# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 2 System Model
### Problem Formulation  
With the addition of virtual vertices and virtual arcs in TEG, the mission execution process in SGCN can be modeled as the flow originated from an observation object vertex and the destinated virtual vertex in TEG. Specifically, the implementing process of mission \(om_{n,i}\) is modeled by the flow set \(F_{n,i}\) in TEG, which is expressed as  
\[F_{n,i}=\{o_{n,i}^{sk}\to u^{ek}\bigg{|}\bigg{|}\left[\frac{st}{\tau}\right] \leq sk\leq\bigg{|}\left[\frac{et}{\tau}\right],ek=\bigg{|}\left[\frac{et}{ \tau}\right]\}. \tag{5}\]  
The set of flows corresponding to all the missions in the network is given by  
\[\mathcal{F}=\bigcup_{1\leq n\leq N}F_{n}=\bigcup_{1\leq n\leq N}\bigcup_{1\leq i \leq|OM_{n}|}F_{n,i}, \tag{6}\]  
where \(F_{n}\) is the flow set corresponding to the missions of the \(n\)-th RSS system. By modeling the missions as the flows of TEG, the coordinated resource scheduling problem can be transformed into the multicommodity flow problem. Let the Boolean variable \(z_{n,i}\in 0,1\) represent whether the mission \(om_{n,i}\) is successfully scheduled. As coordinated resource scheduling aims at maximizing the number of successfully scheduled missions, the target of the corresponding multicommodity flow problem is expressed follows:  
\[\max\sum_{1\leq n\leq N}\sum_{1\leq i\leq|OM_{n}|}z_{n,i} \tag{7}\]  
Let \(x(f)\) denote the value of flow \(f\in\mathcal{F}\) on TEG. Let \(x(a,f)>0\) denote the amount of flow \(f\) on arc \(a\in A\). The flows in set \(\mathcal{F}\) should satisfy the following constraints:  
(1) Data Volume Constraint  
The data volume constraint guarantees that for a successfully scheduled mission \(om_{n,i}\), the total amount of data acquired and transmitted by RSSs within the scheduling horizon satisfies mission demands \(b_{n,i}\), which is given by  
\[\sum_{f\in F_{n,i}}x(f)=z_{n,i}\cdot b_{n,i},\forall 1\leq i\leq|OM_{n}|,\,1 \leq n\leq N \tag{8}\]Equation (8) imposes that if the mission \(om_{n,i}\) is successfully executed, the sum volume of the corresponding flow in TEG is \(b_{n,i}\), and 0 otherwise. (2) Flow Conservation Constraint  
For each flow \(f\in\mathcal{F}\) of TEG, the value entering any vertex except its source and sink equals the value out of this vertex. That is,  
\[\sum_{h(a)=v}x(a,f)-\sum_{t(a)=v}x(a,f)=\left\{ \begin{array}{cc}-x(f)&v=s(f), f\in\mathcal{F}\\ 0&v\in V-\{s(f),d(f)\},f\in\mathcal{F}\\ x(f)&v=d(f),f\in\mathcal{F}\end{array} \right. \tag{9}\]  
where \(h(a)\) and \(t(a)\) denote the head and tail of arc \(a\) in TEG, respectively. \(s(f)\) and \(d(f)\), respectively, represent the source and destination vertex of the flow \(f\) in TEG. (3) Conflict Constraints  
Due to the limited service capability of antenna/imager and the restriction of satellite platform attitude, there are conflicts between the schedule of the same observation and communication resources in SGCN. For example, each RSS can only observe one target in each time slot. To model this kind of conflict, we introduce a set of Boolean variables \(y(os^{k}_{n,i,j})\in\{0,1\}\) to indicate whether the observation resources of \(s_{n,j}\) are scheduled to observe \(o_{n,i}\) in the \(k\)-th time slot. Specifically, \(y(o^{k}_{n,j},s^{k}_{n,i})=1\) means that the observation resources of \(s_{n,j}\) are scheduled to observe \(o_{n,i}\) in the \(k\)-th time slot, and 0 otherwise. The conflicts of observation resources can be expressed as  
\[\sum_{i:os^{k}_{n,i,j}\in A_{O}}y(os^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k\leq K,1\leq j\leq|S_{n}|,1\leq n\leq N. \tag{10}\]  
To avoid the waste of observation resources, it is restricted that each observation object can only be observed once during the scheduling horizon, which can be formulated as  
\[\sum_{j:os^{k}_{n,i,j}\in A_{O}}y(os^{k}_{n,i,j})\leq 1,\quad\forall 1\leq i\leq| OM_{n}|,1\leq n\leq N. \tag{11}\]  
Each RSS has two transceivers to communicate with the ground stations and the DRSs, respectively.