# A Hierarchical Mission Planning Method for Simultaneous Arrival of Multi-UAV Coalition
## 2 Description of the Multi-UAV Cooperative Attack Problem
### Objective Function and Constraints  
The objective of the task allocation problem is to assign the targets to the UAV team to obtain the maximum system utility. To satisfy the resource requirement of targets, there may be some UAVs forming a coalition to attack one target. The reward to destroy targets and the cost of the UAV coalitions were considered in the objective function, so, two criteria were selected to quantify the objective function of this mission planning problem: The reward to destroy targets and the cumulative path length of all the UAV coalitions. Thus, the objective function to be maximized is as follows:  
\[J=\sum_{j=1}^{M}(w_{1}\times Value_{j}\times e^{-\beta_{j}}-w_{2}\times\sum_{ i=1}^{|C_{j}|}L_{ij}) \tag{6}\]  
where \(w_{1},w_{2}\) respectively denotes the weights of the reward and cost. \(C_{j}\) represents the UAV coalition assigned to attack the target \(T_{j}\), \(\left|C_{j}\right|\) denotes the number of UAVs in the coalition \(C_{j}\), \(L_{ij}\) denotes the path length from the initial location of UAV \(U_{i}\) to the target \(T_{j}\). Assuming \(A^{(i)}\) is the task sequence of UAV \(U_{i}\), tasks before \(T_{j}\) should be considered in calculating \(L_{ij}\). \(t_{j}\) is the estimated time of arrival (ETA) of the latest UAV in the coalition assigned to the target \(T_{j}\):  
\[t_{j}=\max_{i\in C_{j}}(ETA(i,j)) \tag{7}\]  
The constraints considered in this paper can be categorized into UAV-related constraints, target-related constraints and environment-related constraints.  
(1) UAV-related constraints  
First of all, UAVs have limited fuel, so a maximum flying distance constraint should be satisfied:  
\[L(A^{(i)})\leq L_{\max}^{i},\ i=1,\ldots,N \tag{8}\]  
where \(L(A^{(i)})\) is the total path length of UAV \(U_{i}\), \(L_{\max}^{i}\) denotes the maximum flight distance of UAV \(U_{i}\).  
Each UAV can carry limited resources that deplete with use, so each UAV cannot consume more resources than it carries.  
\[R(A^{(i)})\leq R^{U_{i}} \tag{9}\]  
where \(R(A^{(i)})\) represents the total resource consumption for performing the targets assigned to \(U_{i}\).  
A flyable curve [24] is required to satisfy the UAV's kinematic constraint. In a two-dimensional plane, a curve is determined by its curvature. Let \(\kappa_{i}(q)\) be the curvature of UAV \(U_{i}\) at point \(q\), the curvature of the path should be continuous and bounded as follows:  
\[\kappa_{i}(q)\leq 1/R_{\min}^{i} \tag{10}\]  
(2) Target-related constraints  
Each target is required to be performed by a UAV coalition simultaneously, and the resource requirement should be satisfied:  
\[(\sum_{l\in C_{j}}R_{p}^{U_{j}})\geq R_{p}^{T_{j}},\ p=1,\ldots,m;\ j=1,\ldots M \tag{11}\]  
(3) Environment-related constraints  
Let \(\Omega\) be the two-dimensional space in which UAVs fly. Let \(\Theta\) denote the set of obstacles in the environment. The collision avoidance between UAVs and obstacles must be satisfied:  
\[(x(t),y(t))\cap\mathrm{int}(\Theta)=\varnothing \tag{12}\]  
(4) Simultaneous arrival constraints  
According to the resource model, each target may need a coalition of UAVs. To maximize the system performance, the UAVs in the same coalition are required to arrive to the target simultaneously as follows:  
\[t_{i,j}^{path}=t_{i,j}^{path},\ i,l\in\left\{1,\ldots,\left|C_{j}\right|\right\} ;i\
eq l \tag{13}\]  
where \(U_{i}\) and \(U_{l}\) are two different UAVs in coalition \(C_{j}\), \(T_{j}\) is the target assigned to the coalition \(C_{j}\).  
Thus, the mission planning problem was provided as the objective function (Equation (6)) with constraints (Equations (8)-(13))