## 2 Problem Description
### Mathematical Formulation  
Based on the assumptions above, the objective function and constraints for agile satellite scheduling are as follows. We define \(x_{ij}\) as a binary decision variable for determining whether to select the visible time window \(tw_{ij}\) of a task \(r_{i}\). The value of \(x_{ij}\) is equal to \(1\) if and only if the task \(r_{i}\) is scheduled and selects the corresponding visible time window \(tw_{ij}\) and the observation window \(ow_{i}\in tw_{ij}\) of \(r_{i}\). If all VTWs of the task are not selected, the observation of the pair fails.  
This problem aims to maximize the cumulative revenue derived from completing observation tasks. The objective function is defined as follows (see Equation (1)):  
\[\max\sum_{i=1}^{N}\sum_{j=1}^{|w_{i}|}x_{ij}p_{i} \tag{1}\]  
Based on the description and constraint analysis of agile satellite scheduling, we establish the following model to satisfy satellite task scheduling constraints:  
\[\sum_{j=1}^{|w_{i}|}x_{ij}\leq 1 \tag{2}\] \[st_{ij}\leq u_{ij}\leq u_{ij}+l_{i}\leq et_{ij}\text{, if }x_{ij}=1 \forall t_{i}\in T\] (3) \[\Delta g=|\gamma^{u_{ij}}-\gamma^{u_{ij}\gamma}|+|\pi^{u_{ij}}- \pi^{u_{ij}\gamma}|+|\varphi^{u_{ij}}|\qquad\forall t_{i},t_{i^{*}}\in T\] (4) \[\Delta(t_{i},t_{i^{*}})=\begin{cases}11.66,&\text{if }\Delta g \leq 10\\ a_{1}+\frac{\Delta g}{\gamma_{1}},&\text{if }10<\Delta g\leq 30\\ a_{2}+\frac{\Delta g}{\gamma_{2}},&\text{if }30<\Delta g\leq 60\\ a_{3}+\frac{\Delta g}{\gamma_{3}},&\text{if }60<\Delta g\leq 90\\ a_{4}+\frac{\Delta g}{\gamma_{4}},&\text{if }\Delta g>90.\end{cases}\] (5) \[u_{ij}+l_{i}+\Delta(t_{i},t_{i^{*}})\leq u_{i^{*}j^{*}}\text{, if } \rho_{w_{ij}w_{j^{*}}}=1 \forall t_{j},t_{i^{*}}\in T,i\
eq i^{*},w_{ij}\in w_{i},w_{i^{*}j^{*}}\in w _{i}^{*} \tag{6}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Explanation** \\ \hline \(E\) & The total satellite power; \\ \(M\) & The total satellite memory; \\ \(\gamma^{t}\) & The rolling angles at time \(t\); \\ \(\pi^{t}\) & The pitching angles at time \(t\); \\ \(\mu^{t}\) & The yawing angles at time \(t\); \\ \(|w_{i}|\) & The number of VTW for task \(r_{i}\); \\ \(st_{ij}\) & The start time; \\ \(et_{ij}\) & The end time; \\ \(e^{f}\) & The fixed power consumption per satellite attitude adjustment; \\ \(e^{s}\) & The power consumption per degree of satellite rotation; \\ \hline \hline \end{tabular}
\end{table}
Table 2: Symbol of satellite properties.  
\[\sum_{i=1}^{N}\sum_{j=1}^{|w_{i}|}x_{ij}l_{i}m_{i}\leq M \tag{7}\] \[\sum_{i=1}^{N}\sum_{j=1}^{|w_{i}|}(x_{ij}|e_{i}+\rho w_{ij}w_{i^{ \prime}j^{\prime}}e^{f}+\rho w_{ij}w_{i^{\prime}j^{\prime}}\Delta ge^{s})\leq E \qquad\forall t_{i^{*}}\in T,w_{i^{*}j^{*}}\in w_{i^{*}} \tag{8}\]  
The meanings of constraints are as follows.  
* Constraint (2) states that each task can only be executed once;
* Constraint (3) specifies that the executed task must fall within the visible time window;
* Constraints (4) and (5) specify the method of calculating transition time. Among them, \(v_{1}\), \(v_{2}\), \(v_{3}\), and \(v_{4}\) represent four different satellite attitude conversion speeds. Here, \(\Delta g\) represents the total angle change between two observations, calculated as \(\Delta g=\Delta\gamma+\Delta\pi+\Delta\mu\), where \(\Delta\gamma\), \(\Delta\pi\), and \(\Delta\mu\) represent the satellite's pitch angle, yaw angle, and roll angle, respectively. For the satellites referred to in the article, \(v_{1}=1.5^{\circ}/s\), \(v_{2}=2^{\circ}/s\), \(v_{3}=25^{\circ}/s\), \(v_{4}=3^{\circ}/s\), \(a_{1}=5\), \(a_{2}=10\), \(a_{3}=16\), \(a_{4}=22\), respectively.
* Constraint (6) dictates that the transition time between two tasks must not exceed the specified limit.
* Constraint (7) indicates that the memory usage must not exceed the maximum memory capacity, denoted as \(M\).
* Constraint (8) specifies that the power consumption should not surpass the maximum power limit, denoted as \(E\).