# Deep Reinforcement Learning for the Agile Earth Observation Satellite Scheduling Problem
## 2 Problem Description
### Mathematical Formulation  
\[max\sum_{i=0}^{n_{tsk}}\frac{n_{tsk}}{j=0}x_{ij}pri_{j} \tag{1}\]  
\[wb_{i}\leq tb_{i}\leq te_{i}\leq we_{i}\forall i\in\{0,1,2,\ldots,n_{tsk}\} \tag{2}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Description** \\ \hline \(n_{tsk}\) & The number of tasks \\ \(i,j\) & The index of tasks, \(i,j=0,1,...,n_{tsk}\). 0 means a virtual task \\ \(pri_{i}\) & The profit of \(task_{i}\), \(pri_{0}=0\) \\ \(\theta_{i,t}\) & The roll angle of the satellite to \(task_{i}\) at time \(t\) \\ \(\varphi_{i,t}\) & The pitch angle of the satellite to \(task_{i}\) at time \(t\) \\ \(\psi_{i,t}\) & The yaw angle of the satellite to \(task_{i}\) at time \(t\) \\ \(wb_{i}\) & The VTW start time for \(task_{i}\), \(wb_{0}=0\) \\ \(we_{i}\) & The VTW end time for \(task_{i}\), \(we_{0}=0\) \\ \(tb_{i}\) & The OW start time of \(task_{i}\), \(tb_{0}=0\) is the initial state time \\ \(te_{i}\) & The OW end time of \(task_{i}\), \(te_{0}=0\) is the initial state time \\ \(ct_{i}\) & The observation duration of \(task_{i}\), \(ct_{0}=0\) \\ \(ba_{i}\) & The satellite attitude at \(tb_{i}\), which is determined by the angle of roll \(\theta_{i,tb_{i}}\), \\  & pitch \(\varphi_{i,tb_{i}}\) and yaw \(\psi_{i,tb_{i}}\). \(ba_{0}\) is the initial attitude of the satellite \\ \(ea_{i}\) & The satellite attitude at \(te_{i}\), which is determined by the angle of roll \(\theta_{i,te_{i}}\), \\  & pitch \(\varphi_{i,te_{i}}\) and yaw \(\psi_{i,te_{i}}\). \(ea_{0}\) is the initial attitude of the satellite \\ \(\rho_{ij}\) & The attitude transition angle between \(task_{i}\) and \(task_{j}\) \\ \(\text{trans}\big{(}ea_{i},ba_{j}\big{)}\) & The attitude transition time between \(task_{i}\) and \(task_{j}\) \\ \(x_{ij}\) & Binary decision variable, indicating whether \(task_{i}\) is a former task for \(task_{j}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter description.  
\[tb_{i}+ct_{i}=te_{i_{i}}\forall i\in\{0,1,2,\ldots,n_{tsk}\} \tag{3}\]  
\[te_{i}+trans\big{(}ea_{i},ba_{j}\big{)}-tb_{j}\leq 0,\forall x_{ij}=1 \tag{4}\]  
\[\sum_{i=0}^{n_{tsk}}\sum_{j=0}^{n_{tsk}}x_{ij}\big{(}ct_{j}\cdot uie+ute\cdot trans \big{(}ea_{i},ba_{j}\big{)}\big{)}\leq(1-\zeta)\cdot Egy_{max} \tag{5}\]  
\[\sum_{i=0}^{n_{tsk}}x_{ij}\leq 1,\forall j\in\{0,1,2,\ldots,n_{tsk}\} \tag{6}\]  
\[\sum_{j=0}^{n_{tsk}}x_{ij}\leq 1,\forall i\in\{0,1,2,\ldots,n_{tsk}\} \tag{7}\]  
\[x_{ii}=0,\forall i\in\{0,1,2,\ldots,n_{tsk}\} \tag{8}\]  
\[x_{ij}\in\{0,1\},\forall i,j\in\{0,1,2,\ldots,n_{tsk}\} \tag{9}\]  
Equation (1) is the optimization objective function of the AEOSSP, which is to maximize the sum of completed task priorities; Equation (2) represents the time window constraint of the satellite, which means the task must be observed within the VTW of the satellite; Equation (3) represents the relationship between the start, end time, and duration of the task; Equation (4) represents the time-dependent transformation time constraint between the tasks; Equation (5) represents the the consumed power constraint of the satellite; Equation (6) indicates that there exists at most one former task for each task; Equation (7) indicates that there exists at most one latter task for each task; Equation (8) indicates that a task can neither be its former task nor its latter task; Equation (9) represents the value of the decision variable.