# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 2 System Model
### Problem Formulation
Because of using single-access antenna, an RSS can establish a communication link with at most one DRS in a time slot. Let Boolean variables \(y(sr^{k}_{n,i,j})\in\{0,1\}\) denote whether link \((s_{n,i},r_{j})\) are scheduled in the \(k\)-th time slot. Then, the conflict of communication resources of RSSs can be expressed as  
\[\sum_{j:sr^{k}_{n,i,j}\in A_{RL}}y(sr^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k \leq K,1\leq i\leq|S_{n}|,1\leq n\leq N. \tag{12}\]  
Likewise, an RSS can only communicate with at most one ground station in a time slot, even if there are multiple ground stations in its coverage range, which is formulated as  
\[\sum_{j:sg^{k}_{n,i,j}\in A_{GL}}y(sg^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k \leq K,1\leq i\leq|S_{n}|,1\leq n\leq N, \tag{13}\]  
where the Boolean variable \(y(sg^{k}_{n,i,j})\in\{0,1\}\) denotes whether link \((s_{n,i},g_{n,j})\) is scheduled in the \(k\)-th time slot. Similarly, a DRS/ground station can communicate with only one RSS in the same time slot, which can be expressed as  
\[\sum_{n,i:sr^{k}_{n,i,j}\in A_{RL}}y(sr^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k \leq K,1\leq j\leq|R|, \tag{14}\]\[\sum_{isg^{k}_{n,i,j}\in A_{GL}}y(sg^{k}_{n,i,j})\leq 1,\quad\forall 1\leq k\leq K,1\leq j\leq|G_{n}|,1\leq n\leq N. \tag{15}\]  
(4) Capacity Constraints  
The capacity constraint models the constraints of the capacity of observation, communication, and storage resources on the mission execution process by restricting the flow volume on the arcs in TEG, which are given by  
\[\sum_{f\in F_{n}}x(ss^{k}_{n,i},f)\leq C(ss^{k}_{n,i}),\quad\forall ss^{k}_{n,i }\in A_{5}, \tag{16}\]  
\[\sum_{f\in F_{n,i}}x(os^{k}_{n,i,j},f)\leq C(os^{k}_{n,i,j})\cdot y(os^{k}_{n, i,j}),\quad\forall os^{k}_{n,i,j}\in A_{O}, \tag{17}\]  
\[\sum_{f\in F_{n}}x(sg^{k}_{n,i,j},f)\leq C(sg^{k}_{n,i,j})\cdot y(sg^{k}_{n, i,j}),\quad\forall sg^{k}_{n,i,j}\in A_{GL}, \tag{18}\]  
\[\sum_{f\in F_{n}}x(sr^{k}_{n,i,j},f)\leq C(sr^{k}_{n,i,j})\cdot y(sr^{k}_{n, i,j}),\quad\forall sr^{k}_{n,i,j}\in A_{RL}. \tag{19}\]  
To be specific, the capacity constraint of the storage arc in Equation (16) imposes that the amount of data cached in RSSs in any time slot does not exceed its storage volume. The capacity constraint of the observation arc in Equation (17) shows that if the observation resource \(os^{k}_{n,i,j}\) is scheduled, the total volume of the flows on arc \(os^{k}_{n,i,j}\) does not exceed its capacity; otherwise, the flow volume of all flows on arc \(os^{k}_{n,i,j}\) is zero. This constraint ensures that the observation resource of RSS observes the target only in the slots being scheduled, and the amount of data acquired in each slot does not exceed the capacity of the corresponding observation resource. Similarly, Equations (18) and (19) enforce that the links between RSS to ground stations/DRSs transmit only in the slots being scheduled, and the amount of data transmitted in each slot does not exceed the capacity of the corresponding link. In summary, the coordinated resource scheduling problem can be formulated as the following optimization problem:  
\[\sum_{j:os^{k}_{n,i,j}\in A_{O}}y(os^{k}_{n,i,j})\leq 1,\quad\forall 1\leq i\leq| OM_{n}|,1\leq n\leq N. \tag{11}\]0 \[\sum_{j:os^{k}_{n,i,j}\in A_{O}}y(os^{k}_{n,i,j})\leq 1,\quad\forall 1\leq i\leq| OM_{n}|,1\leq n\leq N. \tag{11}\]1  
It can be observed from P1 that the objective and all the constraints satisfy linear conditions. \(x(f)\) and \(x(a,f)\) are continuous variables, and \(z_{n,i},y(os^{k}_{n,i,j}),y(sg^{k}_{n,i,j})\), and \(y(sr^{k}_{n,i,j})\) are integer variables. Therefore, problem P1 is a mixed-integer linear programming problem [23]. Moreover, P1 is a centralized optimization model; directly solving P1 not only requires global mission and resource information but also suffers from high complexity. In order to distributedly solve the resource scheduling in polynomial time under the coordination of the RSS and DRS operation control center, P1 is transformed and decomposed in the next section.