## 3 System Model and Problem Statement
### Evaluation Metrics  
As stated in Section 3.1, the UAV swarm is aimed at collecting maximum information over Pols as long as possible. UAVs should avoid collisions with obstacles and borders during movement, and recharge in time when power is low. Following [PERSON] et al. [37] and [PERSON] et al. [45], we propose three global evaluation metrics to evaluate the effectiveness of the joint cooperation of the UAV swarm in this SAG-MCS task. These metrics are ultimately used to evaluate the DRL policy we have trained. The first metric is _Data Coverage Index_, which describes the average data amounts collected by the whole UAV swarm per timestep, as:  
\[c_{t}=\frac{\sum_{k=1}^{K}w_{t}(k)d(k)}{Kt},\quad t=1,\dots,T. \tag{3}\]  
where \(w_{t}(k)\) denotes the number of timesteps when PoI \(k\) was successfully collected from timestep 1 till \(t\). \(d(k)\) denotes the data amount carried by PoI \(k\) and \(K\) is the number of Pols. We noticed that in some cases, isolated Pols in rural areas may not be covered even when the data coverage index is quite high; however, isolated or sparse PoIs in remote areas can carry valuable information in certain scenarios such as disaster relief. Considering the comprehensiveness of the data collection task, we propose the second global metric _Geographical Fairness Index_ to evaluate the exploration ability of the UAV team, as:  
\[f_{t}=\frac{\left(\sum_{k=1}^{K}w_{t}(k)d(k)\right)^{2}}{K\sum_{k=1}^{K}(w_{t }(k)d(k))^{2}},\quad t=1,\dots,T. \tag{4}\]  
where \(w_{t}(k)\) and \(d(k)\) are defined the same as Equation (3). When all Pols are evenly covered, Equation (4) gives \(f_{t}=1\). In addition, the third metric _Energy Consumption Index_ is used to indicate the energy-saving status of the UAV swarm. In order to further simulate the energy consumed by multi-rotor UAV in reality, we adopt an equation of power on the flight speed [46], as:  
\[P_{T}=\frac{1}{2}C_{D}A\rho v^{3}+\frac{W^{2}}{\rho b^{2}v}, \tag{5}\]  
where \(C_{D}\) is the aerodynamic drag coefficient, \(\rho\) is the density of air and \(v\) is the current flying speed. Parameter \(A\), \(W\), \(b\) denote UAV's front facing area, total weight, and width, respectively. For simplicity, we adopt a general UAV model and specific values are omitted in this paper. In timestep \(t\), we assume the consumed energy \(e_{t}^{u}\) by UAV \(u\) is linear to its battery power, as:  
\[e_{t}^{u}=e_{0}+\eta_{e}P_{T}^{u}, \tag{6}\]  
where \(e_{0}\) represents hovering energy consumption and \(\eta_{e}\) is an energy coefficient. \(P_{T}^{u}\) refers to the output power of UAV \(u\) in timestep \(t\). Equations (5) and (6) reveal that UAV's battery is more efficient at an optimal cruising flight speed, while hovering or flying at maximum speed will consume more power. Note that energy consumed during flight is mainly from rotors and embedded sensors, and we ignore the communication budgets in the ad-hoc network. Therefore, we define the energy consumption index by taking the average of all \(U\) UAVs in \(T\) timesteps:  
\[e_{t}=\frac{1}{t\times U}\sum_{t=1}^{t}\sum_{u=1}^{U}e_{\tau}^{u},\quad t=1, \dots,T. \tag{7}\]After a complete simulation episode, we calculate the metrics mentioned above as _final global metrics_, denoted as \(\{c_{T},f_{T},e_{T}\}=\{c_{t},f_{t},e_{t}\}_{t=T}\). We hope to maximize the coverage and fairness index for sensing data adequately, while minimize the energy consumption index for energy-saving. Therefore, following [PERSON] et al. [37], we define the overall objective _coverage-fairness-energy score_ (CFE score) by a DRL policy \(\pi\):  
\[CFE_{t}(\pi)=\frac{c_{t}\times f_{t}}{e_{t}},\quad t=1,\dots,T. \tag{8}\]  
Obviously, our objective is to optimize the policy \(\pi\) to maximize \(CFE_{T}(\pi)\) of the whole episode. As our SAG-MCS is a practical partially observable scenario, UAV agents cannot be aware of these global metrics of the whole swarm. They can only make actions according to the decentralized policy \(\pi_{u},\forall u\in\mathcal{U}\) and self-owned information.