ZHANG Tianjie,et al.:A modified consensus algorithm for multi-UAV formations based on 第4期 pigeon-inspired optimization with a slow diving strategy .577. 4 Simulation results and analysis from this perspective,however,it is important to clarify the fact that the trajectories do not cross and the To validate the SD-PIO-based consensus, UAVs do not crash in the sky. numerical simulations were conducted using MATLAB 2012a on a PC with an i5-480 m,2.67 GHz CPU and UAV Windows 10 operating system.SD-PIO was compared 40 with consensus itself and consensus with two other 号20 optimizations,particle swarm optimization (PSO)and UAV 6005004003002001000分m UAV PIO.The results prove the effectiveness of SD-PIO. m The main parameters in Egs.(7),(9),and Fig.5 Trajectories of 9 UAVs by consensus (18)are listed in Table 2 below.The time step for the simulation is T=0.05 s. To demonstrate the advantage of SD-PIO-based Table 2 Simulation parameters consensus,PSO-and PIO-based consensus algorithms were adopted to compare their performances with that R Tule/s Tae/s of the modified algorithm.In Fig.6,most trajectories 50 4 6 are apparently better than those in Fig.5,as the Nine UAVs were initialized with random positions trajectories are gentle at the beginning and turn without in a 3-dimensional environment as shown in Table 3. diving sharply.UAV3,UAVs,andUAV,serve as a set Table 3 Initial positions of 9 UAVs m of convincing simulations that support the conclusions UAV number above.In Figs.7 and 8,PIO and SD-PIO work and UAV, 20.804 30.971 30.477 further improve the performance.In Fig.7,the UAV2 12.685 37.633 31.252 overshoot almost disappears,but there is still UAV3 22.260 38.622 24.661 fluctuation in the trajectories from this perspective, UAVa 0.318 27.929 9.703 especially for UAV3,UAV,and UAVs.In Fig.8,the UAVs trajectories become much smoother and the overshoot 21.859 15.846 6.076 UAVo can be ignored compared with Figs.5 to 7.This is 16.942 13.999 6.644 reflected in the lowest parts of the trajectories of UAV7 19.293 20.723 17.662 UAV3,UAV,and UAVs. UAVs 24.806 23.061 9.317 UAVo 37.298 12.688 7.861 UAV These heights,the numbers on the z-axis,are 号20 UAV relative positions corresponding to the height of origin UAV of the common moving coordinate system in Fig.2. 0 UAV 60050040030020010006 Each UAV had the same initial position in the four x/m UAV 3/m simulations,i.e.,consensus,PSO-based consensus, Fig.6 Trajectories of 9 UAVs by PSO-based consensus PIO-based consensus,and SD-PIO-based consensus. As known from Sections 2 and 3,the performance of consensus cannot satisfy the requirements,which is 401 UAV UAV UAV clearly displayed in Fig.5.Because of communication UAV 20 -UAV restrictions and lack of forecasting,all the UAVs tend ≠9UAV。 UAVs to dive sharply and climb quickly.This causes great 6005040300200100六m0AV m UAV overload at the turning point,which may damage the UAV's structure and result in difficulties for flight Fig.7 Trajectories of 9 UAVs by PIO-based consensus control.Their routes seem to tie a knot when viewed４ Ｓｉｍｕｌａｔｉｏｎ ｒｅｓｕｌｔｓ ａｎｄ ａｎａｌｙｓｉｓ Ｔｏ ｖａｌｉｄａｔｅ ｔｈｅ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ， ｎｕｍｅｒｉｃａｌ ｓｉｍｕｌａｔｉｏｎｓ ｗｅｒｅ ｃｏｎｄｕｃｔｅｄ ｕｓｉｎｇ ＭＡＴＬＡＢ ２０１２ａ ｏｎ ａ ＰＣ ｗｉｔｈ ａｎ ｉ５⁃４８０ ｍ， ２．６７ ＧＨｚ ＣＰＵ ａｎｄ Ｗｉｎｄｏｗｓ １０ ｏｐｅｒａｔｉｎｇ ｓｙｓｔｅｍ． ＳＤ⁃ＰＩＯ ｗａｓ ｃｏｍｐａｒｅｄ ｗｉｔｈ ｃｏｎｓｅｎｓｕｓ ｉｔｓｅｌｆ ａｎｄ ｃｏｎｓｅｎｓｕｓ ｗｉｔｈ ｔｗｏ ｏｔｈｅｒ ｏｐｔｉｍｉｚａｔｉｏｎｓ， ｐａｒｔｉｃｌｅ ｓｗａｒｍ ｏｐｔｉｍｉｚａｔｉｏｎ （ＰＳＯ） ａｎｄ ＰＩＯ． Ｔｈｅ ｒｅｓｕｌｔｓ ｐｒｏｖｅ ｔｈｅ ｅｆｆｅｃｔｉｖｅｎｅｓｓ ｏｆ ＳＤ⁃ＰＩＯ． Ｔｈｅ ｍａｉｎ ｐａｒａｍｅｔｅｒｓ ｉｎ Ｅｑｓ． （ ７ ）， （ ９ ）， ａｎｄ （１８） ａｒｅ ｌｉｓｔｅｄ ｉｎ Ｔａｂｌｅ ２ ｂｅｌｏｗ． Ｔｈｅ ｔｉｍｅ ｓｔｅｐ ｆｏｒ ｔｈｅ ｓｉｍｕｌａｔｉｏｎ ｉｓ Ｔ ＝ ０．０５ ｓ． Ｔａｂｌｅ ２ Ｓｉｍｕｌａｔｉｏｎ ｐａｒａｍｅｔｅｒｓ γ Ｒ Ｔｌａｂｅｌ ／ ｓ Ｔｃｈｇ ／ ｓ ５ ５０ ４ ６ Ｎｉｎｅ ＵＡＶｓ ｗｅｒｅ ｉｎｉｔｉａｌｉｚｅｄ ｗｉｔｈ ｒａｎｄｏｍ ｐｏｓｉｔｉｏｎｓ ｉｎ ａ ３⁃ｄｉｍｅｎｓｉｏｎａｌ ｅｎｖｉｒｏｎｍｅｎｔ ａｓ ｓｈｏｗｎ ｉｎ Ｔａｂｌｅ ３． Ｔａｂｌｅ ３ Ｉｎｉｔｉａｌ ｐｏｓｉｔｉｏｎｓ ｏｆ ９ ＵＡＶｓ ｍ ＵＡＶ ｎｕｍｂｅｒ ｘ ｙ ｚ ＵＡＶ１ ２０．８０４ ３０．９７１ ３０．４７７ ＵＡＶ２ １２．６８５ ３７．６３３ ３１．２５２ ＵＡＶ３ ２２．２６０ ３８．６２２ ２４．６６１ ＵＡＶ４ ０．３１８ ２７．９２９ ９．７０３ ＵＡＶ５ ２１．８５９ １５．８４６ ６．０７６ ＵＡＶ６ １６．９４２ １３．９９９ ６．６４４ ＵＡＶ７ １９．２９３ ２０．７２３ １７．６６２ ＵＡＶ８ ２４．８０６ ２３．０６１ ９．３１７ ＵＡＶ９ ３７．２９８ １２．６８８ ７．８６１ Ｔｈｅｓｅ ｈｅｉｇｈｔｓ， ｔｈｅ ｎｕｍｂｅｒｓ ｏｎ ｔｈｅ ｚ⁃ａｘｉｓ， ａｒｅ ｒｅｌａｔｉｖｅ ｐｏｓｉｔｉｏｎｓ ｃｏｒｒｅｓｐｏｎｄｉｎｇ ｔｏ ｔｈｅ ｈｅｉｇｈｔ ｏｆ ｏｒｉｇｉｎ ｏｆ ｔｈｅ ｃｏｍｍｏｎ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ ｉｎ Ｆｉｇ． ２． Ｅａｃｈ ＵＡＶ ｈａｄ ｔｈｅ ｓａｍｅ ｉｎｉｔｉａｌ ｐｏｓｉｔｉｏｎ ｉｎ ｔｈｅ ｆｏｕｒ ｓｉｍｕｌａｔｉｏｎｓ， ｉ． ｅ．， ｃｏｎｓｅｎｓｕｓ， ＰＳＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ， ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ， ａｎｄ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ． Ａｓ ｋｎｏｗｎ ｆｒｏｍ Ｓｅｃｔｉｏｎｓ ２ ａｎｄ ３， ｔｈｅ ｐｅｒｆｏｒｍａｎｃｅ ｏｆ ｃｏｎｓｅｎｓｕｓ ｃａｎｎｏｔ ｓａｔｉｓｆｙ ｔｈｅ ｒｅｑｕｉｒｅｍｅｎｔｓ， ｗｈｉｃｈ ｉｓ ｃｌｅａｒｌｙ ｄｉｓｐｌａｙｅｄ ｉｎ Ｆｉｇ． ５． Ｂｅｃａｕｓｅ ｏｆ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｒｅｓｔｒｉｃｔｉｏｎｓ ａｎｄ ｌａｃｋ ｏｆ ｆｏｒｅｃａｓｔｉｎｇ， ａｌｌ ｔｈｅ ＵＡＶｓ ｔｅｎｄ ｔｏ ｄｉｖｅ ｓｈａｒｐｌｙ ａｎｄ ｃｌｉｍｂ ｑｕｉｃｋｌｙ． Ｔｈｉｓ ｃａｕｓｅｓ ｇｒｅａｔ ｏｖｅｒｌｏａｄ ａｔ ｔｈｅ ｔｕｒｎｉｎｇ ｐｏｉｎｔ， ｗｈｉｃｈ ｍａｙ ｄａｍａｇｅ ｔｈｅ ＵＡＶ’ ｓ ｓｔｒｕｃｔｕｒｅ ａｎｄ ｒｅｓｕｌｔ ｉｎ ｄｉｆｆｉｃｕｌｔｉｅｓ ｆｏｒ ｆｌｉｇｈｔ ｃｏｎｔｒｏｌ． Ｔｈｅｉｒ ｒｏｕｔｅｓ ｓｅｅｍ ｔｏ ｔｉｅ ａ ｋｎｏｔ ｗｈｅｎ ｖｉｅｗｅｄ ｆｒｏｍ ｔｈｉｓ ｐｅｒｓｐｅｃｔｉｖｅ， ｈｏｗｅｖｅｒ， ｉｔ ｉｓ ｉｍｐｏｒｔａｎｔ ｔｏ ｃｌａｒｉｆｙ ｔｈｅ ｆａｃｔ ｔｈａｔ ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｄｏ ｎｏｔ ｃｒｏｓｓ ａｎｄ ｔｈｅ ＵＡＶｓ ｄｏ ｎｏｔ ｃｒａｓｈ ｉｎ ｔｈｅ ｓｋｙ． Ｆｉｇ．５ Ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ｃｏｎｓｅｎｓｕｓ Ｔｏ ｄｅｍｏｎｓｔｒａｔｅ ｔｈｅ ａｄｖａｎｔａｇｅ ｏｆ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ， ＰＳＯ⁃ ａｎｄ ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍｓ ｗｅｒｅ ａｄｏｐｔｅｄ ｔｏ ｃｏｍｐａｒｅ ｔｈｅｉｒ ｐｅｒｆｏｒｍａｎｃｅｓ ｗｉｔｈ ｔｈａｔ ｏｆ ｔｈｅ ｍｏｄｉｆｉｅｄ ａｌｇｏｒｉｔｈｍ． Ｉｎ Ｆｉｇ． ６， ｍｏｓｔ ｔｒａｊｅｃｔｏｒｉｅｓ ａｒｅ ａｐｐａｒｅｎｔｌｙ ｂｅｔｔｅｒ ｔｈａｎ ｔｈｏｓｅ ｉｎ Ｆｉｇ． ５， ａｓ ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ａｒｅ ｇｅｎｔｌｅ ａｔ ｔｈｅ ｂｅｇｉｎｎｉｎｇ ａｎｄ ｔｕｒｎ ｗｉｔｈｏｕｔ ｄｉｖｉｎｇ ｓｈａｒｐｌｙ． ＵＡＶ３ ，ＵＡＶ５ ， ａｎｄＵＡＶ９ ｓｅｒｖｅ ａｓ ａ ｓｅｔ ｏｆ ｃｏｎｖｉｎｃｉｎｇ ｓｉｍｕｌａｔｉｏｎｓ ｔｈａｔ ｓｕｐｐｏｒｔ ｔｈｅ ｃｏｎｃｌｕｓｉｏｎｓ ａｂｏｖｅ． Ｉｎ Ｆｉｇｓ． ７ ａｎｄ ８， ＰＩＯ ａｎｄ ＳＤ⁃ＰＩＯ ｗｏｒｋ ａｎｄ ｆｕｒｔｈｅｒ ｉｍｐｒｏｖｅ ｔｈｅ ｐｅｒｆｏｒｍａｎｃｅ． Ｉｎ Ｆｉｇ． ７， ｔｈｅ ｏｖｅｒｓｈｏｏｔ ａｌｍｏｓｔ ｄｉｓａｐｐｅａｒｓ， ｂｕｔ ｔｈｅｒｅ ｉｓ ｓｔｉｌｌ ｆｌｕｃｔｕａｔｉｏｎ ｉｎ ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｆｒｏｍ ｔｈｉｓ ｐｅｒｓｐｅｃｔｉｖｅ， ｅｓｐｅｃｉａｌｌｙ ｆｏｒ ＵＡＶ３ ，ＵＡＶ４ ， ａｎｄ ＵＡＶ５ ． Ｉｎ Ｆｉｇ． ８， ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｂｅｃｏｍｅ ｍｕｃｈ ｓｍｏｏｔｈｅｒ ａｎｄ ｔｈｅ ｏｖｅｒｓｈｏｏｔ ｃａｎ ｂｅ ｉｇｎｏｒｅｄ ｃｏｍｐａｒｅｄ ｗｉｔｈ Ｆｉｇｓ． ５ ｔｏ ７． Ｔｈｉｓ ｉｓ ｒｅｆｌｅｃｔｅｄ ｉｎ ｔｈｅ ｌｏｗｅｓｔ ｐａｒｔｓ ｏｆ ｔｈｅ ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ＵＡＶ３ ，ＵＡＶ４ ， ａｎｄ ＵＡＶ５ ． Ｆｉｇ． ６ Ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ＰＳＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ Ｆｉｇ． ７ Ｔｒａｊｅｃｔｏｒｉｅｓ ｏｆ ９ ＵＡＶｓ ｂｙ ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ·５７７· 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