# Revising the Observation Satellite Scheduling Problem Based on Deep Reinforcement Learning
## 2 Problem Description
### Task Scheduling Problem
#### 2.2.2 Constraint Conditions  
In this paper, the VTW is seen as the allocated resource, and the OWs for tasks are continuous decision variables to decide when to start the observation. The solution of the EOSSP model aims to schedule an observation sequence and maximize the observation profit, subject to corresponding constraints. In practical engineering scenarios, the following constraints are usually taken into account [24]:  
(1) VTW constraint  
The VTW constraint ensures that the observation tasks can be executed within the VTW of EOSs in the observation process.  
For \(\forall t_{u}^{i}\in\{t_{i}\mid i=1,2,\ldots,N\}\), where \(N\) is the number of tasks,  
\[\left\{\begin{array}{l}ObvS_{u}^{i}\geq TWS_{u}^{i}\\ ObvE_{u}^{i}+d_{u}\leq TWE_{u}^{i}\end{array}\right. \tag{6}\]  
where \(d_{u}\) represents the observation duration time of task \(t_{u}\), \(d_{u}=ObvE_{u}^{i}-ObvS_{u}^{i}\) is the observation start time of \(t_{u}\) in the \(i\)th orbit, \(ObvE_{u}^{i}\) is defined as the observation end time. \(TWS_{u}^{i}\) and \(TWE_{u}^{i}\) represent the start and end time of the VTW for task \(t_{u}^{i}\).  
(2) Conflict constraint for task execution  
The conflict constraint for task execution means that there is no crossover between any two tasks as the optical sensor cannot perform two observation tasks at the same time:  
\[\sum_{v=1,v\
eq u}^{N}x_{uv}\leq 1 \tag{7}\]  
where \(x_{uv}\) is a decision variable and denotes whether to transform execution from task \(t_{u}\) to task \(t_{v}\). \(x_{uv}=1\) means that \(t_{v}\) will be executed after \(t_{u}\).  
(3) Task conversion time constraint  
Between any two sequential tasks, enough preparation time is required, mainly including slewing maneuvering time and sensor shutdown-restart setup time [38], which could be described as the following formula:  
For \(\forall u,v\in N\) and \(u<v\),  
\[ObvS_{v}^{i}-ObvE_{u}^{i}\geq s_{uv}+transT_{uv} \tag{8}\]  
where \(s_{uv}\) is the preparation time for restarting the sensor and \(tranT_{uv}\) is the slewing maneuver time from task \(t_{u}\) to \(t_{v}\), and the slewing maneuver time can be calculated as the following formula:  
\[transT_{uv}=\frac{|\theta_{v}-\theta_{u}|}{v_{s}} \tag{9}\]  
In the above formula, \(\theta_{u}\) and \(\theta_{v}\) represent the observation slewing angle of \(t_{u}\) and \(t_{v}\). \(v_{s}\) denotes the angular velocity of the satellite slewing maneuver.  
(4) Optical sensor boot time constraint  
According to the power constraint of the optical payload, the observation time of a task cannot exceed the maximum operating time of the optical sensor,  
\[\max T\geq x_{u}d_{u}(u=1,2,\cdots,N) \tag{10}\]  
(5) Storage size constraint  
Limited by the total storage size in the satellite, the constraint could be described as the following equation,  
\[\sum_{u=1}^{N}x_{u}c_{i}d_{u}\leq M \tag{11}\]  
where \(M\) represents the total data storage capacity of the satellite in one orbit. \(c_{i}\) is the storage consumption per unit observation time in one orbit.  
(6) Power consumption constraint  
In each orbit, the energy to be consumed is limited by the maximum capacity, and the corresponding energy consumed by the sensor operation and slewing maneuver is mainly considered in this paper as:  
\[\sum_{u=1}^{N}x_{u}e_{i}d_{u}+\sum_{u=1}^{N}\sum_{v=1,\varphi\
eq u}^{N}x_{uv}(s _{uv}+\text{tran}\ T_{uv})\varepsilon_{uv}\leq E \tag{12}\]  
In this formula, \(e_{i}\) represents the energy consumption per unit time of observation operation. \(\varepsilon_{uv}\) represents the energy consumption per unit time of the slewing maneuver from \(t_{u}\) to \(t_{v}\). \(E\) is the total energy available for observation activities in one orbit.