# Research on the Efficient Space Debris Observation Method Based on Optical Satellite Constellations
## 3 Constellation Observation Optimal Scheduling
### Principles of Observation Task Scheduling  
Each detection device has its own independent requirements and tasks. Task scheduling is to allocate time for each device to observe the space objects every day. In brief, the satellite network observation task scheduling problem refers to scheduling different satellite observation equipment to observe space objects in the observable time period under certain constraints [27; 28]. That is, the detection scheduling result is better under certain expected effects (such as the largest number of observation targets and the highest equipment utilization). The principle of task optimization scheduling is to track as many space objects as possible [29]. At the same time, the turning angle of the telescope should be made smaller and the idle time of the telescope should be minimized. Observation is given to objects with high priority and the space objects with short arcs are observed first. The objects with small phase angles are also observed first. followed by the targets with small changes, etc. Suppose there are a total of \(N\) space objects to be observed. For any space object \(n\in\{1,\ldots,N\}\), the space object is divided into \(R\) classes, and the level of each space object is expressed as \(l_{n}\), \(l_{n}\in\{1,\ldots,R\}\). Each space object can only be observed within a time window, expressed as \([O_{n},D_{n}]\), where \(O_{n}\) is the initial time when the space object can be observed and \(D_{n}\) is the cut-off time. Suppose there are a total of \(M\) optical observation satellites, for which any satellite \(m\in\{1,\ldots,M\}\). An optical satellite can observe only one space object at a time, and likewise, each space object can only be assigned to one optical satellite for observation. The definition of the (0, 1) variables \(x_{n,m}\), \(x_{n,m}=1\) indicates that the space object to be observed \(n\) is assigned to the optical satellite \(m\), otherwise \(x_{n,m}=0\). The observation start time of the space object \(n\) is \(S_{n}\) and the observation end time is \(C_{n}\), which is expressed as the following formula. \[C_{n}=\left\{ \begin{array}{ll}S_{n}+t_{\min},&if\ D_{n}-S_{n}\geq t_{\min}\\ D_{n},&\text{Otherwise}\end{array} \right. \tag{13}\]  
From \(S_{n}\) and \(C_{n}\), the observation time \(P_{n}\) can be calculated as the following. \[P_{n}=C_{n}-S_{n} \tag{14}\]  
The time it takes for an optical satellite to observe a space object must be \(\geq\)\(t_{\min}\) seconds to be counted as a successful observation. The observation result \(k_{n}\) of the space object \(n\) is expressed as the following. \[k_{n}=\left\{ \begin{array}{ll}1,&P_{n}\geq t_{\min}\\ 0,&\text{Otherwise}\end{array} \right. \tag{15}\]  
where \(k_{n}\) is the (0, 1) variable, \(k_{n}=1\) indicates that the observation of the space object \(n\) is successful, and \(k_{n}=0\) indicates that the observation failed. In order to observe as many high-priority space objects as possible, the purpose of observation scheduling is the total weighted number \(K_{NL}\) of successful observations of space objects, which can be expressed as the following. \[K_{NL}=\sum_{n=1}^{N}\sum_{m=1}^{M}x_{n,m}\cdot k_{n}\cdot l_{n} \tag{16}\]  
Therefore, the constellation observation scheduling model is established as follows. \[\max\ K_{NL}=\sum_{n=1}^{N}\sum_{m=1}^{M}x_{n,m}\cdot k_{n}\cdot l_{n} \tag{17}\]The greedy algorithm based on the multi-dimensional list is used for constellation observation task scheduling, and the algorithm is introduced below. 1. Algorithm Fundamentals  
The multi-dimensional dynamic list programming algorithm is a fast heuristic approach for solving the match between the tasks and resources. It is a combination of a set of algorithms, which can be roughly divided into two parts, namely [30; 31]:  
1. To determine the priority function of the task. 2. To determine the priority function of the observation platform.