## 5 Conclusions  
In this paper, an efficient method of collision prediction and avoidance maneuvering is represented with a rolling optimization algorithm based on predictive state space. A team of cooperative and homogeneous UAVs is considered to fly in the common airspace, which is described by grid model, and the UAV waypoints are limited to be the crossing points among cubes. Under some restrictions, optimal trajectories with the shortest total trajectories of the UAVs involved are generated. The simulation results have validated the effectiveness of the proposed method.  
The main contributions of this paper include:  
* A novel and effective collision avoidance maneuver is presented, and the scenarios have validated the effectiveness of the proposed algorithm. The collision-free trajectories can be obtained in a short time, and performance of the better results depends on the run duration of the algorithm.
* The variables in the ROA can adjust freely according to the specific tasks.
* Predictive state space is used to reduce the dimension of the waypoints of UAVs in the next move, which has decreased the computation complexity greatly and improved the efficiency of the proposed algorithm.
* The rolling optimization algorithm makes good performance by getting rid of local optimal value of the objective function, obtaining better results as the algorithm continues.  
The main future work will consider the problem that real UAVs may make fine tuning adjustments when following the generated trajectories to cause potential collisions. A diagrammatic sketch is shown in Figure 8. In the figure, the blue solid line is the trajectories generated by ROA, and the red dashed line denotes the real trajectories of UAVs. The collision (distance shorter than the safe distance) might happen when the UAVs do not fly as planned. In the future work, we will research what the real trajectory of UAVs would be after the generated trajectory is received. What is more, we will research a trajectory smoothing algorithm to produce more robust trajectories for UAVs.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Total Distance of UAVs (m)** & **Average Velocity (m/s)** & **Run Duration (s)** & **Rd** \\ \hline PN Method [23] & 20,000 & 176 & 119.66 & 1.0530 \\ RRT Method [27] & 6000 & 40 & 148.70 & 0.9913 \\ ROA1 & 2200 & 2.5 & 2.38 & 0.0027045 \\ ROA5000 & 2200 & 2.5 & 11900 & 13.523 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of evaluation index comprised of other methods. PN: proportional navigation; RRT: rapidly-exploring random tree.  
**Acknowledgments:** The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (71601181), and the State Key Laboratory of Management and Control for Complex Systems-Open Project (20160109).  
**Author Contributions:** [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments, [PERSON] and [PERSON] helped in the experiment; [PERSON] and [PERSON] analyzed the data; [PERSON] and [PERSON] wrote the paper.  
**Conflicts of Interest:** The authors declare no conflicts of interest.