## 2 Problem Description  
In this paper, the objective of a path-planning algorithm is to generate an optimum route based on a given risk map of the flying area, which should also be suitable for all the UAVs to follow and with the lowest possible flying cost. As shown in Figure 1, a risk map is provided in the form of grids. The size of this ground risk map is 30 km \(\times\) 20 km. The different colors of the grids show different risk levels of the ground areas, whose total number is 600. The grids with higher risk levels should not be in the way of the UAV'soperation, which means that the ground areas with higher risk should be avoided in the UAV's route. In this way, determining a path from a given starting point \(S\) to a destination point \(D\) with the lowest risk as well as the lowest flying cost becomes a key problem.  
As illustrated in Figure 1, a complete UAV flying path can be composed of many waypoints. For example, there are seven waypoints in total in this figure, which are \(WP_{1}\), \(WP_{2}\), \(WP_{3}\), \(WP_{4}\), \(WP_{5}\), \(WP_{6}\), and \(WP_{7}\). By using these seven waypoints, a flying path for UAVs can be formulated. The number of waypoints is variable and can be set differently based on the UAV's operation scenario. Therefore, determining the positions of waypoints with the lowest flying cost as well as the lowest ground risk becomes the target of a path-planning algorithm. Meanwhile, a generated UAV route should be guaranteed to be smooth enough for UAVs to follow.  
According to the analysis above, a ground risk map would be set as an input for the path-planning model in the proposed BSO method. It is assumed that the ground risk map can be provided in advance, which will be used directly as a constraint condition to make the overall risk level of a flying path as low as possible. Besides the ground risk, some other factors are taken into account, including the requirements of UAVs themselves, as shown in Figure 2. In this paper, the most basic method is to generate collision-free paths in two-dimensional environments with the lowest path risk. It is known that the complexity analysis is quite important, which can predict the execution time of the algorithm as a function of the problem's input size. In the proposed path-planning algorithm, the input size mainly consists of two variables, which represent the complexity of the UAVs and low-altitude airspace, respectively. In this condition, the complexity of a UAV path-planning algorithm would increase significantly compared with the traditional path-planning problems. Usually, the spatial pose of a UAV or its configuration is uniquely defined by its degrees of freedom. Correspondingly, the degrees of freedom of UAVs have possibly become the most crucial parameter in determining the complexity of the path-planning problem. So, the diverse range of air-borne movements a UAV may utilize in the airspace introduces an exponential growth in the potential paths available, which further presents significant complexity. In order to reduce the complexity, some assumptions are made in the proposed model. It is generally accepted that all the UAVs are assumed to be some mass points, ideally, in which condition elevation levels and taking off/landing trajectories are reasonably neglected. Based on the analysis and calculations, the time complexity level would be above \(O(n^{2})\). Figure 2 gives the overall framework of the proposed method. Firstly, when the UAVs are following a determined flying path, the turning angles between any two sequential waypoints should satisfy their operation limits. Subsequently, the length of a generated path should be as short as possible to reduce energy consumption and extend the UAV's lifetime further. As a result, the path risk constraint, path turn constraint, and path length constraint are included in the proposed path-planning model, which is shown in Figure 2.  
Figure 1: Problem description.