## 2 Problem Description and Modelling  
In this section, we introduce the observing schedule problem in the AEOS constellation and propose the corresponding mathematical model. As shown in Figure 2, the AEOS constellation consists of \(Z_{o}\) adjacent orbital planes with \(Z_{s}\) satellites in each orbital plane, i.e., it contains \(N_{s}=Z_{o}\times Z_{s}\) satellites. Each AEOS in the constellation is connected to four adjacent satellites through ISLs, two of which are the closest in the same orbital plane, and two are the closest in adjacent orbital planes. Although all satellites move at high speed, their ISL-based connections always exist. For example, as illustrated in Figure 2, AEOS \(\lessapprox\) connects not only \(s_{2}\) and \(s_{8}\) (in the same plane), but also the closest hetero-orbiting satellites \(s_{4}\) and \(s_{6}\). In this paper, AEOS \(s\) is called a neighbor of \(c\) if it is connected by an ISL. A group of AEOSs in the constellation, denoted by \(S=\{s_{1},s_{2},\ldots,s_{m}\}\), \(m\leq N_{s}\), is equipped with cameras to observe a set of ground _spot_ targets denoted by \(G=\{g_{1},g_{2},\ldots,g_{n}\}\). The geographic positions, observation times \(d_{g}\), and profits \(\phi_{g}\) for each target \(g\in G\) are given by users in advance. We use an \(m\times m\) matrix \(H\) to describe the ISL connections among AEOSs, where \(H[s,c]=1\) represents that satellites \(s\) and \(c\) can exchange messages via ISL, and otherwise \(H[s,c]=0\). Some useful notations are summarized in Table 1.  
During the scheduling horizon, an AEOS \(s\) may circle the earth several times and a target can be observed by \(s\) in different orbits. Herein, an orbit is referred to the time period that the satellite flies through the sunshine area while circling the Earth one time. Let \(R\) be the set of all orbits of AEOSs in \(S\). According to the satellite movement parameters and camera parameters, in an orbit \(k\in R\), a target \(g\in G\) is visible for a satellite \(s\in S\) during a visible time window (VTW) denoted by \(W_{sg}^{k}=[st_{sg}^{k},et_{sg}^{k}]\), where \(st_{sg}^{k}\) (respectively, \(et_{sg}^{k}\)) is the start (respectively, end) time. Moreover, for each time \(t\in W_{sg}^{k}\), the look angle of \(s\) on three axes (roll, pitch, and yaw) is provided as \(\Psi_{sg}(t)=\varsigma\gamma_{sg}[t]\), \(\pi_{sg}[t]\), \(\psi_{sg}[t]\)\(\succ\). All VTWs of AEOS \(s\) regarded with target \(g\) can be denoted by \(\mathbf{W}_{sg}=\{W_{sg}^{k}\}\,|\,k\in R\). Note that each target \(g\) can be observed at most once by one satellite during one VTW of \(\{\mathbf{W}_{sg}\,|\,s\in S\}\). Owing to the finite camera resources and visible constraints, only a set of targets can be observed in the specific horizon. The AEOS observation schedule aims to assign targets for different AEOSs in \(S\) such that the objective function is somehow optimized while satisfying the VTW limitations and transition time constraints. Firstly, based on the work [29], we make certain assumptions to simplify the scheduling problem. Figure 2: (**a**) An AEOS constellation with 36 AEOSs; (**b**) ISLs connections where \(s_{5}\) connects the closest AEOSs \(s_{2}\) and \(s_{8}\) are in the same plane but with \(s_{4}\) and \(s_{6}\) also being in adjacent orbital planes. **Assumption:** (1) Each satellite has enough onboard power and memory in every orbit, and hence, we do not consider the power and memory constraints; (2) each target is a spot one, which can be observed by AEOS with one pass; (3) we focus on the semiagile satellite AS-01 whose look angles are unchanged when observing a target. We intend to schedule a target sequence \(\mathbf{a}_{s}=\)\(<\)\(\mathbf{\mathit{S}}_{g1}\), \(\mathbf{\mathit{S}}_{g2}\),..., \(\mathbf{\mathit{S}}_{\mathit{S}}\)\(\mid\)\(\mathbf{\mathit{AS}}\)\(\mid\)\(>\) for each AEOS \(s\in S\) such that targets in \(\mathbf{a}_{s}\) are observed sequentially by \(s\).