## 2 Materials and Methods
### Methods
#### 2.2.2 Mathematical Model  
In the cooperation scheme of GEO-LEO satellites, autonomous planning is performed on the basis of an initial observation schedule generated by ground control center. The GEO meteorological satellite dynamically adjusts the observation tasks of multiple LEO optical imaging satellites based on real-time cloud cover information to obtain a new observation scheme. Due to the complex temporal constraints and high coupling between observation tasks, local adjustments may have influence on subsequent tasks, resulting in a chain reaction. How to design autonomous planning algorithms with small perturbations and high efficiency is one of the key issues. Based on the above analysis, we establish the mathematical model of the autonomous planning problem. A summary of notations is presented as Table 1.  
We developed two decision variables in the model, which are as follows:  
\[x_{ki}^{n}=\left\{ \begin{array}{ll}1,&\text{if task $t_{i}$ is assigned at the $k$th orbit of satellite $s_{n}$}\\ 0,&\text{otherwise}\end{array} \right. \tag{1}\]  
\[y_{ij}^{nk}=\left\{ \begin{array}{ll}1,&\text{if task $t_{j}$ is the immediate successor of task $t_{i}$}\\ &\text{at the $k$th orbit of satellite $s_{n}$}\\ 0,&\text{otherwise}\end{array} \right. \tag{2}\]  
The objective of optimization is maximizing the schedule profits of all selected tasks. \[F=max\sum_{n=1}^{N_{S}}\sum_{k=1}^{N_{0}^{n}}\sum_{i=1}^{N_{T}}x_{ki}^{n}\cdot p _{i} \tag{3}\]  
where \(F\) means maximizing the schedule profits of all selected tasks. The operations of EOSs should satisfy following constraints:  
1. Each task can be observed at most once. \[\forall t_{i}\in T:\sum_{n=1}^{N_{S}}\sum_{k=1}^{N_{t}^{n}}\sum_{i=1}^{N_{T}}x_ {ki}^{n}\leq 1\] (4) \[\forall t_{i},t_{j}\in T,i\
eq j:\sum_{n=1}^{N_{S}}\sum_{k=1}^{N_{T}}\sum_{ i=1}^{N_{T}}x_{ki}^{n}=\sum_{n=1}^{N_{S}}\sum_{k=1}^{N_{T}^{n}}\sum_{i=1,j=2}^{N _{T}}y_{ij}^{nk}\] (5)
2. Each task must be observed within its available time windows. \[\forall t_{i}\in T,x_{ki}^{n}=1:b_{ki}^{n}\leq o_{ki}^{n}\leq q_{ki}^{n}\] (6)
3. Setup time between contiguous tasks must be sufficient for transition. \[\forall t_{i},t_{j}\in T,i\
eq j,x_{ki}^{n}=y_{ij}^{nk}=1:o_{kj}^{n}\geq o_{ki} ^{n}+d_{i}+u_{ij}^{nk}+t_{st}\] (7)
4. The memory consumption cannot exceed maximum memory capacity for each orbit. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Definitions** \\ \hline \(T\) & Observation tasks set, \(T=\left\{t_{1},t_{2},\ldots,t_{N_{T}}\right\}\),\(N_{T}\) represents the number of tasks \\ \(S\) & EOS resources set, \(S=\left\{s_{1},s_{2},\ldots,s_{N_{S}}\right\}\),\(N_{S}\) represents the number of EOSs \\ \(C^{n}\) & Orbits set of \(s_{n},C^{n}=\left\{c_{1},c_{2},\ldots,c_{N_{C}^{n}}\right\},\)\(N_{C}^{n}\) represents the number of orbits \\ \(b_{ki}^{n}\) & Observation start time of task \(t_{i}\) at the \(k\)th orbit of satellite \(s_{n}\) \\ \(q_{ki}^{n}\) & Observation end time of task \(t_{i}\) at the \(k\)th orbit of satellite \(s_{n}\) \\ \(d_{i}^{n}\) & Observation time of task \(t_{i}\) at the \(k\)th orbit of satellite \(s_{n}\) \\ \(d_{i}\) & Observation duration of task \(t_{i}\) \\ \(p_{i}\) & Priority of task \(t_{i}\) \\ \(u_{ij}^{nk}\) & Attitude maneuver duration from task \(t_{i}\) to \(t_{j}\) at the \(k\)th orbit of satellite \(s_{n}\) \\ \(e_{u}\) & Energy consumption of satellite attitude maneuver per unit time \\ \(t_{st}\) & Satellite attitude stabilization time \\ \(e_{st}\) & Energy consumption of satellite attitude stabilization \\ \(m_{o}\) & Memory consumption of observation per unit time \\ \(e_{o}\) & Energy consumption of observation per unit time \\ \(M_{n}\) & Memory capacity per orbit \\ \(E_{n}\) & Energy capacity per orbit \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of the problem’s notations. \[\forall s_{n}\in S,\forall c_{k}\in C^{n}:\sum\limits_{i=1}^{N_{T}}m_{ obj}\cdot d_{i}\cdot x_{ki}^{n}\leq M_{n} \tag{8}\]
5. The energy consumption cannot exceed maximum energy capacity for each orbit. \[