宗群 , 王丹丹 , 邵士凯 , 等 . 多无人机协同编队飞行控制研究现状及发展 [J ] . 哈尔滨工业大学学报 , 2017 , 49 ( 3 ): 1 - 14 .

ZONG Q , WANG D D , SHAO S K , et al. Research status and development of multi-UAV cooperative formation flight control [J ] . Journal of Harbin Institute of Technology , 2017 , 49 ( 3 ): 1 - 14 . (in Chinese)

张哲 , 吴剑 , 代冀阳 , 等 . 基于改进A * 算法的多无人机协同战术规划 [J ] . 兵工学报 , 2020 , 41 ( 12 ): 2530 - 2539 . DOI: 10.3969/j.issn.1000-1093.2020.12.019 http://doi.org/10.3969/j.issn.1000-1093.2020.12.019 多无人机协同作战是未来无人机作战方式的重要发展趋势。为增强多无人机系统的任务执行能力，提高系统整体作战效能并实现高效资源分配和调度，提出一种基于改进A^*算法的多无人机协同战术规划方法。按照离线规划和重规划两方面，设计战役层和战术层的作战目标迭代优化方案；建立编队协同作战的数学模型，以编队成员间的时间协同和碰撞协同代价为变量，得到多约束条件下的综合编队目标函数；结合多层变步长搜索策略和单步扩展的搜索方式，基于改进A^*算法，用于求解复杂战场环境下的多无人机编队协同作战航路。分别利用改进A^*算法和传统A^*算法进行对比仿真实验。仿真结果表明，多无人机协同战术规划方法能够较好地完成作战任务，改进A^*算法能够获得更优的航路，从而验证了所提算法的有效性。

ZHANG Z , WU J , DAI J Y , et al. Cooperativetactical planning of multiple UAVs based on improved A * algorithm [J ] . Acta Armamentarii , 2020 , 41 ( 12 ): 2530 - 2539 . (in Chinese)

任智 , 张栋 , 唐硕 . 基于强化学习的改进三维A * 算法在线航迹规划 [J ] . 系统工程与电子技术 , 2023 , 45 ( 1 ): 193 - 201 . DOI: 10.12305/j.issn.1001-506X.2023.01.23 http://doi.org/10.12305/j.issn.1001-506X.2023.01.23 针对飞行器在线航迹规划对算法实时性与结果最优性要求高的问题，基于强化学习方法改进三维A^*算法。首先，引入收缩因子改进代价函数的启发信息加权方法提升算法时间性能；其次，建立算法实时性与结果最优性的性能变化度量模型，结合深度确定性策略梯度方法设计动作-状态与奖励函数，对收缩因子进行优化训练；最后，在多场景下对改进后的三维A^*算法进行仿真验证。仿真结果表明，改进算法能够在保证航迹结果最优性的同时有效提升算法时间性能。

REN Z , ZHANG D , TANG S . Improved 3D A * algorithm for online track planning based on reinforcement learning [J ] . Systems Engineering & Electronics , 2023 , 45 ( 1 ): 193 - 201 . (in Chinese)

韩晓微 , 韩震 , 岳高峰 , 等 . 救灾无人机的优化A * 航迹规划算法 [J ] . 计算机工程与应用 , 2021 , 57 ( 6 ): 232 - 238 . DOI: 10.3778/j.issn.1002-8331.1912-0304 http://doi.org/10.3778/j.issn.1002-8331.1912-0304 针对抢险救灾中无人机派遣量及空间航迹规划最短路径相制约的问题，提出了一种优化A*的航迹算法。通过设计的蛇形割圆法对圆形巡查区域进行路径规划，通过提取感兴趣区域的方法选择较佳搜索方向，提高搜索效率，采用加权评估法优化自然威胁权重系数，重定义航迹估计函数。将提出的方法在灾情巡查和生命勘测实际问题中进行性能检测。仿真结果表明，该算法能够合理分配无人机数量且能快速规划出较优飞行轨迹，实现巡查覆盖率达88.96%。

HAN X W , HAN Z , YUE G F , et al. Optimal A * track planning Algorithm for Disaster relief UAV [J ] . Computer Engineering and Applications , 2021 , 57 ( 6 ): 232 - 238 . (in Chinese)

程凝怡 , 刘志乾 , 李昱奇 . 一种基于Dijkstra的多约束条件下智能飞行器航迹规划算法 [J ] . 西北工业大学学报 , 2020 , 38 ( 6 ): 1284 - 1290 .

CHENG N Y , LIU Z Q , LI Y Q . A trajectory planning algorithm for intelligent aircraft under multiple constraints based on Dijkstra [J ] . Journal of Northwestern Polytechnical University , 2020 , 38 ( 6 ): 1284 - 1290 . (in Chinese) DOI: 10.1051/jnwpu/20203861284 http://doi.org/10.1051/jnwpu/20203861284 https://www.jnwpu.org/10.1051/jnwpu/20203861284 https://www.jnwpu.org/10.1051/jnwpu/20203861284 Aiming at the rapid planning of the optimal flight path of the intelligent aircraft, considering the error constraints and correction probability constraints, a model for intelligent aircraft path planning under multiple constraints is constructed, and a global search algorithm based on Dijkstra algorithm is proposed to solve the model. By calculating the residual error and restricts flight distance, the basic Dijkstra algorithm is improved to make it more adaptable to solve the path planning under multiple constraints. At the same time, simulation experiment is conducted with the optimal goal of the shortest track length and satisfying the error constraints. The experimental results show that the aircraft passed a total of 18 correction points when it reached the destination. The total track length was 144 287.932 m, the vertical position error was 17.254 units, and the horizontal position error was 6.420 units. The results meet the error requirements. The results show that the intelligent aircraft path planning model and Dijkstra-based global search algorithm with multiple constraints are reasonable in solving such problems.

郑弈 , 谢亚琴 . 基于Dijkstra算法改进的飞行器航迹快速规划算法 [J ] . 电子测量技术 , 2022 , 45 ( 12 ): 73 - 79 .

ZHENG Y , XIE Y Q . Fast flight path planning algorithm for aircraft based on Dijkstra algorithm [J ] . Electronic Measurement Technology , 2022 , 45 ( 12 ): 73 - 79 . (in Chinese)

傅嘉晨 , 付润定 , 张亚 . 基于分布式遗传算法和改进人工势场法的导弹反探测航迹规划 [J ] . 东南大学学报(自然科学版) , 2023 , 53 ( 4 ): 709 - 717 .

FU J C , FU R D , ZHANG Y . Trajectory planning of missile counter detection based on distributed genetic algorithm and improved artificial potential field method [J ] . Journal of Southeast University (Natural Science Edition) , 2023 , 53 ( 4 ): 709 - 717 . (in Chinese)

王庆禄 , 吴冯国 , 郑成辰 , 等 . 基于优化人工势场法的无人机航迹规划 [J ] . 系统工程与电子技术 , 2023 , 45 ( 5 ): 1461 - 1468 . DOI: 10.12305/j.issn.1001-506X.2023.05.22 http://doi.org/10.12305/j.issn.1001-506X.2023.05.22 针对传统人工势场(traditional artificial potential field，TAPF)法在无人机航迹规划中存在的局部极小值、斥力过大、无效避障等问题，提出一种优化人工势场法。首先将障碍物斥力进行分解，避免了局部极小值情况；其次重构合力计算方式，避免无人机在障碍密集区域所受斥力过大；最后引入二次碰撞预测方法，减少无人机无效避障的同时保证航迹平滑。在考虑无人机物理约束条件下进行航迹规划实验。仿真结果表明，该方法相较于TAPF法，不仅缩短了规划航线长度，且在航迹平滑性上有明显提升。

WANG Q L , WU F G , ZHENG C C , et al. UAV track planning based on optimized artificial potential field method [J ] . Systems Engineering and Electronics , 2023 , 45 ( 5 ): 1461 - 1468 . (in Chinese)

韩尧 , 李少华 . 基于改进人工势场法的无人机航迹规划 [J ] . 系统工程与电子技术 , 2021 , 43 ( 11 ): 3305 - 3311 . DOI: 10.12305/j.issn.1001-506X.2021.11.31 http://doi.org/10.12305/j.issn.1001-506X.2021.11.31 针对传统人工势场(traditional artificial potential field, TAPF)方法在无人机航迹规划时航迹摆动幅度较大且容易陷入局部极小值的问题, 提出了一种改进人工势场法。首先在TAPF方法的基础上, 引入角度与速度调节因子, 模拟更真实的无人机飞行轨迹; 然后再引入辅助避障力, 实现避障的同时平滑轨迹; 最后对改进航迹规划算法与TAPF方法进行仿真实验。结果表明, 相较于传统算法, 改进后的航迹规划算法在航迹平滑性上有显著提升, 并且有效地避开了局部最小值点。

HAN Y , LI S H . UAV track planning based on improved artificial potential field method [J ] . Systems Engineering and Electronics , 2021 , 43 ( 11 ): 3305 - 3311 . (in Chinese)

张丹萌 , 甄子洋 , 陈棪 . 基于改进RRT-Connect的协同航迹规划 [J ] . 电光与控制 , 2021 , 28 ( 9 ): 25 - 29 .

ZHANG D M , ZHEN Z Y , CHEN L . Collaborative track planning based on improved RT-Connect [J ] . Electro-optics & Control , 2021 , 28 ( 9 ): 25 - 29 . (in Chinese)

张康 , 陈建平 . 复杂环境下基于采样空间自调整的航迹规划算法 [J ] . 计算机应用 , 2021 , 41 ( 4 ): 1207 - 1213 . DOI: 10.11772/j.issn.1001-9081.2020060863 http://doi.org/10.11772/j.issn.1001-9081.2020060863 针对具有渐进最优性的快速扩展随机树（RRT^*）算法在面对高维、复杂环境时所表现出的寻路效率低、收敛速度缓慢的问题，在RRT^*的基础上，提出一种基于采样空间自调整的渐进最优快速扩展随机树（AS-RRT^*）无人机（UAV）航迹规划算法。该算法可以自适应调整采样空间，进而引导树更为高效地生长，而这些主要通过有偏采样、节点筛选和节点学习这三种策略来实现。首先，在采样空间中定义向光和背光区域来进行有偏采样，而向光和背光区域的概率权重由当前扩展失败率决定，从而保证算法在搜索初始航迹时同时具有探索性和方向性；然后，在完成初始航迹的搜索后，算法就开始周期性地筛选节点，高质量的节点作为学习样本来产生新的抽样分布，质量最低的节点在算法达到最大节点数量后被新节点替代。在多种不同类型的环境下进行了对比仿真实验，结果表明所提算法在一定程度上改善了采样算法固有的随机性，而且相较于传统的RRT^*算法，该算法在相同环境里使用了更少的寻路时间，在相同时间里生成了更低代价的航迹，且在三维空间里的改进更为明显。

ZHANG K , CHEN J P . Track planning algorithm based on self-adjustment of sampling space in complex environment [J ] . Journal of Computer Applications , 2021 , 41 ( 4 ): 1207 - 1213 . (in Chinese) DOI: 10.11772/j.issn.1001-9081.2020060863 http://doi.org/10.11772/j.issn.1001-9081.2020060863 To overcome low pathfinding efficiency and slow convergence speed of Rapid-exploring Random Tree star(RRT^*) in high-dimensional and complex environment, an Unmanned Aerial Vehicle(UAV) path planning algorithm with self-adjusting sampling space based on RRT^* named Adjust Sampling space-RRT^*(AS-RRT^*) was proposed. In this algorithm, by adjusting the sampling space adaptively, the tree was guided to grow more efficiently, which was realized through three strategies including:biased sampling, node selection and node learning. Firstly, the light and dark areas in the sampling space were defined to performing biased sampling, and the probability weights of the light and dark areas were determined by the current expansion failure rate, so as to ensure that the algorithm was both exploratory and directional when searching for the initial path. Then, once the initial path was found,the nodes were periodically filter,and the high-quality nodes were used as learning samples to generate the new sampling distribution, the lowest-quality nodes were replaced by new nodes after the algorithm reaching the maximum number of nodes. Simulation experiments for comparison were conducted in multiple types of environments. The results show that the proposed algorithm improves the inherent randomness of the sampling algorithm to a certain extent, and compared with the traditional RRT^* algorithms, it has less pathfinding time used in the same environment, lower cost path generated in the same time, and the improvements are more obvious in three-dimensional space.

傅阳光 , 周成平 , 丁明跃 . 基于混合量子粒子群优化算法的三维航迹规划 [J ] . 宇航学报 , 2010 , 31 ( 12 ): 2657 - 2664 .

FU Y G , ZHOU C P , DING M Y . Three-dimensional flight path planning based on hybrid quantum particle swarm optimization [J ] . Journal of Astronautics , 2010 , 31 ( 12 ): 2657 - 2664 . (in Chinese)

方群 , 徐青 . 基于改进粒子群算法的无人机三维航迹规划 [J ] . 西北工业大学学报 , 2017 , 35 ( 1 ): 66 - 73 .

FANG Q , XU Q . UAV 3D track planning based on improved particle swarm optimization [J ] . Journal of Northwestern Polytechnical University , 2017 , 35 ( 1 ): 66 - 73 . (in Chinese)

徐瑞莲 , 周新志 , 宁芊 . 基于改进差分进化算法的多无人机航迹规划 [J ] . 火力与指挥控制 , 2020 , 45 ( 1 ): 169 - 173 ,179.

XU R L , ZHOU X Z , NING Q . Multi-UAV track planning based on improved differential evolution algorithm [J ] . Fire Control & Command Control , 2020 , 45 ( 1 ): 169 - 173 ,179. (in Chinese)

吴文海 , 郭晓峰 , 周思羽 . 基于改进约束差分进化算法的动态航迹规划 [J ] . 控制与决策 , 2020 , 35 ( 10 ): 2381 - 2390 .

WU W H , GUO X F , ZHOU S Y . Dynamic track planning based on improved constrained differentialevolution algorithm [J ] . Control and Decision , 2020 , 35 ( 10 ): 2381 - 2390 . (in Chinese)

马军 . 基于改进人工蜂群算法的多无人机航迹规划技术研究 [D ] . 南京 : 南京航空航天大学 , 2021 .

MA J . Research onmulti-UAV flight path planning technology based on improved artificial bee colony algorithm [D ] . Nanjing : Nanjing University of Aeronautics and Astronautics , 2021 . (in Chinese)

伍鹏飞 , 李涛 , 曹广旭 , 等 . 基于改进混沌蜂群算法的无人战斗机路径规划 [J ] . 中国科技论文 , 2021 , 16 ( 3 ): 301 - 306 .

WU P F , LI T , CAO G X , et al. Path planning of unmanned fighter aircraft based on improved chaotic swarm algorithm [J ] . Chinese Journal of Science and Technology , 2021 , 16 ( 3 ): 301 - 306 . (in Chinese)

李文广 , 胡永江 , 庞强伟 , 等 . 基于改进遗传算法的多无人机协同侦察航迹规划 [J ] . 中国惯性技术学报 , 2020 , 28 ( 2 ): 248 - 255 .

LI W G , HU Y J , PANG Q W , et al. Multi-UAV cooperative reconnaissance track planning based on improved genetic algorithm [J ] . Journal of Chinese Inertial Technology , 2020 , 28 ( 2 ): 248 - 255 . (in Chinese)

程泽新 , 李东生 , 高杨 . 一种改进遗传算法的无人机航迹规划 [J ] . 计算机仿真 , 2019 , 36 ( 12 ): 31 - 35 .

CHENG Z X , LI D S , GAO Y . An improvedgenetic algorithm for UAV track planning [J ] . Computer Simulation , 2019 , 36 ( 12 ): 31 - 35 . (in Chinese)

柳长安 , 王晓鹏 , 刘春阳 , 等 . 基于改进灰狼优化算法的无人机三维航迹规划 [J ] . 华中科技大学学报(自然科学版) , 2017 , 45 ( 10 ): 38 - 42 .

LIU C A , WANG X P , LIU C Y , et al. UAV 3D track planning based on improved grey wolf optimization algorithm [J ] . Journal of Huazhong University of Science and Technology (Natural Science Edition) , 2017 , 45 ( 10 ): 38 - 42 . (in Chinese)

曹建秋 , 张广言 , 徐鹏 . A * 初始化的变异灰狼优化的无人机路径规划 [J ] . 计算机工程与应用 , 2022 , 58 ( 4 ): 275 - 282 . DOI: 10.3778/j.issn.1002-8331.2008-0426 http://doi.org/10.3778/j.issn.1002-8331.2008-0426 无人机（unmanned aerial vehicle，UAV）路径规划问题是无人机任务规划系统的重要组成部分，需要在一个存在威胁区的搜索空间中获得最优路径。为解决灰狼优化算法存在收敛速度慢、容易陷入局部最优等问题，提出了一种基于A*初始化的变异灰狼优化算法。该算法首先将模型离散化，进而使用A*算法进行头狼的初始化，使后续算法有一个较优的起点，随后通过简化后的灰狼优化算法在连续模型上构建和更新种群，在迭代过程中，通过新提出的一种新型修正变异算子优化种群。利用三次B样条平滑后的无人机航迹，符合无人机的性能要求。经实验验证，算法在代价收敛速度、求取的最终路径以及算法稳定性方面均优于粒子群算法（particle swarm optimization，PSO）、灰狼优化算法（gray wolf optimizer，GWO）、共生生物搜索算法（symbiotic organisms search，SOS）算法，在解决无人机路径规划问题上具有较高的应用价值。

CAO J Q , ZHANG G Y , XU P . A * initializedvariant gray wolf optimized UAV path planning [J ] . Computer Engineering and Applications , 2022 , 58 ( 4 ): 275 - 282 . (in Chinese)

尹依伊 , 王晓芳 , 周健 . 基于Q学习的多无人机协同航迹规划方法 [J ] . 兵工学报 , 2023 , 44 ( 2 ): 484 - 495 . DOI: 10.12382/bgxb.2021.0606 http://doi.org/10.12382/bgxb.2021.0606 针对多无人机同时到达目标的航迹规划问题,建立战场环境模型和单无人机航迹规划的马尔可夫决策模型,基于Q学习算法解算航程最短的最优航迹,应用基于Q学习算法得到的经验矩阵快速解算各无人机的最短航迹并计算协同航程,通过调整绕行无人机的动作选择策略,得到各无人机满足时间协同的航迹组。考虑多无人机的避碰问题,通过设计后退参数确定局部重规划区域,基于深度Q学习理论,采用神经网络替代Q_table对局部多无人机航迹进行重规划,避免维度爆炸问题。对于先前未探明的障碍物,参考人工势场法思想设计障碍物Q矩阵,将其叠加至原Q矩阵,实现无人机的避碰。仿真结果表明:所提基于Q学习的多无人机协同航迹规划算法能够得到时间协同与碰撞避免的协同航迹,并对环境建模时所未探明的障碍物进行躲避;与A^*算法相比,针对在线应用问题,新算法具有更高的求解效率。

YIN Y Y , WANG X F , ZHOU J . Multi-uav collaborative track planning method based on Q learning [J ] . Acta Armamentarii , 2023 , 44 ( 2 ): 484 - 495 . (in Chinese)

唐嘉宁 , 杨昕 , 周思达 , 等 . 未知环境下改进DDQN的无人机探索航迹规划研究 [J ] . 电光与控制 , 2023 , 30 ( 4 ): 23 - 27 ,33.

TANG J N , YANG X , ZHOU S D , et al. Research on UAV exploration track planning with improved DDQN in unknown environment [J ] . Electro-optics & Control , 2023 , 30 ( 4 ): 23 - 27 ,33. (in Chinese)

高敬鹏 , 胡欣瑜 , 江志烨 . 改进DDPG无人机航迹规划算法 [J ] . 计算机工程与应用 , 2022 , 58 ( 8 ): 264 - 272 . DOI: 10.3778/j.issn.1002-8331.2106-0054 http://doi.org/10.3778/j.issn.1002-8331.2106-0054 针对无人机飞行过程存在未知威胁使智能算法处理复杂度高，导致航迹实时规划困难，以及深度强化学习中调整DDPG算法参数，存在时间成本过高的问题，提出一种改进DDPG航迹规划算法。围绕无人机航迹规划问题，构建飞行场景模型，根据飞行动力学理论，搭建动作空间，依据非稀疏化思想，设计奖励函数，结合人工蜂群算法，改进DDPG算法模型参数的更新机制，训练网络模型，实现无人机航迹决策控制。仿真结果表明，所提算法整体训练时长仅为原型算法单次平均训练时长的1.98倍，大幅度提升网络训练效率，降低时间成本，且在满足飞行实时性情况下，符合无人机航迹质量需求，为推动深度强化学习在航迹规划的实际应用提供新思路。

GAO J P , HU X Y , JIANG Z Y . Improved DDPG UAV trackplanning algorithm [J ] . Computer Engineering and Applications , 2022 , 58 ( 8 ): 264 - 272 . (in Chinese)

丁强 . 多无人机协同的飞行航迹规划问题研究 [D ] . 杭州 : 浙江大学 , 2018 .

DING Q . Research on flight path planning for multi-UAVcollaboration [D ] . Hangzhou : Zhejiang University , 2018 . (in Chinese)

池海红 , 周明鑫 . 融合强化学习和进化算法的高超声速飞行器航迹规划 [J ] . 控制理论与应用 , 2022 , 39 ( 5 ): 847 - 856 .

CHI H H , ZHOU M X . Trajectory planning ofhypersonic vehicle based on reinforcement learning and evolutionary algorithm [J ] . Control Theory & Applications , 2022 , 39 ( 5 ): 847 - 856 . (in Chinese)

SAREMI S , MIRIALILI S , LEWIS A , et al. Grasshopper optimisation algorithm: theory and application [J ] . Advances in Engineering Software , 2017 , 105 : 30 - 47 . DOI: 10.1016/j.advengsoft.2017.01.004 http://doi.org/10.1016/j.advengsoft.2017.01.004 https://linkinghub.elsevier.com/retrieve/pii/S0965997816305646 https://linkinghub.elsevier.com/retrieve/pii/S0965997816305646

刘琨 . 多无人机协同侦察航迹规划算法研究 [D ] . 南京 : 南京航空航天大学 , 2021 .

LIU K . Research on trajectory planningalgorithm for multi-UAV cooperative reconnaissance [D ] . Nanjing : Nanjing University of Aeronautics and Astronautics , 2021 . (in Chinese)

SAXENA A . A comprehensive study of chaos embedded bridging mechanisms and crossover operators for grasshopper optimization algorithm [J ] . Expert Systems with Applications , 2019 , 132 : 166 - 188 . DOI: 10.1016/j.eswa.2019.04.043 http://doi.org/10.1016/j.eswa.2019.04.043 https://linkinghub.elsevier.com/retrieve/pii/S0957417419302738 https://linkinghub.elsevier.com/retrieve/pii/S0957417419302738

ZHU W Y , XU K L , SUN Y , et al. Logistics distribution route planning with fusion algorithm of petri net and ant colony [J ] . Journal of Zhejiang University , 2011 , 45 ( 12 ): 2229 - 2234 .

罗学义 . 基于智能Petri网的物流配送路径优化算法 [J ] . 计算机工程与设计 , 2011 , 32 ( 7 ): 2381 - 2384 .

LUO Y X . Algorithm of logistics distribution path optimization based on intelligent Petri net [J ] . Computer Engineering and Design , 2011 , 32 ( 7 ): 2381 - 2384 . (in Chinese)