第4期 ZHANG Tianjie,et al.:A modified consensus algorithm for multi-UAV formations based on pigeon-inspired optimization with a slow diving strategy ·579 suitable as commands for multi-UAV formations. the designed trajectory. Performance of the z trajectories can be evaluated 40m UAV by the following criterion:diving slowly and never 30外 UAV;UAV: flying up and down.In Fig.13,all the trajectories 20 UAV quickly go down before t=4 s,this means that the 10UAV,UAV.UAV,UAV, UAVs dive sharply in the beginning,as shown in Fig. UAV o 1520 5.The aim of this work is to overcome this shortcoming. Fig.15 Z trajectories of 9 UAVs by PIO-based consensus The trajectories in Fig.14 clarify the advantage of 40 PSO-based consensus because the modified algorithm UAV can effectively reduce oscillation.UAV,and UAVs 20 AV? clarify UAVs this view as they both slowly decrease I0 AV.DAKV UAV their height following the desired trajectory instead of UAV 0 10 1520 diving down swiftly. 修 40- Fig.16 Z trajectories of 9 UAVs by SD-PIO-based consensus 30 UAV The SD-PIO-based consensus algorithm also -UAV 号20 follows the steps above,but the difference occurs in -UAV UAV the first 8 s as the UAVs descend more slowly and 10 smoothly.UAV,UAV2 and UAV3 can support this A UAV, 0 10 15 20 opinion,as they at the same respective height,but at ti t=5 s,UAV2 and UAV3 by SD-PIO are above 20 m Fig.13 Z trajectories of 9 UAVs by consensus while their counterparts are clearly lower.At t=7 s, 40 the height of UAV,by SD-PIO is 16.04 m and the outcome by PIO is only 15.47 m. 30 UAV From the fitness function Eg.(8),the closer the 2 -UAV:UAV UA UAV UAV;approaches its desired location,the smaller the AV 10 一UAV。UAV, fitness value.In Fig.17,as iteration Nc increases, UAV average fitness values decrease and finally approach 0 5 10 15 20 tis zero,and different optimizers possess different convergence rates.In Fig.17(a),PIO shows the Fig.14 Z trajectories of 9 UAVs by PSO-based consensus fastest speed in the beginning,SD-PIO is a bit slower However,reduplicative climb and dive still exist, and PSO shows the weakest performance of the three. which may induce oscillation.The dangerous However,after Ne 80,PIO becomes slow and is trajectories of UAV3 and UAVs at UAVst [0,10]s finally overtaken by SD-PIO at Ne 93.The poor efficiency of PIO is due to the inadequate local in Fig.14 reveal that they are not the desired ones that optimum. descend slowly at the beginning,then converge to the 12 desired trajectory.This is why there is great demand for -SDPIP 10 ·PIO another optimizer. oPSO In Fig.15,PIO is used to apply a gliding procedure.Before 4s,the map and compass operator takes effect and all the UAVs fly towards the best local position.In the time range,4 to 6 s,the landmark operator drives the UAVs nearby to the same height. When all of them fly to the standard height,the PIO 100 200300400 500 part stops working and consensus continues Nc concentrating the UAVs and guiding them to fly along (a)fitness valueｓｕｉｔａｂｌｅ ａｓ ｃｏｍｍａｎｄｓ ｆｏｒ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎｓ． Ｐｅｒｆｏｒｍａｎｃｅ ｏｆ ｔｈｅ ｚ ｔｒａｊｅｃｔｏｒｉｅｓ ｃａｎ ｂｅ ｅｖａｌｕａｔｅｄ ｂｙ ｔｈｅ ｆｏｌｌｏｗｉｎｇ ｃｒｉｔｅｒｉｏｎ： ｄｉｖｉｎｇ ｓｌｏｗｌｙ ａｎｄ ｎｅｖｅｒ ｆｌｙｉｎｇ ｕｐ ａｎｄ ｄｏｗｎ． Ｉｎ Ｆｉｇ． １３， ａｌｌ ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｑｕｉｃｋｌｙ ｇｏ ｄｏｗｎ ｂｅｆｏｒｅ ｔ ＝ ４ ｓ， ｔｈｉｓ ｍｅａｎｓ ｔｈａｔ ｔｈｅ ＵＡＶｓ ｄｉｖｅ ｓｈａｒｐｌｙ ｉｎ ｔｈｅ ｂｅｇｉｎｎｉｎｇ， ａｓ ｓｈｏｗｎ ｉｎ Ｆｉｇ． ５． Ｔｈｅ ａｉｍ ｏｆ ｔｈｉｓ ｗｏｒｋ ｉｓ ｔｏ ｏｖｅｒｃｏｍｅ ｔｈｉｓ ｓｈｏｒｔｃｏｍｉｎｇ． Ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｉｎ Ｆｉｇ． １４ ｃｌａｒｉｆｙ ｔｈｅ ａｄｖａｎｔａｇｅ ｏｆ ＰＳＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ｂｅｃａｕｓｅ ｔｈｅ ｍｏｄｉｆｉｅｄ ａｌｇｏｒｉｔｈｍ ｃａｎ ｅｆｆｅｃｔｉｖｅｌｙ ｒｅｄｕｃｅ ｏｓｃｉｌｌａｔｉｏｎ． ＵＡＶ２ ａｎｄ ＵＡＶ８ ｃｌａｒｉｆｙ ＵＡＶ８ ｔｈｉｓ ｖｉｅｗ ａｓ ｔｈｅｙ ｂｏｔｈ ｓｌｏｗｌｙ ｄｅｃｒｅａｓｅ ｔｈｅｉｒ ｈｅｉｇｈｔ ｆｏｌｌｏｗｉｎｇ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ ｉｎｓｔｅａｄ ｏｆ ｄｉｖｉｎｇ ｄｏｗｎ ｓｗｉｆｔｌｙ． Ｆｉｇ． １３ Ｚ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ｃｏｎｓｅｎｓｕｓ Ｆｉｇ． １４ Ｚ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ＰＳＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ Ｈｏｗｅｖｅｒ， ｒｅｄｕｐｌｉｃａｔｉｖｅ ｃｌｉｍｂ ａｎｄ ｄｉｖｅ ｓｔｉｌｌ ｅｘｉｓｔ， ｗｈｉｃｈ ｍａｙ ｉｎｄｕｃｅ ｏｓｃｉｌｌａｔｉｏｎ． Ｔｈｅ ｄａｎｇｅｒｏｕｓ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ＵＡＶ３ ａｎｄ ＵＡＶ５ ａｔ ＵＡＶ５ ｔ∈［０， １０］ ｓ ｉｎ Ｆｉｇ．１４ ｒｅｖｅａｌ ｔｈａｔ ｔｈｅｙ ａｒｅ ｎｏｔ ｔｈｅ ｄｅｓｉｒｅｄ ｏｎｅｓ ｔｈａｔ ｄｅｓｃｅｎｄ ｓｌｏｗｌｙ ａｔ ｔｈｅ ｂｅｇｉｎｎｉｎｇ， ｔｈｅｎ ｃｏｎｖｅｒｇｅ ｔｏ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ． Ｔｈｉｓ ｉｓ ｗｈｙ ｔｈｅｒｅ ｉｓ ｇｒｅａｔ ｄｅｍａｎｄ ｆｏｒ ａｎｏｔｈｅｒ ｏｐｔｉｍｉｚｅｒ． Ｉｎ Ｆｉｇ． １５， ＰＩＯ ｉｓ ｕｓｅｄ ｔｏ ａｐｐｌｙ ａ ｇｌｉｄｉｎｇ ｐｒｏｃｅｄｕｒｅ． Ｂｅｆｏｒｅ ４ｓ， ｔｈｅ ｍａｐ ａｎｄ ｃｏｍｐａｓｓ ｏｐｅｒａｔｏｒ ｔａｋｅｓ ｅｆｆｅｃｔ ａｎｄ ａｌｌ ｔｈｅ ＵＡＶｓ ｆｌｙ ｔｏｗａｒｄｓ ｔｈｅ ｂｅｓｔ ｌｏｃａｌ ｐｏｓｉｔｉｏｎ． Ｉｎ ｔｈｅ ｔｉｍｅ ｒａｎｇｅ， ４ ｔｏ ６ ｓ， ｔｈｅ ｌａｎｄｍａｒｋ ｏｐｅｒａｔｏｒ ｄｒｉｖｅｓ ｔｈｅ ＵＡＶｓ ｎｅａｒｂｙ ｔｏ ｔｈｅ ｓａｍｅ ｈｅｉｇｈｔ． Ｗｈｅｎ ａｌｌ ｏｆ ｔｈｅｍ ｆｌｙ ｔｏ ｔｈｅ ｓｔａｎｄａｒｄ ｈｅｉｇｈｔ， ｔｈｅ ＰＩＯ ｐａｒｔ ｓｔｏｐｓ ｗｏｒｋｉｎｇ ａｎｄ ｃｏｎｓｅｎｓｕｓ ｃｏｎｔｉｎｕｅｓ ｃｏｎｃｅｎｔｒａｔｉｎｇ ｔｈｅ ＵＡＶｓ ａｎｄ ｇｕｉｄｉｎｇ ｔｈｅｍ ｔｏ ｆｌｙ ａｌｏｎｇ ｔｈｅ ｄｅｓｉｇｎｅｄ ｔｒａｊｅｃｔｏｒｙ． Ｆｉｇ． １５ Ｚ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ Ｆｉｇ． １６ Ｚ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ Ｔｈｅ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ａｌｓｏ ｆｏｌｌｏｗｓ ｔｈｅ ｓｔｅｐｓ ａｂｏｖｅ， ｂｕｔ ｔｈｅ ｄｉｆｆｅｒｅｎｃｅ ｏｃｃｕｒｓ ｉｎ ｔｈｅ ｆｉｒｓｔ ８ ｓ ａｓ ｔｈｅ ＵＡＶｓ ｄｅｓｃｅｎｄ ｍｏｒｅ ｓｌｏｗｌｙ ａｎｄ ｓｍｏｏｔｈｌｙ． ＵＡＶ１ ， ＵＡＶ２ ａｎｄ ＵＡＶ３ ｃａｎ ｓｕｐｐｏｒｔ ｔｈｉｓ ｏｐｉｎｉｏｎ， ａｓ ｔｈｅｙ ａｔ ｔｈｅ ｓａｍｅ ｒｅｓｐｅｃｔｉｖｅ ｈｅｉｇｈｔ， ｂｕｔ ａｔ ｔ ＝ ５ ｓ，ＵＡＶ２ ａｎｄ ＵＡＶ３ ｂｙ ＳＤ⁃ＰＩＯ ａｒｅ ａｂｏｖｅ ２０ ｍ ｗｈｉｌｅ ｔｈｅｉｒ ｃｏｕｎｔｅｒｐａｒｔｓ ａｒｅ ｃｌｅａｒｌｙ ｌｏｗｅｒ． Ａｔ ｔ ＝ ７ ｓ， ｔｈｅ ｈｅｉｇｈｔ ｏｆ ＵＡＶ１ ｂｙ ＳＤ⁃ＰＩＯ ｉｓ １６． ０４ ｍ ａｎｄ ｔｈｅ ｏｕｔｃｏｍｅ ｂｙ ＰＩＯ ｉｓ ｏｎｌｙ １５．４７ ｍ． Ｆｒｏｍ ｔｈｅ ｆｉｔｎｅｓｓ ｆｕｎｃｔｉｏｎ Ｅｑ． （８）， ｔｈｅ ｃｌｏｓｅｒ ｔｈｅ ＵＡＶｉ ａｐｐｒｏａｃｈｅｓ ｉｔｓ ｄｅｓｉｒｅｄ ｌｏｃａｔｉｏｎ， ｔｈｅ ｓｍａｌｌｅｒ ｔｈｅ ｆｉｔｎｅｓｓ ｖａｌｕｅ． Ｉｎ Ｆｉｇ． １７， ａｓ ｉｔｅｒａｔｉｏｎ Ｎｃ ｉｎｃｒｅａｓｅｓ， ａｖｅｒａｇｅ ｆｉｔｎｅｓｓ ｖａｌｕｅｓ ｄｅｃｒｅａｓｅ ａｎｄ ｆｉｎａｌｌｙ ａｐｐｒｏａｃｈ ｚｅｒｏ， ａｎｄ ｄｉｆｆｅｒｅｎｔ ｏｐｔｉｍｉｚｅｒｓ ｐｏｓｓｅｓｓ ｄｉｆｆｅｒｅｎｔ ｃｏｎｖｅｒｇｅｎｃｅ ｒａｔｅｓ． Ｉｎ Ｆｉｇ． １７ （ ａ ）， ＰＩＯ ｓｈｏｗｓ ｔｈｅ ｆａｓｔｅｓｔ ｓｐｅｅｄ ｉｎ ｔｈｅ ｂｅｇｉｎｎｉｎｇ， ＳＤ⁃ＰＩＯ ｉｓ ａ ｂｉｔ ｓｌｏｗｅｒ ａｎｄ ＰＳＯ ｓｈｏｗｓ ｔｈｅ ｗｅａｋｅｓｔ ｐｅｒｆｏｒｍａｎｃｅ ｏｆ ｔｈｅ ｔｈｒｅｅ． Ｈｏｗｅｖｅｒ， ａｆｔｅｒ Ｎｃ ＝ ８０， ＰＩＯ ｂｅｃｏｍｅｓ ｓｌｏｗ ａｎｄ ｉｓ ｆｉｎａｌｌｙ ｏｖｅｒｔａｋｅｎ ｂｙ ＳＤ⁃ＰＩＯ ａｔ Ｎｃ ＝ ９３． Ｔｈｅ ｐｏｏｒ ｅｆｆｉｃｉｅｎｃｙ ｏｆ ＰＩＯ ｉｓ ｄｕｅ ｔｏ ｔｈｅ ｉｎａｄｅｑｕａｔｅ ｌｏｃａｌ ｏｐｔｉｍｕｍ． （ａ）Ｘ ｆｉｔｎｅｓｓ ｖａｌｕｅ ·５７９· 第 ４ 期 ＺＨＡＮＧ Ｔｉａｎｊｉｅ， ｅｔ ａｌ．：Ａ ｍｏｄｉｆｉｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｆｏｒ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎｓ ｂａｓｅｄ ｏｎ ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄ ｏｐｔｉｍｉｚａｔｉｏｎ ｗｉｔｈ ａ ｓｌｏｗ ｄｉｖｉｎｇ ｓｔｒａｔｅｇｙ