# A Hierarchical Mission Planning Method for Simultaneous Arrival of Multi-UAV Coalition
## 4 Path Estimation and Simultaneous Arrival Path Planning
### Path Planning for Simultaneous Arriual  
When the coalition \(C_{j}\), \(T_{j}\in T\) is determined to attack the target \(T_{j}\), the UAVs in the coalition \(C_{j}\) should arrive at the target simultaneously to maximize the system performance. The simultaneous arrival problem for the UAVs in the coalition \(C_{j}\) can be regarded as a constrained optimization problem:  
\[ \begin{array}{ll}\min&J(x_{i})\quad i=1,\ldots,\left|C_{j}\right|\\ s.t.&Path\_i\
ot\in\text{ obstacles}\\ &\kappa(Path\_i)\leq\kappa_{\max,i}\\ &t_{i,j}^{path}=t_{i,j}^{path},i,1\in\left\{1,\ldots,\left|C_{j}\right|\right\} ;i\
eq l;j=1,\ldots,M\end{array}  \tag{28}\]The cost function and the first two constraints are the same as Equation (24). For the simultaneous arrival problem, the third constraint requires the UAVs in the same coalition to realize the simultaneous arrival. It should be noted that, in this paper, the UAVs and targets are modeled by their resources. Each target may require a UAV coalition, and each UAV can attack a set of targets. So, the task sequence of UAVs should be considered in the third constraint, which means that the target assigned before \(T_{j}\) should be considered in the simultaneous arrival paths. The following two cases illustrate how to obtain the simultaneous arrival paths for the UAVs in the coalition \(C_{j}\). (1) If the target \(T_{j}\) is the first assigned target. If the target \(T_{j}\) is the first assigned target, which means the target \(T_{j}\) is the first target in the task sequence for the UAVs in the coalition \(C_{j}\). A cooperative particle swarm optimization (CPSO) algorithm in our previous study [31] can be used to solve the simultaneous arrival problem. Here is a brief explanation: A sub-swarm is generated for each UAV, the positions and velocities of the particles in each sub-swarm are initialized. Each particle's fitness was calculated using Equation (25) or Equation (26; if the particle violates the first two constraints). Then, each sub-swarm selects its best global position as its representative particle. The particles in each sub-swarm modifies their fitness by communicating with other sub-swarms' representative particles. After several rounds of iteration, the UAVs in the coalition \(C_{j}\) can obtain the paths of equal length. (2) If the target \(T_{j}\) is not the first assigned target. If the target \(T_{j}\) is not the first assigned target, the UAVs in the coalition \(C_{j}\) may have tasks before the target \(T_{j}\). For the simultaneous arrival problem, the UAVs in the coalition \(C_{j}\) should consider the targets performed before the target \(T_{j}\), a modified CPSO algorithm is presented here. In the modified CPSO algorithm, a sub-swarm is initialized for each UAV in the coalition \(C_{j}\). The positions and velocities of the particles in sub-swarm \(i\) can be represented as \(z_{ik}=(h_{ik,0},h_{ik,1})\) and \(v_{ik}=(v_{ik,0},v_{ik,1})\), respectively. Where \(i=1,\ldots,\left|C_{j}\right|\), \(k=1,\ldots,M_{p}\), \(M_{p}\) denotes the population size of each sub-swarm. It should be noted that each particle in sub-swarm \(i\) determines a PH path from the location of target \(T_{m}\) to \(T_{j}\), where \(T_{m}\) denotes the target performed by the UAV \(U_{i}\) just before the target \(T_{j}\). The fitness of the particles in each sub-swarm were calculated using Equation (25) or Equation (26). However, as the target \(T_{j}\) was not the first assigned target, the target assigned before \(T_{j}\) should be considered in the cost function. For example, for the particles in sub-swarm \(i\), the cost function for path length is modified as:  
\[J_{length}=L_{ij}=L_{im}^{before}+L_{mj} \tag{29}\]  
where \(L_{im}^{before}\) denotes the accurate path length from the current location of UAV \(U_{i}\) to the target \(T_{m}\), which can be obtained in the previous allocation round. \(L_{mj}\) denotes the path length from the location of the target \(T_{m}\) to \(T_{j}\).