## 2 Mathematical Model of Multi-Satellite Scheduling
### Problem Description and Basic Assumptions  
The multi-satellite scheduling problem can be described as that M satellites cooperatively observing R targets in a scheduling cycle so that the objective function is optimal. The final output of the scheduling plan is mainly the allocation scheme of the observation mission. For a satellite, the distribution result can be expressed as a six-element array as follows:  
\[[m,r,\ ST_{Tm},\ ET_{Tm},\ D_{rm},\ V_{rm}]\]  
Considering the actual satellite system, some reasonable simplifications and basic assumptions for multi-satellite scheduling problems are as follows:  
1. The observation of each target by the satellite requires a certain image scanning time; that is, the observation activity has a certain duration;
2. The satellite resources involved in the scheduling process are the satellites with side-swing capability carrying only one space-borne remote sensor;
3. The satellite needs to maintain a stable attitude during the execution of the observation mission. After completing an observation task, it needs to adjust the posture so that the observation task for the next target can be performed smoothly. From the start of the attitude adjustment to the stable attitude of the satellite, the time taken for this  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(S=\{S_{1},S_{2},\ldots,S_{m},\ldots,S_{M}\}\) & Satellite collection, a total of \(M\) satellites. \\ \hline \(T=\{t_{1},t_{2},\ldots,t_{r},\ldots,t_{R}\}\) & Target collection, a total of \(R\) targets. \\ \hline \(ST_{Tm}\), \(ET_{Tm}\) & The start time and end time of the satellite \(S_{m}\) observing the target \(t_{r}\). \\ \hline \(WinST_{Tm}\), \(WinET_{Tm}\) & Start time and end time of the visible time window of the satellite \(S_{m}\) to the target \(t_{r}\). \\ \hline \(D_{rm}\) & The duration of the satellite \(S_{m}\) observing the target \(t_{r}\). \\ \hline \(V_{rm}\) & The benefit of the target \(t_{r}\) observed by the satellite \(S_{m}\). \\ \hline \(M_{ij}\) & Time required for attitude maneuvering, i.e., satellite attitude adjustment time between missions. \\ \hline \(DS_{mfree}\) & Satellite \(S_{m}\) currently available storage capacity. \\ \hline \(DS_{rm}\) & The satellite \(S_{m}\)storage space capacity that will be occupied by the observation task for the target \(t_{r}\). \\ \hline \(T_{gk^{\prime}}T_{dk}\) & The start time of the illumination period of the \(k\)th circle and the start time of the shadow period of the \(k\)th circle. \\ \hline \(P_{0}\) & Ground shadow period constant load power. \\ \hline \(P_{C}\) & Charging power. \\ \hline \(E_{B}\), \(\eta_{D}\), \(\eta_{C}\), \(\eta_{Wh}\) & Battery parameters. Where \(E_{B}\)is the battery capacity, \(\eta_{D}\) is the discharge controller or discharge regulator efficiency, \(\eta_{C}\) is the charge controller or charge regulator efficiency, \(\eta_{Wh}\) is the watt-hour efficiency of the battery pack. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbol definition.  
process is the satellite attitude adjustment time (also called the attitude maneuver stabilization time).