,576 智能系统学报 第12卷 can only converge to a local optimal solution because where T is the moment when consensus replaces SD- each pigeon can only receive its neighbor's information instead of global information. PI0,±×arctan(R×(t-T)+0.5,the For the second part of SD-PIO,all the pigeons inverse trigonometric function,is utilized to modify the converge to the flock's center.Their trajectories can only output of the SD-PIO and consensus,it works as a contract and approach each other.In formation processes this part serves only as a transition,and as the traces do switch key to smoothly connect the two parts of the not diverge this second part satisfies the demands. trajectory.f((t),(t))is the output velocity of SD- Multi-UAV formations with a SD- PI0 from Eqs.(7）and(9）.The implementation PIO-based consensus algorithm procedure of the SD-PIO-based consensus algorithm is given as follows: The first three sections introduced consensus and 1)Input the number of UAVs in the formation SD-PIO,now this section uses them to optimize the trajectory of an UAV.When consensus is only applied task,give the expectation formation and consider the to the planned routes of all the UAVs,there is communication restriction. unavoidable overshoot because of the distributed control 2)Design a suitable communication network plan itself.Every UAV has its own limited topology and acquire its adjacency matrix and the communication relationship,which means it cannot Laplace matrix. always receive the desired trajectory.For example, 3)Randomly initialize the positions and velocities UAV can never acquire the planned trajectory of all the UAVs along with the parameters involved in directly,it relies on UAV,to pass on the information. If UAV3's initial altitude is much too high,it must the equations used in Eq.(18). dive to meet the desired trajectory.Because of inertia 4)Calculate the acceleration input,consensus, and a lack of forecasting,UAV,can only produce and the velocity input using Eq.(18) positive additions to its height when it goes below the 5)Update each UAV's position and velocity using desired trajectory and this is the reason for overshoot. Eq.(18) However,as for UAV,only when its leader UAV3 6)Draw the UAV's trajectory according to its flies below the proposed height does it begin to position in each iteration. recognize that it should fly a bit higher,then its velocity is changed by Eq.(7),followed by its The process is summarized in the following position.This process is much slower than a change in process flow chart: the desired route.As a result,the UAVs'trajectories Design the desired dive sharply and appear to overshoot,then they formation and give the converge to the desired trajectory.SD-PIO provides a communication topology gentle and slowly changing trajectory for each UAV at Acquire the adjacency matrix the start,then,as soon as the UAVs fly down and and the Laplace martix meet the desired trajectory,consensus starts to operate. Calculate the acceleration Initialize the positions and input by the second stage The formula of the combination of these two velocities of all the UAVs formula of SD-PIO,Eq.(18) algorithms can be described as: [x(t+1)=x(t)+T·((t+1)+,(t+1))》 Calculate the acceleration Update the velocity and input by the first stage v(t+1)=v(t)+T·a(t+1) position of each UAV formula of SD-PIO.Eq.(18) a(t+1)=-(L☒3)·x(t+1)+ Ift>Tlabel Update the velocity and N y·(L☒I3)·(t+1))· position of each UAV Y 三×arctan(R×(t-Ths)+0.5) <Ift>Tlabel Draw the trajectories of UAVs 2(t+1)=f(x(t),v(t))· N ×arctan(R×(t-Te)+0.5) Fig.4 The process of the SD-PIO based consensus (18) algorithm for a multi-UAV formationｃａｎ ｏｎｌｙ ｃｏｎｖｅｒｇｅ ｔｏ ａ ｌｏｃａｌ ｏｐｔｉｍａｌ ｓｏｌｕｔｉｏｎ ｂｅｃａｕｓｅ ｅａｃｈ ｐｉｇｅｏｎ ｃａｎ ｏｎｌｙ ｒｅｃｅｉｖｅ ｉｔｓ ｎｅｉｇｈｂｏｒ’ｓ ｉｎｆｏｒｍａｔｉｏｎ ｉｎｓｔｅａｄ ｏｆ ｇｌｏｂａｌ ｉｎｆｏｒｍａｔｉｏｎ． Ｆｏｒ ｔｈｅ ｓｅｃｏｎｄ ｐａｒｔ ｏｆ ＳＤ⁃ＰＩＯ， ａｌｌ ｔｈｅ ｐｉｇｅｏｎｓ ｃｏｎｖｅｒｇｅ ｔｏ ｔｈｅ ｆｌｏｃｋ’ｓ ｃｅｎｔｅｒ． Ｔｈｅｉｒ ｔｒａｊｅｃｔｏｒｉｅｓ ｃａｎ ｏｎｌｙ ｃｏｎｔｒａｃｔ ａｎｄ ａｐｐｒｏａｃｈ ｅａｃｈ ｏｔｈｅｒ． Ｉｎ ｆｏｒｍａｔｉｏｎ ｐｒｏｃｅｓｓｅｓ ｔｈｉｓ ｐａｒｔ ｓｅｒｖｅｓ ｏｎｌｙ ａｓ ａ ｔｒａｎｓｉｔｉｏｎ， ａｎｄ ａｓ ｔｈｅ ｔｒａｃｅｓ ｄｏ ｎｏｔ ｄｉｖｅｒｇｅ ｔｈｉｓ ｓｅｃｏｎｄ ｐａｒｔ ｓａｔｉｓｆｉｅｓ ｔｈｅ ｄｅｍａｎｄｓ． ３ Ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎｓ ｗｉｔｈ ａ ＳＤ⁃ ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ Ｔｈｅ ｆｉｒｓｔ ｔｈｒｅｅ ｓｅｃｔｉｏｎｓ ｉｎｔｒｏｄｕｃｅｄ ｃｏｎｓｅｎｓｕｓ ａｎｄ ＳＤ⁃ＰＩＯ， ｎｏｗ ｔｈｉｓ ｓｅｃｔｉｏｎ ｕｓｅｓ ｔｈｅｍ ｔｏ ｏｐｔｉｍｉｚｅ ｔｈｅ ｔｒａｊｅｃｔｏｒｙ ｏｆ ａｎ ＵＡＶ． Ｗｈｅｎ ｃｏｎｓｅｎｓｕｓ ｉｓ ｏｎｌｙ ａｐｐｌｉｅｄ ｔｏ ｔｈｅ ｐｌａｎｎｅｄ ｒｏｕｔｅｓ ｏｆ ａｌｌ ｔｈｅ ＵＡＶｓ， ｔｈｅｒｅ ｉｓ ｕｎａｖｏｉｄａｂｌｅ ｏｖｅｒｓｈｏｏｔ ｂｅｃａｕｓｅ ｏｆ ｔｈｅ ｄｉｓｔｒｉｂｕｔｅｄ ｃｏｎｔｒｏｌ ｐｌａｎ ｉｔｓｅｌｆ． Ｅｖｅｒｙ ＵＡＶ ｈａｓ ｉｔｓ ｏｗｎ ｌｉｍｉｔｅｄ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｒｅｌａｔｉｏｎｓｈｉｐ， ｗｈｉｃｈ ｍｅａｎｓ ｉｔ ｃａｎｎｏｔ ａｌｗａｙｓ ｒｅｃｅｉｖｅ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ． Ｆｏｒ ｅｘａｍｐｌｅ， ＵＡＶ１ ｃａｎ ｎｅｖｅｒ ａｃｑｕｉｒｅ ｔｈｅ ｐｌａｎｎｅｄ ｔｒａｊｅｃｔｏｒｙ ｄｉｒｅｃｔｌｙ， ｉｔ ｒｅｌｉｅｓ ｏｎ ＵＡＶ３ ｔｏ ｐａｓｓ ｏｎ ｔｈｅ ｉｎｆｏｒｍａｔｉｏｎ． Ｉｆ ＵＡＶ３ ’ ｓ ｉｎｉｔｉａｌ ａｌｔｉｔｕｄｅ ｉｓ ｍｕｃｈ ｔｏｏ ｈｉｇｈ， ｉｔ ｍｕｓｔ ｄｉｖｅ ｔｏ ｍｅｅｔ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ． Ｂｅｃａｕｓｅ ｏｆ ｉｎｅｒｔｉａ ａｎｄ ａ ｌａｃｋ ｏｆ ｆｏｒｅｃａｓｔｉｎｇ， ＵＡＶ３ ｃａｎ ｏｎｌｙ ｐｒｏｄｕｃｅ ｐｏｓｉｔｉｖｅ ａｄｄｉｔｉｏｎｓ ｔｏ ｉｔｓ ｈｅｉｇｈｔ ｗｈｅｎ ｉｔ ｇｏｅｓ ｂｅｌｏｗ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ ａｎｄ ｔｈｉｓ ｉｓ ｔｈｅ ｒｅａｓｏｎ ｆｏｒ ｏｖｅｒｓｈｏｏｔ． Ｈｏｗｅｖｅｒ， ａｓ ｆｏｒ ＵＡＶ１ ， ｏｎｌｙ ｗｈｅｎ ｉｔｓ ｌｅａｄｅｒ ＵＡＶ３ ｆｌｉｅｓ ｂｅｌｏｗ ｔｈｅ ｐｒｏｐｏｓｅｄ ｈｅｉｇｈｔ ｄｏｅｓ ｉｔ ｂｅｇｉｎ ｔｏ ｒｅｃｏｇｎｉｚｅ ｔｈａｔ ｉｔ ｓｈｏｕｌｄ ｆｌｙ ａ ｂｉｔ ｈｉｇｈｅｒ， ｔｈｅｎ ｉｔｓ ｖｅｌｏｃｉｔｙ ｉｓ ｃｈａｎｇｅｄ ｂｙ Ｅｑ． （ ７ ）， ｆｏｌｌｏｗｅｄ ｂｙ ｉｔｓ ｐｏｓｉｔｉｏｎ． Ｔｈｉｓ ｐｒｏｃｅｓｓ ｉｓ ｍｕｃｈ ｓｌｏｗｅｒ ｔｈａｎ ａ ｃｈａｎｇｅ ｉｎ ｔｈｅ ｄｅｓｉｒｅｄ ｒｏｕｔｅ． Ａｓ ａ ｒｅｓｕｌｔ， ｔｈｅ ＵＡＶｓ’ ｔｒａｊｅｃｔｏｒｉｅｓ ｄｉｖｅ ｓｈａｒｐｌｙ ａｎｄ ａｐｐｅａｒ ｔｏ ｏｖｅｒｓｈｏｏｔ， ｔｈｅｎ ｔｈｅｙ ｃｏｎｖｅｒｇｅ ｔｏ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ． ＳＤ⁃ＰＩＯ ｐｒｏｖｉｄｅｓ ａ ｇｅｎｔｌｅ ａｎｄ ｓｌｏｗｌｙ ｃｈａｎｇｉｎｇ ｔｒａｊｅｃｔｏｒｙ ｆｏｒ ｅａｃｈ ＵＡＶ ａｔ ｔｈｅ ｓｔａｒｔ， ｔｈｅｎ， ａｓ ｓｏｏｎ ａｓ ｔｈｅ ＵＡＶｓ ｆｌｙ ｄｏｗｎ ａｎｄ ｍｅｅｔ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｊｅｃｔｏｒｙ， ｃｏｎｓｅｎｓｕｓ ｓｔａｒｔｓ ｔｏ ｏｐｅｒａｔｅ． Ｔｈｅ ｆｏｒｍｕｌａ ｏｆ ｔｈｅ ｃｏｍｂｉｎａｔｉｏｎ ｏｆ ｔｈｅｓｅ ｔｗｏ ａｌｇｏｒｉｔｈｍｓ ｃａｎ ｂｅ ｄｅｓｃｒｉｂｅｄ ａｓ： ｘ（ｔ ＋ １） ＝ ｘ（ｔ） ＋ Ｔ·（ｖ（ｔ ＋ １） ＋ ｖ２（ｔ ＋ １）） ｖ（ｔ ＋ １） ＝ ｖ（ｔ） ＋ Ｔ·ａ（ｔ ＋ １） ａ（ｔ ＋ １） ＝ － （（Ｌ 􀱋 Ｉ３ ）·ｘ（ｔ ＋ １） ＋ γ·（Ｌ 􀱋 Ｉ３ ）·ｖ（ｔ ＋ １））· （ １ π × ａｒｃｔａｎ（Ｒ × （ｔ － Ｔｃｈｇ）） ＋ ０．５） ｖ２（ｔ ＋ １） ＝ ｆ（ｘ（ｔ），ｖ（ｔ））· （ － １ π × ａｒｃｔａｎ（Ｒ × （ｔ － Ｔｃｈｇ）） ＋ ０．５） ì î í ï ï ï ï ï ï ï ï ï ï ï ï （１８） ｗｈｅｒｅ Ｔｃｈｇ ｉｓ ｔｈｅ ｍｏｍｅｎｔ ｗｈｅｎ ｃｏｎｓｅｎｓｕｓ ｒｅｐｌａｃｅｓ ＳＤ⁃ ＰＩＯ， ± １ π × ａｒｃｔａｎ （Ｒ × （ｔ － Ｔｃｈｇ）） ＋ ０．５， ｔｈｅ ｉｎｖｅｒｓｅ ｔｒｉｇｏｎｏｍｅｔｒｉｃ ｆｕｎｃｔｉｏｎ， ｉｓ ｕｔｉｌｉｚｅｄ ｔｏ ｍｏｄｉｆｙ ｔｈｅ ｏｕｔｐｕｔ ｏｆ ｔｈｅ ＳＤ⁃ＰＩＯ ａｎｄ ｃｏｎｓｅｎｓｕｓ， ｉｔ ｗｏｒｋｓ ａｓ ａ ｓｗｉｔｃｈ ｋｅｙ ｔｏ ｓｍｏｏｔｈｌｙ ｃｏｎｎｅｃｔ ｔｈｅ ｔｗｏ ｐａｒｔｓ ｏｆ ｔｈｅ ｔｒａｊｅｃｔｏｒｙ． ｆ（ｘ（ｔ），ｖ（ｔ）） ｉｓ ｔｈｅ ｏｕｔｐｕｔ ｖｅｌｏｃｉｔｙ ｏｆ ＳＤ⁃ ＰＩＯ ｆｒｏｍ Ｅｑｓ． （ ７ ） ａｎｄ （ ９ ）． Ｔｈｅ ｉｍｐｌｅｍｅｎｔａｔｉｏｎ ｐｒｏｃｅｄｕｒｅ ｏｆ ｔｈｅ ＳＤ⁃ＰＩＯ⁃ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｉｓ ｇｉｖｅｎ ａｓ ｆｏｌｌｏｗｓ： １） Ｉｎｐｕｔ ｔｈｅ ｎｕｍｂｅｒ ｏｆ ＵＡＶｓ ｉｎ ｔｈｅ ｆｏｒｍａｔｉｏｎ ｔａｓｋ， ｇｉｖｅ ｔｈｅ ｅｘｐｅｃｔａｔｉｏｎ ｆｏｒｍａｔｉｏｎ ａｎｄ ｃｏｎｓｉｄｅｒ ｔｈｅ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｒｅｓｔｒｉｃｔｉｏｎ． ２ ） Ｄｅｓｉｇｎ ａ ｓｕｉｔａｂｌｅ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｎｅｔｗｏｒｋ ｔｏｐｏｌｏｇｙ ａｎｄ ａｃｑｕｉｒｅ ｉｔｓ ａｄｊａｃｅｎｃｙ ｍａｔｒｉｘ ａｎｄ ｔｈｅ Ｌａｐｌａｃｅ ｍａｔｒｉｘ． ３）Ｒａｎｄｏｍｌｙ ｉｎｉｔｉａｌｉｚｅ ｔｈｅ ｐｏｓｉｔｉｏｎｓ ａｎｄ ｖｅｌｏｃｉｔｉｅｓ ｏｆ ａｌｌ ｔｈｅ ＵＡＶｓ ａｌｏｎｇ ｗｉｔｈ ｔｈｅ ｐａｒａｍｅｔｅｒｓ ｉｎｖｏｌｖｅｄ ｉｎ ｔｈｅ ｅｑｕａｔｉｏｎｓ ｕｓｅｄ ｉｎ Ｅｑ． （１８）． ４） Ｃａｌｃｕｌａｔｅ ｔｈｅ ａｃｃｅｌｅｒａｔｉｏｎ ｉｎｐｕｔ， ｃｏｎｓｅｎｓｕｓ， ａｎｄ ｔｈｅ ｖｅｌｏｃｉｔｙ ｉｎｐｕｔ ｕｓｉｎｇ Ｅｑ． （１８） ５）Ｕｐｄａｔｅ ｅａｃｈ ＵＡＶ’ｓ ｐｏｓｉｔｉｏｎ ａｎｄ ｖｅｌｏｃｉｔｙ ｕｓｉｎｇ Ｅｑ． （１８） ６） Ｄｒａｗ ｔｈｅ ＵＡＶ’ ｓ ｔｒａｊｅｃｔｏｒｙ ａｃｃｏｒｄｉｎｇ ｔｏ ｉｔｓ ｐｏｓｉｔｉｏｎ ｉｎ ｅａｃｈ ｉｔｅｒａｔｉｏｎ． Ｔｈｅ ｐｒｏｃｅｓｓ ｉｓ ｓｕｍｍａｒｉｚｅｄ ｉｎ ｔｈｅ ｆｏｌｌｏｗｉｎｇ ｐｒｏｃｅｓｓ ｆｌｏｗ ｃｈａｒｔ： Ｆｉｇ． ４ Ｔｈｅ ｐｒｏｃｅｓｓ ｏｆ ｔｈｅ ＳＤ⁃ＰＩＯ ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｆｏｒ ａ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ ·５７６· 智 能 系 统 学 报 第 １２ 卷