# Multi-Satellite Observation-Relay Transmission-Downloading Coupling Scheduling Method
## 2 Problem Description
\begin{array}{l}t_{\textit{Outskli}}\geq t_{\textit{rnsklp}}\wedge t _{\textit{Outskli}}\leq t_{\textit{rneklp}}\\ t_{\textit{rInskli}}\geq t_{\textit{rnsklp}}\wedge t_{\textit{rInkli}}\leq t_{ \textit{rneklp}}\end{array} \right.\quad\exists p,1\leq p\leq n_{\textit{ rnsklp}}\,, \tag{4}\]  
\[t_{\textit{dskmi}}\geq t_{\textit{duskmq}}\wedge t_{\textit{dokmi}}\leq t_{ \textit{dwekmq}}\quad\exists q,1\leq q\leq n_{\textit{dukkm}}, \tag{5}\]  
\[\left\{\begin{array}{l}[t_{\textit{Outskli}},t_{\textit{Outkli}})\cap\left[ t_{\textit{Outsklj}},t_{\textit{Outkli}}\right]\right)=\phi&\forall l,i,j,i\
eq j\\ \left[t_{\textit{Outskli}_{1}}-\Delta t_{\textit{prkli}_{1}},t_{\textit{Outsk li}_{1}}\right)\cap\left[t_{\textit{Outskli}_{2j}}-\Delta t_{\textit{prkli}_{2}},t_{ \textit{Outkli}_{2j}}\right)=\phi&\forall l_{1},l_{2},l_{1}\
eq l_{2}\,,\\ \left\{ \begin{array}{l}[t_{\textit{dskmi}},t_{\textit{dkemi}})\cap\left[t_{ \textit{dskmj}},t_{\textit{dekmj}}\right)=\phi&\forall m,i,j,i\
eq j\\ \left[t_{\textit{dskm}_{1}}-t_{\textit{pdk}},t_{\textit{dekm}_{1}}\right]\cap \left[t_{\textit{dskm}_{2j}}-t_{\textit{pdk}},t_{\textit{dekm}_{2}}\right)= \phi&\forall m_{1},m_{2},m_{1}\
eq m_{2}\,,\\ \end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}y_{ik,l}\leq x_{ki}\\ y_{ik,k_{2},k_{3}}\leq y_{ik_{1},k_{2}}\\ z_{ikm}\leq x_{ki}\\ z_{ikm}\leq y_{il,k}\end{array} \right., \tag{8}\]  
\[t_{\textit{Outsk2 kj}}\geq t_{\textit{rInck1k2i}}, \tag{9}\]  
\[\left\{ \begin{array}{l}t_{\textit{dskmi}}\geq t_{\textit{ooki}}\\ t_{\textit{dskmi}}\geq t_{\textit{rInelki}}\end{array} \right., \tag{10}\]  
\[0\leq W_{k}\leq W_{\textit{Totalk}}, \tag{11}\]  
\[0\leq M_{k}\leq M_{\textit{Totalk}}. \tag{12}\]  
Equation (2) represents the observation time window constraint, which means that a satellite's observation must occur within the observation time window. Equation (3) is the attitude maneuver constraint, which means that the observation time interval between two adjacent missions of a single satellite needs to be greater than the attitude maneuver conversion time of the observation attitude of the two observation missions. Equation (4) represents the inter-satellite data transmission time window constraint. It signifies that the inter-satellite data transmission must occur within the inter-satellite data transmission time window. Equation (5) represents the satellite-ground data transmission time window constraint. It dictates that a satellite's data downloading operations must align with the requirements of the satellite-ground data transmission time window. Equations (6) and (7) correspond to the data transmission switch time constraint, which means that when a satellite is performing data transmission to different satellites, there should be an interval between two transmission missions for data transmission equipment preparation. Similarly, for data downloading to ground stations, if the data is transmitted to two different ground stations, an interval must exist between the two transmission missions to accommodate the data transmission equipment preparation time, where \(\Delta t_{prkl}=\text{max}\Big{(}\Delta t_{prk},\Delta t_{prl}\Big{)}\). Equation (8) represents the data transmission logical constraint, signifying that the premise of data forwarding and downloading is the completion of the observation or data reception. Equations (9) and (10) are the timing constraints of data transmission, which means that the start time of inter-satellite data transmission or satellite-ground data downloading must be greater than the end time of observation or data reception. Equation (11) represents the electricity constraint. In this equation, \(W_{k}\) represents the current electric energy of satellite \(k\), and the formula for \(W_{k}\) calculation is  
\[W_{k}=W_{Initialk}+\int(P_{Ink}-P_{Camk}-P_{GroundDataTransk}-P_{ SatelliteDataTransk}-P_{Attitudek})dt, \tag{13}\]  
where \(W_{Initialk}\) represents the initial electricity energy of satellite \(k\). \(P_{Ink}\) stands for the charging power of satellite \(k\).