# Optimization Dubins Path of Multiple UAVs for Post-Earthquake Rapid-Assessment
## 2 Problem Modeling
### Heterogeneous UAVs  
Let \(\mathbf{U}=\{1,\cdots,k,\cdots,K\}\) be a set of \(K\) UAVs which are heterogeneous. The UAV heterogeneity is manifested in terms of different airspeeds \(v_{k}\), different turning radii \(\rho_{k}\), different endurances \(T_{\max}^{k}\) and different detection errors of the carried sensors \(p_{k}\). Since UAVs are relatively scarce disaster-relief resources, we assume that each rescue team has only one UAV and that each UAV is launched from a different depot. In reality, most UAVs are equipped with automatic obstacle avoidance devices and stabilizing devices. Although the wind gusts and obstacles will affect the UAVs' motion trajectory, making the UAVs' actual flight trajectory slightly different from the pre-planned path, it will not cause the UAVs to fail to perform the rapid assessment task. Therefore, we also assume that the UAVs have \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & \multicolumn{1}{c}{**Explanation**} \\ \hline \(\mathbf{U}\) & set of UAVs, \(\mathbf{U}=\{1,\cdots,k,\cdots,K\}\). \\ \hline \(k\) & index of UAVs, \(k\in\mathbf{U}\). \\ \hline \(\mathbf{T}\) & set of potential targets, \(\mathbf{T}=\{1,\cdots,i,\cdots,N\}\). \\ \hline \(i,j\) & index of potential targets, \(i,j\in\mathbf{T}\). \\ \hline \(\mathbf{D}\) & set of UAV launching depots, \(\mathbf{D}=\{D_{1},\cdots,D_{k^{\prime}},\cdots,D_{K}\}\). \\ \hline \(\mathbf{D^{\prime}}\) & set of UAV landing depots, \(\mathbf{D^{\prime}}=\{D_{1},\cdots,D_{k},\cdots,D_{K^{\prime}}\}\). \\ \hline \(\mathbf{V}\) & set of all vertices, \(\mathbf{V}=\mathbf{T}\,\cup\mathbf{D}\cup\mathbf{D^{\prime}}\). \\ \hline \(H\) & set of heading angle discretization, \(H=\left\{\theta_{h}:\theta_{h}=\frac{2\pi h}{N_{0}},h=0,1,\ldots,N_{0}-1\right\}\). \\ \hline \(q_{i}^{k}\) & the configuration of UAV \(k\) when accessing targets \(i\), \(q_{i}^{k}=(x_{i},y_{i},\theta_{h})\). \\ \hline \(R_{k}\) & feasible route of UAV \(k\), a configuration sequence of vertices visited by UAV \(k\), \(R_{k}=\left\{q_{D_{k}},q_{i}^{k},\cdots,q_{j}^{k},q_{D_{k^{\prime}}}\right\}\), \(i,j\in\mathbf{T}k\in\mathbf{U}\). \\ \hline _Scheme_ & feasible routing scheme for the MURARP, \(\mathit{Scheme}=[R_{1};\ldots;R_{k};\ldots;R_{K}],k\in\mathbf{U}\). \\ \hline \(v_{k}\) & airspeed of UAV \(k\), \(k\in\mathbf{U}\). \\ \hline \(p_{k}\) & turning radius of UAV \(k\), \(k\in\mathbf{U}\). \\ \hline \(T_{\max}^{k}\) & endurance of UAV \(k\), \(k\in\mathbf{U}\). \\ \hline \(w_{i}\) & weight of a potential target \(i\), \(i\in\mathbf{T}\). \\ \hline \(p_{k}\) & error probability of the sensor carried by UAV \(k\), \(p_{k}\in(0,1)\). \\ \hline \(P_{i}\) & probability of successfully obtaining a profit for visiting potential target \(i\), \(i\in\mathbf{T}\). \\ \hline \(d_{ij}^{k}\) & distance between vertices \(i\) and \(j\) under the turning radius of UAV \(k\), \(i,j\in\mathbf{V}k\in\mathbf{U}\). \\ \hline \(t_{ij}^{k}\) & travel time between vertices \(i\) and \(j\) for UAV \(k\), \(i,j\in\mathbf{V}k\in\mathbf{U}\). \\ \hline \(d_{R_{k}}\) & total flight distance of route \(R_{k}\), \(k\in\mathbf{U}\). \\ \hline \(t_{R_{k}}\) & total flight time of route \(R_{k}\), \(k\in\mathbf{U}\). \\ \hline \(y_{i}^{k}\) & times that potential target \(i\) is visited by UAV \(k\), \(i\in\mathbf{T}k\in\mathbf{U}\). \\ \hline \(x_{ij}^{k}\) & binary variable representing whether UAV \(k\) travels from point \(i\) and \(j\), \(i,j\in\mathbf{A},k\in\mathbf{U}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of major notations of sets, indices, parameters, and variables of MURARP. the ability to avoid obstacles automatically and can perform rapid assessment tasks safely, and the resulting path deviation from the total flight path length is very small and can be ignored. In this paper, the UAV motion state is described by the Dubins car model [9]. The state of a Dubins car is represented by \(q=(x,y,\theta)\),which consists of its position in the plane \((x,y)\) and its heading angle \(\theta\).