection quality while minimizing energy consumption. Suppose the system has a total of \(N_{A}\) intelligent satellites denoted as \(A=\{A_{1},A_{2},\ldots,A_{N_{A}}\}\). The detection method of the intelligent satellites follows that described in the first part. The objective function is \(S_{m}^{L}\), where \(m\in\{1,2\}\) represents two types of detection scenarios, and \(L\in\{Aprpr,Dur\}\) represents the mission phases, namely the approach phase and the sustaining phase. When \(S_{m}^{L}\) takes a value of 0, it indicates mission success. For an intelligent satellite, successful missions in both the approach phase and the sustaining phase result in \(S_{m}^{Aprpr}\) and \(S_{m}^{Dur}\) both being 0. There are \(N_{O}\) target satellites denoted as \(O=\{O_{1},O_{2},\ldots,O_{N_{\psi}}\}\). When an intelligent satellite detects a target satellite, its task failure rate \(f_{c1}\) is defined as:  
\[f_{c1}=1-\frac{\sum_{i,j}I\Big{\{}\Big{(}S_{m}^{Aprpr}=0\Big{)}\;and\;\big{(} S_{m}^{Dur}=0\Big{)}\Big{\}}s_{ij}}{\sum_{i,j}s_{ij}} \tag{13}\]  
Here, \(s_{ij}\) is a 0-1 matrix, where the horizontal axis and vertical axis represent the indices of the intelligent satellites and the target satellites, respectively. The higher the value of \(f_{c1}\), the lower the mission success rate. Thus, minimizing \(f_{c1}\) maximizes the mission success rate. Figure 3: Flow of the co-evolutionary algorithm. Time cost of detection is the maximum time taken by any intelligent satellite to complete its task. By normalizing this time cost by dividing it by the maximum endurance time \(t_{f}^{max}\) of the intelligent satellites, the time cost \(f_{c2}\) is obtained as follows:  
\[f_{e2}=\frac{max}{t_{f}^{max}}\frac{(s_{ij}t_{ij})}{t_{f}^{max}} \tag{14}\]  
The application of impulse velocity changes by intelligent satellites consumes fuel. The total amount of velocity change can be used to reflect the fuel consumption. Similar to the approach for handling time cost, the maximum impulse amount is used for normalization, resulting in the total fuel consumption \(f_{c3}\) as follows:  
\[f_{c3}=\frac{\sum_{i=1}^{N_{a}}\sum_{j=1}^{N_{a}}s_{ij}\big{|}\Delta v_{ij} \big{|}}{\sum_{i=1}^{N_{a}}\Delta v_{max,i}} \tag{15}\]  
Here, \(\Delta v_{max,i}\) represents the total remaining impulse capacity of the \(i\)-th intelligent satellite, where \(\Delta v_{ij}\) represents the total velocity impulse required for the \(i\)-th intelligent satellite to maneuver and detect the \(j\)-th target satellite. For a task assignment strategy \(s_{ij}\), covering a greater number of target satellites is a crucial objective. The proportion of target satellites not covered by the strategy, denoted as \(f_{e4}\), can be defined as follows:  
\[f_{e4}=1-\frac{\sum_{j=1}^{N_{b}}max_{i}s_{ij}}{N_{o}} \tag{16}\]  
The comprehensive effectiveness \(f_{e}\) is obtained by calculating the weighted average of all effectiveness values using certain weights \(\varphi_{i}\), as follows:  
\[f_{e}=\sum_{i=1}^{4}\varphi_{i}f_{ei} \tag{17}\]  
The objective of the planning is to minimize \(f_{e}\).