# Multi-Satellite Observation-Relay Transmission-Downloading Coupling Scheduling Method
## 2 Problem Description  
The multi-satellite observation-relay transmission-downloading process entails satellites using cameras to observe target information and storing observation data in storage. Subsequently, during overpasses of ground stations, they download stored data. Alternatively, they may relay the observation data to other satellites for subsequent downloading. The observation target type in this research is the point targets that are fixed on the Earth's surface. The mission scheduling process involves multi-satellite observation, mission allocation, and observation data transmission path planning. In the problem studied in this paper, each satellite in the satellite cluster can engage in observation operations as well as receive, transmit, and download target observation data, serving as a relay data transmission satellite when necessary. This creates a coupling of the entire multi-satellite observation-relay transmission-downloading process. The working process of the satellite cluster is shown in Figure 1. The multi-satellite observation-relay transmission-downloading scheduling problem is defined as a mixed-integer programming problem. Parameters that are involved in the scheduling problem are shown in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline \(n_{t}\) & The number of targets \\ \hline \(n_{r}\) & The number of satellites \\ \hline \(n_{s}\) & The number of ground stations \\ \hline \(i,j\) & The index of targets \\ \hline \(k,l\) & The index of satellites \\ \hline \(m\) & The index of ground stations \\ \hline \(t_{ss},t_{c}\) & The start time and end time of the scheduling \\ \hline \(t_{outskip},t_{outskip}\) & Satellite \(k\)’s start time and end time for the \(o\)th  observation time window of target \(i\), with the  total number of observation time windows  being \(n_{outskip}\) \\ \hline \(t_{truskip},t_{truckip}\) & Satellite \(k\)’s start time and end time for the \(p\)th  inter-satellite data transmission time window  to satellite \(l\). The total number of inter-satellite  data transmission time windows is \(n_{truk},\) and  the windows between two satellites are mutual;  thus, \(n_{truk}=n_{truk},\)\(t_{truskip}=t_{truskip},\)\(t_{truckip}=t_{truskip}.\) \\ \hline \(t_{davslm},t_{davclm}\) & Satellite \(k\)’s start time and end time for the \(q\)th  satellite-ground data transmission time  window to ground station \(m\). The total number  of satellite-ground data transmission time  windows is \(n_{dtuk}\). \\ \hline \(t_{oski},t_{oeli}\) & Satellite \(k\)’s observation start time and end time  for target \(i\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in the problem description. Figure 1: Satellite cluster observation-inter-transmission-downloading schematic diagram. The decision variables are as follows:  
\(x_{ki}\in\{0,1\}\quad i=1,2,\ldots,n_{t},k=1,2,\ldots,n_{r}\) represents whether satellite \(k\) observes target \(i\). \(g_{ki,j}\in\{0,1\}\quad i,j=1,2,\ldots,n_{t},k=1,2,\ldots n_{r}\) represents whether satellite \(k\)'s observation order for targets \(i\) and \(j\), where \(g_{ki,j}=1\) indicates that satellite \(k\) observes target \(i\) and target \(j\) in an adjacent order, with target \(j\) observed after target \(i\). \(y_{ik,l}\in\{0,1\}\quad i=1,2,\ldots,n_{t},k,l=1,2,\ldots,n_{r}\) represents whether satellite \(k\) transfers data of target \(i\) to satellite \(l\), with the direction of transmission from satellite \(k\) to satellite \(l\). \(z_{ikm}\in\{0,1\}\quad i=1,2,\ldots,n_{t},k=1,2,\ldots,n_{r},m=1,2,\ldots,n_{s}\) represents whether satellite \(k\) transfers data of target \(i\) to ground station \(m\). The objective function of the problem can be expressed as  
\[J=\sum_{i=1}^{n_{t}}\left(v_{i}\sum_{k=1}^{n_{r}}\sum_{m=1}^{n_{s}}z_{ikm} \right), \tag{1}\]  
where \(J\) is the objective function value, the profit \(v_{i}\) of target \(i\) can be described as its value to the users and its importance to the targets.