# Fault Reconstruction for a Giant Satellite Swarm Based on Hybrid Multi-Objective Optimization
## 2 Construction of On-Orbit Reconstruction Indicators for a Giant Satellite Swarm
### The Introduction of the Reconstruction Indicators  
Because satellites carry limited fuel, fuel consumption, and energy consumption need to be considered during on-orbit reconstruction. Meanwhile, the health state also needs to be considered, so the following parameters are used to establish a reconstruction indicator system:  
(1) Satellite swarm health state  
The satellite swarm health state refers to the durability of individual satellites and topological structures in the satellite swarm and the ability of the satellite swarm to meet mission requirements [32]. The satellite swarm health state is an important indicator that measures the overall health of the satellite swarm and detects faults effectively. The health state is used to measure the effect of satellite swarm reconstruction. Considering the nonlinear composition characteristics of the satellite swarm, such as the multiple dimensions of a single satellite, inter-satellite communication link, and mission effectiveness in the satellite swarm, the health state at time \(H_{swarm}\) is expressed as:  
\[ \begin{array}{l}H_{swarm}=H_{sat}^{k_{1}}\cdot H_{link}^{k_{2}}\cdot E_{swarm }^{k_{3}}\\ H_{sat}=\sum\limits_{i=1}^{N}R_{sat,i}\omega_{sat,i}\\ H_{link}=\sum\limits_{i=1}^{M}R_{link,i}\omega_{link,i}\end{array}  \tag{5}\]  
In the formula, \(H_{sat}\) is the weighted sum of the health states of all individual satellites in the swarm, where the single-satellite reliability of satellite \(i\) is \(R_{sat,i}\) and its weight is \(\omega_{sat,i}\), in the same way, \(H_{link}\) is the weighted sum of the health status of the inter-satellite communication links, where the link reliability of link \(i\) is \(R_{link,i}\) and its weight is \(\omega_{link,j}\), \(k_{1},k_{2},k_{3}\) are the influencing parameters, which represent the relative influence of each part of the satellite swarm system. \(E_{swarm}\) is the satellite swarm performance, indicating the ability of the satellite swarm to achieve a task. Different tasks have different forms, which are determined by the specific circumstances of the task. This paper considers a satellite swarm that performs space telescope missions. According to the requirements of optical imaging, the fewer faulty satellites are located near the center of the mirror, the better the imaging performance, i.e., the number of effective satellites can be used to measure the satellite swarm performance. Therefore, the satellites are divided into different layers according to the distance from the satellite to the center of the surface. Satellite swarm effectiveness is determined by the number of working satellites in each layer and the weight values of each layer:\[E_{swarm}=\sum\limits_{i=1}^{layer}N_{layer\_i}e_{layer\_i} \tag{6}\]  
In the formula, \(\omega_{sat\_i}\) is the weight value of layer \(i\); \(N_{layer\_i}\) indicates the number of nonfaulty satellites contained in the \(i\)-th layer and satisfies \(\sum\limits_{i=1}^{layer}N_{layer\_i}=N_{H}\), where \(N_{H}\) is the number of nonfaulty satellites in the satellite swarm. The total energy consumption of reconstruction is the sum of the energy consumed by all satellites participating in the reconstruction to complete orbital maneuvers. Due to the limited fuel carried by each satellite, to satisfy the satellite swarm's requirements for operating life and mission performance, the lower the total energy consumption is, the better. Therefore, the total energy consumption of reconstruction is expressed as:  
\[J_{sum}=\sum\limits_{i=1}^{k}J_{i},i=1,2,\cdots,k \tag{7}\]  
In this formula, \(J_{i}\) indicates the energy consumption of the orbital maneuver of the \(i\)-th satellite participating in the reconstruction, and \(k\) indicates the number of satellites participating in the reconstruction. Fuel consumption balance [33]  
To reduce the reconstruction time, the principle of nearest replacement is adopted.