【智能系统】A modified consensus algorithm for multi-UAV formations based on pigeon-inspired optimization with a slow diving strategy

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第12卷第4期智能系统学报 Vol.12 No.4 2017年8月 CAAI Transactions on Intelligent Systems Aug.2017 D0I:10.11992/is.201604006 网s络出版地址：http:/kns.cnki.net/kcms/detail/23.1538.TP.20170630.1420.004.html. A modified consensus algorithm for multi-UAV formations based on pigeon-inspired optimization with a slow diving strategy ZHANG Tianjie,DUAN Haibin (Science and Technology on Aircraft Control Laboratory,School of Automation Science and Electrical Engineering,Beihang University (BUAA),Beijing 100191,China) Abstract:This paper considers the formation control problem for a group of unmanned aerial vehicles (UAVs) employing consensus with different optimizers.A group of UAVs can never accomplish difficult tasks without formation because if disordered they do not work any better than a single vehicle,and a single vehicle is limited by its undeveloped intelligence and insufficient load.Among the many formation methods,consensus has attracted much attention because of its effectiveness and simplicity.However,at the beginning of convergence,overshoot and oscillation are universal because of the limitation of communication and a lack of forecasting,which are inborn shortcomings of consensus.It is natural to modify this method with lots of optimizers.In order to reduce overshoot and smooth trajectories,this paper first adopted particle swarm optimization (PSO),then pigeon-inspired optimization (PIO)to modify the consensus.PSO is a very popular optimizer,while PIO is a new method,both work but still retain disadvantages such as residual oscillation.As a result,it was necessary to modify PIO,and a pigeon-inspired optimization with a slow diving strategy (SD-PIO)is proposed.Convergence analysis was performed on the SD-PIO based on the Banach fixed-point theorem and conditions sufficient for stability were achieved. Finally,a series of comparative simulations were conducted to verify the feasibility and effectiveness of the proposed approach. Keywords:unmanned aerial vehicle (UAV);formation;consensus;pigeon-inspired optimization (PIO);Banach fixed-point theorem 中图分类号：TP18:TP273文献标志码：A文章编号：1673-4785(2017)04-0570-12 英文引用格式：ZHANG Tianjie,,DUAN Haibin.A modified consensus algorithm for multi-UAV formations based on pigeon-inspired optimization with a slow diving strategy[J].CAAI transactions on intelligent systems,2017,12(4):570-581. Formation control of unmanned aerial vehicles leader-follower approach artificial potential field (UAVs)can accomplish a task better than a single and sliding mode controlts]. UAV and has been studied using various methods. Among these approaches,consensus is an UAVs are widely adopted in various situationst-2]. effective and easy method for distributed controlt9. However,the requirements nowadays have already Using several simple consensus equations,UAVs can overtaken the current abilities of a single UAV.In fly along complex trajectories.Consensus originates order to satisfy the requirements,different plans for the from the natural behavior of animals such as fish and cooperation of many UAVs have been proposed by birds.A single bird or fish is small and weak,but a Chaumette3]and Raschet4,and the design of a population can hold a complicated formation.It is method for multi-UAV formation is importantts). consensus that plays an essential role in this Different methods have been proposed such as the phenomenon.Vicsek and Ren-]have achieved many results in this field.This paper adopts this 收稿日期：2017-04-11.网络出版日期：2017-06-30. method to design a controller for a multi-UAV 基金项目：Natural Science Foundation of China under Grant(61333004). 通信作者：ZHANG Tianjie..E-mail:11031148@buaa.edu.cn. formation

第１２卷第４期智能系统学报Ｖｏｌ．１２ №．４２０１７年８月ＣＡＡＩＴｒａｎｓａｃｔｉｏｎｓｏｎＩｎｔｅｌｌｉｇｅｎｔＳｙｓｔｅｍｓＡｕｇ．２０１７ＤＯＩ：１０．１１９９２／ｔｉｓ．２０１６０４００６网络出版地址：ｈｔｔｐ：／／ｋｎｓ．ｃｎｋｉ．ｎｅｔ／ｋｃｍｓ／ｄｅｔａｉｌ／２３．１５３８．ＴＰ．２０１７０６３０．１４２０．００４．ｈｔｍｌ．Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙＺＨＡＮＧＴｉａｎｊｉｅ，ＤＵＡＮＨａｉｂｉｎ（ＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｎＡｉｒｃｒａｆｔＣｏｎｔｒｏｌＬａｂｏｒａｔｏｒｙ，ＳｃｈｏｏｌｏｆＡｕｔｏｍａｔｉｏｎＳｃｉｅｎｃｅａｎｄＥｌｅｃｔｒｉｃａｌＥｎｇｉｎｅｅｒｉｎｇ，ＢｅｉｈａｎｇＵｎｉｖｅｒｓｉｔｙ（ＢＵＡＡ），Ｂｅｉｊｉｎｇ１００１９１，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｃｏｎｓｉｄｅｒｓｔｈｅｆｏｒｍａｔｉｏｎｃｏｎｔｒｏｌｐｒｏｂｌｅｍｆｏｒａｇｒｏｕｐｏｆｕｎｍａｎｎｅｄａｅｒｉａｌｖｅｈｉｃｌｅｓ（ＵＡＶｓ）ｅｍｐｌｏｙｉｎｇｃｏｎｓｅｎｓｕｓｗｉｔｈｄｉｆｆｅｒｅｎｔｏｐｔｉｍｉｚｅｒｓ．ＡｇｒｏｕｐｏｆＵＡＶｓｃａｎｎｅｖｅｒａｃｃｏｍｐｌｉｓｈｄｉｆｆｉｃｕｌｔｔａｓｋｓｗｉｔｈｏｕｔｆｏｒｍａｔｉｏｎｂｅｃａｕｓｅｉｆｄｉｓｏｒｄｅｒｅｄｔｈｅｙｄｏｎｏｔｗｏｒｋａｎｙｂｅｔｔｅｒｔｈａｎａｓｉｎｇｌｅｖｅｈｉｃｌｅ，ａｎｄａｓｉｎｇｌｅｖｅｈｉｃｌｅｉｓｌｉｍｉｔｅｄｂｙｉｔｓｕｎｄｅｖｅｌｏｐｅｄｉｎｔｅｌｌｉｇｅｎｃｅａｎｄｉｎｓｕｆｆｉｃｉｅｎｔｌｏａｄ．Ａｍｏｎｇｔｈｅｍａｎｙｆｏｒｍａｔｉｏｎｍｅｔｈｏｄｓ，ｃｏｎｓｅｎｓｕｓｈａｓａｔｔｒａｃｔｅｄｍｕｃｈａｔｔｅｎｔｉｏｎｂｅｃａｕｓｅｏｆｉｔｓｅｆｆｅｃｔｉｖｅｎｅｓｓａｎｄｓｉｍｐｌｉｃｉｔｙ．Ｈｏｗｅｖｅｒ，ａｔｔｈｅｂｅｇｉｎｎｉｎｇｏｆｃｏｎｖｅｒｇｅｎｃｅ，ｏｖｅｒｓｈｏｏｔａｎｄｏｓｃｉｌｌａｔｉｏｎａｒｅｕｎｉｖｅｒｓａｌｂｅｃａｕｓｅｏｆｔｈｅｌｉｍｉｔａｔｉｏｎｏｆｃｏｍｍｕｎｉｃａｔｉｏｎａｎｄａｌａｃｋｏｆｆｏｒｅｃａｓｔｉｎｇ，ｗｈｉｃｈａｒｅｉｎｂｏｒｎｓｈｏｒｔｃｏｍｉｎｇｓｏｆｃｏｎｓｅｎｓｕｓ．Ｉｔｉｓｎａｔｕｒａｌｔｏｍｏｄｉｆｙｔｈｉｓｍｅｔｈｏｄｗｉｔｈｌｏｔｓｏｆｏｐｔｉｍｉｚｅｒｓ．Ｉｎｏｒｄｅｒｔｏｒｅｄｕｃｅｏｖｅｒｓｈｏｏｔａｎｄｓｍｏｏｔｈｔｒａｊｅｃｔｏｒｉｅｓ，ｔｈｉｓｐａｐｅｒｆｉｒｓｔａｄｏｐｔｅｄｐａｒｔｉｃｌｅｓｗａｒｍｏｐｔｉｍｉｚａｔｉｏｎ（ＰＳＯ），ｔｈｅｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎ（ＰＩＯ）ｔｏｍｏｄｉｆｙｔｈｅｃｏｎｓｅｎｓｕｓ．ＰＳＯｉｓａｖｅｒｙｐｏｐｕｌａｒｏｐｔｉｍｉｚｅｒ，ｗｈｉｌｅＰＩＯｉｓａｎｅｗｍｅｔｈｏｄ，ｂｏｔｈｗｏｒｋｂｕｔｓｔｉｌｌｒｅｔａｉｎｄｉｓａｄｖａｎｔａｇｅｓｓｕｃｈａｓｒｅｓｉｄｕａｌｏｓｃｉｌｌａｔｉｏｎ．Ａｓａｒｅｓｕｌｔ，ｉｔｗａｓｎｅｃｅｓｓａｒｙｔｏｍｏｄｉｆｙＰＩＯ，ａｎｄａｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ（ＳＤ⁃ＰＩＯ）ｉｓｐｒｏｐｏｓｅｄ．ＣｏｎｖｅｒｇｅｎｃｅａｎａｌｙｓｉｓｗａｓｐｅｒｆｏｒｍｅｄｏｎｔｈｅＳＤ⁃ＰＩＯｂａｓｅｄｏｎｔｈｅＢａｎａｃｈｆｉｘｅｄ⁃ｐｏｉｎｔｔｈｅｏｒｅｍａｎｄｃｏｎｄｉｔｉｏｎｓｓｕｆｆｉｃｉｅｎｔｆｏｒｓｔａｂｉｌｉｔｙｗｅｒｅａｃｈｉｅｖｅｄ．Ｆｉｎａｌｌｙ，ａｓｅｒｉｅｓｏｆｃｏｍｐａｒａｔｉｖｅｓｉｍｕｌａｔｉｏｎｓｗｅｒｅｃｏｎｄｕｃｔｅｄｔｏｖｅｒｉｆｙｔｈｅｆｅａｓｉｂｉｌｉｔｙａｎｄｅｆｆｅｃｔｉｖｅｎｅｓｓｏｆｔｈｅｐｒｏｐｏｓｅｄａｐｐｒｏａｃｈ．Ｋｅｙｗｏｒｄｓ：ｕｎｍａｎｎｅｄａｅｒｉａｌｖｅｈｉｃｌｅ（ＵＡＶ）；ｆｏｒｍａｔｉｏｎ；ｃｏｎｓｅｎｓｕｓ；ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎ（ＰＩＯ）；Ｂａｎａｃｈｆｉｘｅｄ⁃ｐｏｉｎｔｔｈｅｏｒｅｍ收稿日期：２０１７－０４－１１．网络出版日期：２０１７－０６－３０．基金项目：ＮａｔｕｒａｌＳｃｉｅｎｃｅＦｏｕｎｄａｔｉｏｎｏｆＣｈｉｎａｕｎｄｅｒＧｒａｎｔ（６１３３３００４）．通信作者：ＺＨＡＮＧＴｉａｎｊｉｅ．Ｅ⁃ｍａｉｌ：１１０３１１４８＠ｂｕａａ．ｅｄｕ．ｃｎ．中图分类号：ＴＰ１８；ＴＰ２７３文献标志码：Ａ文章编号：１６７３－４７８５（２０１７）０４－０５７０－１２英文引用格式：ＺＨＡＮＧＴｉａｎｊｉｅ，ＤＵＡＮＨａｉｂｉｎ．Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ［Ｊ］．ＣＡＡＩｔｒａｎｓａｃｔｉｏｎｓｏｎｉｎｔｅｌｌｉｇｅｎｔｓｙｓｔｅｍｓ，２０１７，１２（４）：５７０－５８１．Ｆｏｒｍａｔｉｏｎｃｏｎｔｒｏｌｏｆｕｎｍａｎｎｅｄａｅｒｉａｌｖｅｈｉｃｌｅｓ（ＵＡＶｓ）ｃａｎａｃｃｏｍｐｌｉｓｈａｔａｓｋｂｅｔｔｅｒｔｈａｎａｓｉｎｇｌｅＵＡＶａｎｄｈａｓｂｅｅｎｓｔｕｄｉｅｄｕｓｉｎｇｖａｒｉｏｕｓｍｅｔｈｏｄｓ．ＵＡＶｓａｒｅｗｉｄｅｌｙａｄｏｐｔｅｄｉｎｖａｒｉｏｕｓｓｉｔｕａｔｉｏｎｓ［１－２］．Ｈｏｗｅｖｅｒ，ｔｈｅｒｅｑｕｉｒｅｍｅｎｔｓｎｏｗａｄａｙｓｈａｖｅａｌｒｅａｄｙｏｖｅｒｔａｋｅｎｔｈｅｃｕｒｒｅｎｔａｂｉｌｉｔｉｅｓｏｆａｓｉｎｇｌｅＵＡＶ．Ｉｎｏｒｄｅｒｔｏｓａｔｉｓｆｙｔｈｅｒｅｑｕｉｒｅｍｅｎｔｓ，ｄｉｆｆｅｒｅｎｔｐｌａｎｓｆｏｒｔｈｅｃｏｏｐｅｒａｔｉｏｎｏｆｍａｎｙＵＡＶｓｈａｖｅｂｅｅｎｐｒｏｐｏｓｅｄｂｙＣｈａｕｍｅｔｔｅ［３］ａｎｄＲａｓｃｈｅ［４］，ａｎｄｔｈｅｄｅｓｉｇｎｏｆａｍｅｔｈｏｄｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｉｓｉｍｐｏｒｔａｎｔ［５］．Ｄｉｆｆｅｒｅｎｔｍｅｔｈｏｄｓｈａｖｅｂｅｅｎｐｒｏｐｏｓｅｄｓｕｃｈａｓｔｈｅｌｅａｄｅｒ⁃ｆｏｌｌｏｗｅｒａｐｐｒｏａｃｈ［６］，ａｒｔｉｆｉｃｉａｌｐｏｔｅｎｔｉａｌｆｉｅｌｄ［７］，ａｎｄｓｌｉｄｉｎｇｍｏｄｅｃｏｎｔｒｏｌ［８］．Ａｍｏｎｇｔｈｅｓｅａｐｐｒｏａｃｈｅｓ，ｃｏｎｓｅｎｓｕｓｉｓａｎｅｆｆｅｃｔｉｖｅａｎｄｅａｓｙｍｅｔｈｏｄｆｏｒｄｉｓｔｒｉｂｕｔｅｄｃｏｎｔｒｏｌ［９］．Ｕｓｉｎｇｓｅｖｅｒａｌｓｉｍｐｌｅｃｏｎｓｅｎｓｕｓｅｑｕａｔｉｏｎｓ，ＵＡＶｓｃａｎｆｌｙａｌｏｎｇｃｏｍｐｌｅｘｔｒａｊｅｃｔｏｒｉｅｓ．Ｃｏｎｓｅｎｓｕｓｏｒｉｇｉｎａｔｅｓｆｒｏｍｔｈｅｎａｔｕｒａｌｂｅｈａｖｉｏｒｏｆａｎｉｍａｌｓｓｕｃｈａｓｆｉｓｈａｎｄｂｉｒｄｓ．Ａｓｉｎｇｌｅｂｉｒｄｏｒｆｉｓｈｉｓｓｍａｌｌａｎｄｗｅａｋ，ｂｕｔａｐｏｐｕｌａｔｉｏｎｃａｎｈｏｌｄａｃｏｍｐｌｉｃａｔｅｄｆｏｒｍａｔｉｏｎ．Ｉｔｉｓｃｏｎｓｅｎｓｕｓｔｈａｔｐｌａｙｓａｎｅｓｓｅｎｔｉａｌｒｏｌｅｉｎｔｈｉｓｐｈｅｎｏｍｅｎｏｎ．Ｖｉｃｓｅｋ［１０］ａｎｄＲｅｎ［９，１１－１２］ｈａｖｅａｃｈｉｅｖｅｄｍａｎｙｒｅｓｕｌｔｓｉｎｔｈｉｓｆｉｅｌｄ．Ｔｈｉｓｐａｐｅｒａｄｏｐｔｓｔｈｉｓｍｅｔｈｏｄｔｏｄｅｓｉｇｎａｃｏｎｔｒｏｌｌｅｒｆｏｒａｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎ．

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 ·571- There are two main parts of the consensus proposed. algorithm[.First,the formation is treated as a The remainder of this paper is organized as virtual rigid body,every UAV has a desired position follows.In Section 2,details of the application of and this acts as a point on the body.Using consensus, consensus in a multi-UAV formation are given.The the UAVs'desired positions concentrate together and construction of SD-PIO and the convergence analysis is create the desired formation.However,a distance error considered in Section 3.In Section 4,the SD-PIO- always exists between the desired position and the real based consensus algorithm is proposed for a multi-UAV position for each UAV,and the second part of the formation,followed by numerical simulation and results algorithm deals with this problem by considering this analysis in Section 5.Finally,several conclusions error as a variable in consensus.By devising a useful concerning the use of optimal algorithms in consensus algorithm,the error closes to zero as time tends to theory are drawn. infinity.The real position for each UAV is the sum of the desired position and the error.In this way,UAVs 1 Controller design based on consensus with random initial locations always concentrate and fly 1.1 Multi-UAV formation controller design along their predesigned routes.These two parts of the It is supposed that each UAV possesses a moving algorithm added to the communication networks of the coordinate system with three axes (x,y,and z).The individual vehicles comprise the whole UAV shape of the UAV is ignored and a mass point model is system.However,consensus can only ensure that used instead.This is because the time delay of an all the variables are close to the given state and does aircraft control loop is much shorter than that of the not know whether the values of variables at each navigation and guidance loops.Therefore,when moment are optimal.It is a good idea to improve dealing with formation control,the UAV model can be consensus with optimal algorithms. simplified as a point,which means there is no need to Pigeon-inspired optimization (PIO)is a novel consider each UAV's attitude.Under this assumption, swarm intelligence optimizer proposed by Duan and all the UAV's x,y,and z axes are parallel. Qiao in 2014015).The method imitates pigeons' Consensus means that multiple UAVs reach an behavior when seeking a way back to their loft.It is agreement on a common state,e.g.,the UAV's believed that pigeons adopt different tools at different position.In detail,every UAV has its initial 3- stages when returning home.In the beginning,compass dimensional position (x,y,z)in an inertial and map factors are adopted,each pigeon's velocity coordinate system,namely the ground coordinate is updated by the linear combination of its neighbors' system.The velocity of the original UAV's moving including itself)best locations and its former velocity. coordinate system converges to the standard state of the When pigeons approach home,some smart ones with virtual leader UAVo by consensus.The UAVo's state good memories can recognize familiar landmarks which consists of a time-varying position and predesigned guide them,the others just need to follow these velocity.In this way,all the UAVs'coordinates leaders.The second part of PIO is designed to become one common moving coordinate system.After concentrate the average location of all the pigeons.It is adding the relative position of each UAV in the moving widely known that each optimizer fits one or several coordinate system,the desired time-varying position is problems,and PIO was first proposed for air robot path achieved for each UAV.When viewed as a mass of plannings],image processing,and orbit moving points,all these UAVs constitute a virtual formations(20].When it comes to the optimization of body. consensus mentioned above,disadvantages appear such The formation method consists of both single-and as the trajectory closes to the desired trace with double-integrator consensus algorithms.The latter aims oscillation to an unsatisfactory degree(This means to concentrate the UAVs with random initial positions that PIO needs to be modified and a pigeon-inspired and ensure that the desired positions of all the UAVs optimization with slow diving strategy (SD-PIO)is converge to points on the predesigned trajectory.The

Ｔｈｅｒｅａｒｅｔｗｏｍａｉｎｐａｒｔｓｏｆｔｈｅｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍ［１３－１４］．Ｆｉｒｓｔ，ｔｈｅｆｏｒｍａｔｉｏｎｉｓｔｒｅａｔｅｄａｓａｖｉｒｔｕａｌｒｉｇｉｄｂｏｄｙ，ｅｖｅｒｙＵＡＶｈａｓａｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎａｎｄｔｈｉｓａｃｔｓａｓａｐｏｉｎｔｏｎｔｈｅｂｏｄｙ．Ｕｓｉｎｇｃｏｎｓｅｎｓｕｓ，ｔｈｅＵＡＶｓ’ ｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｓｃｏｎｃｅｎｔｒａｔｅｔｏｇｅｔｈｅｒａｎｄｃｒｅａｔｅｔｈｅｄｅｓｉｒｅｄｆｏｒｍａｔｉｏｎ．Ｈｏｗｅｖｅｒ，ａｄｉｓｔａｎｃｅｅｒｒｏｒａｌｗａｙｓｅｘｉｓｔｓｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎａｎｄｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｆｏｒｅａｃｈＵＡＶ，ａｎｄｔｈｅｓｅｃｏｎｄｐａｒｔｏｆｔｈｅａｌｇｏｒｉｔｈｍｄｅａｌｓｗｉｔｈｔｈｉｓｐｒｏｂｌｅｍｂｙｃｏｎｓｉｄｅｒｉｎｇｔｈｉｓｅｒｒｏｒａｓａｖａｒｉａｂｌｅｉｎｃｏｎｓｅｎｓｕｓ．Ｂｙｄｅｖｉｓｉｎｇａｕｓｅｆｕｌａｌｇｏｒｉｔｈｍ，ｔｈｅｅｒｒｏｒｃｌｏｓｅｓｔｏｚｅｒｏａｓｔｉｍｅｔｅｎｄｓｔｏｉｎｆｉｎｉｔｙ．ＴｈｅｒｅａｌｐｏｓｉｔｉｏｎｆｏｒｅａｃｈＵＡＶｉｓｔｈｅｓｕｍｏｆｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎａｎｄｔｈｅｅｒｒｏｒ．Ｉｎｔｈｉｓｗａｙ，ＵＡＶｓｗｉｔｈｒａｎｄｏｍｉｎｉｔｉａｌｌｏｃａｔｉｏｎｓａｌｗａｙｓｃｏｎｃｅｎｔｒａｔｅａｎｄｆｌｙａｌｏｎｇｔｈｅｉｒｐｒｅｄｅｓｉｇｎｅｄｒｏｕｔｅｓ．ＴｈｅｓｅｔｗｏｐａｒｔｓｏｆｔｈｅａｌｇｏｒｉｔｈｍａｄｄｅｄｔｏｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｎｅｔｗｏｒｋｓｏｆｔｈｅｉｎｄｉｖｉｄｕａｌｖｅｈｉｃｌｅｓｃｏｍｐｒｉｓｅｔｈｅｗｈｏｌｅＵＡＶｓｙｓｔｅｍ［１３－１４］．Ｈｏｗｅｖｅｒ，ｃｏｎｓｅｎｓｕｓｃａｎｏｎｌｙｅｎｓｕｒｅｔｈａｔａｌｌｔｈｅｖａｒｉａｂｌｅｓａｒｅｃｌｏｓｅｔｏｔｈｅｇｉｖｅｎｓｔａｔｅａｎｄｄｏｅｓｎｏｔｋｎｏｗｗｈｅｔｈｅｒｔｈｅｖａｌｕｅｓｏｆｖａｒｉａｂｌｅｓａｔｅａｃｈｍｏｍｅｎｔａｒｅｏｐｔｉｍａｌ．Ｉｔｉｓａｇｏｏｄｉｄｅａｔｏｉｍｐｒｏｖｅｃｏｎｓｅｎｓｕｓｗｉｔｈｏｐｔｉｍａｌａｌｇｏｒｉｔｈｍｓ．Ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎ（ＰＩＯ）ｉｓａｎｏｖｅｌｓｗａｒｍｉｎｔｅｌｌｉｇｅｎｃｅｏｐｔｉｍｉｚｅｒｐｒｏｐｏｓｅｄｂｙＤｕａｎａｎｄＱｉａｏｉｎ２０１４［１５］．Ｔｈｅｍｅｔｈｏｄｉｍｉｔａｔｅｓｐｉｇｅｏｎｓ􀆳 ｂｅｈａｖｉｏｒｗｈｅｎｓｅｅｋｉｎｇａｗａｙｂａｃｋｔｏｔｈｅｉｒｌｏｆｔ．Ｉｔｉｓｂｅｌｉｅｖｅｄｔｈａｔｐｉｇｅｏｎｓａｄｏｐｔｄｉｆｆｅｒｅｎｔｔｏｏｌｓａｔｄｉｆｆｅｒｅｎｔｓｔａｇｅｓｗｈｅｎｒｅｔｕｒｎｉｎｇｈｏｍｅ．Ｉｎｔｈｅｂｅｇｉｎｎｉｎｇ，ｃｏｍｐａｓｓａｎｄｍａｐｆａｃｔｏｒｓａｒｅａｄｏｐｔｅｄ［１６］，ｅａｃｈｐｉｇｅｏｎ􀆳ｓｖｅｌｏｃｉｔｙｉｓｕｐｄａｔｅｄｂｙｔｈｅｌｉｎｅａｒｃｏｍｂｉｎａｔｉｏｎｏｆｉｔｓｎｅｉｇｈｂｏｒｓ’ （ｉｎｃｌｕｄｉｎｇｉｔｓｅｌｆ）ｂｅｓｔｌｏｃａｔｉｏｎｓａｎｄｉｔｓｆｏｒｍｅｒｖｅｌｏｃｉｔｙ．Ｗｈｅｎｐｉｇｅｏｎｓａｐｐｒｏａｃｈｈｏｍｅ，ｓｏｍｅｓｍａｒｔｏｎｅｓｗｉｔｈｇｏｏｄｍｅｍｏｒｉｅｓｃａｎｒｅｃｏｇｎｉｚｅｆａｍｉｌｉａｒｌａｎｄｍａｒｋｓｗｈｉｃｈｇｕｉｄｅｔｈｅｍ［１７］，ｔｈｅｏｔｈｅｒｓｊｕｓｔｎｅｅｄｔｏｆｏｌｌｏｗｔｈｅｓｅｌｅａｄｅｒｓ．ＴｈｅｓｅｃｏｎｄｐａｒｔｏｆＰＩＯｉｓｄｅｓｉｇｎｅｄｔｏｃｏｎｃｅｎｔｒａｔｅｔｈｅａｖｅｒａｇｅｌｏｃａｔｉｏｎｏｆａｌｌｔｈｅｐｉｇｅｏｎｓ．Ｉｔｉｓｗｉｄｅｌｙｋｎｏｗｎｔｈａｔｅａｃｈｏｐｔｉｍｉｚｅｒｆｉｔｓｏｎｅｏｒｓｅｖｅｒａｌｐｒｏｂｌｅｍｓ，ａｎｄＰＩＯｗａｓｆｉｒｓｔｐｒｏｐｏｓｅｄｆｏｒａｉｒｒｏｂｏｔｐａｔｈｐｌａｎｎｉｎｇ［１５，１８］，ｉｍａｇｅｐｒｏｃｅｓｓｉｎｇ［１７，１９］，ａｎｄｏｒｂｉｔｆｏｒｍａｔｉｏｎｓ［２０］．Ｗｈｅｎｉｔｃｏｍｅｓｔｏｔｈｅｏｐｔｉｍｉｚａｔｉｏｎｏｆｃｏｎｓｅｎｓｕｓｍｅｎｔｉｏｎｅｄａｂｏｖｅ，ｄｉｓａｄｖａｎｔａｇｅｓａｐｐｅａｒｓｕｃｈａｓｔｈｅｔｒａｊｅｃｔｏｒｙｃｌｏｓｅｓｔｏｔｈｅｄｅｓｉｒｅｄｔｒａｃｅｗｉｔｈｏｓｃｉｌｌａｔｉｏｎｔｏａｎｕｎｓａｔｉｓｆａｃｔｏｒｙｄｅｇｒｅｅ［２１］．ＴｈｉｓｍｅａｎｓｔｈａｔＰＩＯｎｅｅｄｓｔｏｂｅｍｏｄｉｆｉｅｄａｎｄａｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ（ＳＤ⁃ＰＩＯ）ｉｓｐｒｏｐｏｓｅｄ．Ｔｈｅｒｅｍａｉｎｄｅｒｏｆｔｈｉｓｐａｐｅｒｉｓｏｒｇａｎｉｚｅｄａｓｆｏｌｌｏｗｓ．ＩｎＳｅｃｔｉｏｎ２，ｄｅｔａｉｌｓｏｆｔｈｅａｐｐｌｉｃａｔｉｏｎｏｆｃｏｎｓｅｎｓｕｓｉｎａｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎａｒｅｇｉｖｅｎ．ＴｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆＳＤ⁃ＰＩＯａｎｄｔｈｅｃｏｎｖｅｒｇｅｎｃｅａｎａｌｙｓｉｓｉｓｃｏｎｓｉｄｅｒｅｄｉｎＳｅｃｔｉｏｎ３．ＩｎＳｅｃｔｉｏｎ４，ｔｈｅＳＤ⁃ＰＩＯ⁃ ｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｉｓｐｒｏｐｏｓｅｄｆｏｒａｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎ，ｆｏｌｌｏｗｅｄｂｙｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎａｎｄｒｅｓｕｌｔｓａｎａｌｙｓｉｓｉｎＳｅｃｔｉｏｎ５．Ｆｉｎａｌｌｙ，ｓｅｖｅｒａｌｃｏｎｃｌｕｓｉｏｎｓｃｏｎｃｅｒｎｉｎｇｔｈｅｕｓｅｏｆｏｐｔｉｍａｌａｌｇｏｒｉｔｈｍｓｉｎｃｏｎｓｅｎｓｕｓｔｈｅｏｒｙａｒｅｄｒａｗｎ．１Ｃｏｎｔｒｏｌｌｅｒｄｅｓｉｇｎｂａｓｅｄｏｎｃｏｎｓｅｎｓｕｓ１．１Ｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｃｏｎｔｒｏｌｌｅｒｄｅｓｉｇｎＩｔｉｓｓｕｐｐｏｓｅｄｔｈａｔｅａｃｈＵＡＶｐｏｓｓｅｓｓｅｓａｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｗｉｔｈｔｈｒｅｅａｘｅｓ（ｘ，ｙ，ａｎｄｚ）．ＴｈｅｓｈａｐｅｏｆｔｈｅＵＡＶｉｓｉｇｎｏｒｅｄａｎｄａｍａｓｓｐｏｉｎｔｍｏｄｅｌｉｓｕｓｅｄｉｎｓｔｅａｄ．Ｔｈｉｓｉｓｂｅｃａｕｓｅｔｈｅｔｉｍｅｄｅｌａｙｏｆａｎａｉｒｃｒａｆｔｃｏｎｔｒｏｌｌｏｏｐｉｓｍｕｃｈｓｈｏｒｔｅｒｔｈａｎｔｈａｔｏｆｔｈｅｎａｖｉｇａｔｉｏｎａｎｄｇｕｉｄａｎｃｅｌｏｏｐｓ．Ｔｈｅｒｅｆｏｒｅ，ｗｈｅｎｄｅａｌｉｎｇｗｉｔｈｆｏｒｍａｔｉｏｎｃｏｎｔｒｏｌ，ｔｈｅＵＡＶｍｏｄｅｌｃａｎｂｅｓｉｍｐｌｉｆｉｅｄａｓａｐｏｉｎｔ，ｗｈｉｃｈｍｅａｎｓｔｈｅｒｅｉｓｎｏｎｅｅｄｔｏｃｏｎｓｉｄｅｒｅａｃｈＵＡＶ’ｓａｔｔｉｔｕｄｅ．Ｕｎｄｅｒｔｈｉｓａｓｓｕｍｐｔｉｏｎ，ａｌｌｔｈｅＵＡＶ’ｓｘ，ｙ，ａｎｄｚａｘｅｓａｒｅｐａｒａｌｌｅｌ．ＣｏｎｓｅｎｓｕｓｍｅａｎｓｔｈａｔｍｕｌｔｉｐｌｅＵＡＶｓｒｅａｃｈａｎａｇｒｅｅｍｅｎｔｏｎａｃｏｍｍｏｎｓｔａｔｅ，ｅ．ｇ．，ｔｈｅＵＡＶ’ ｓｐｏｓｉｔｉｏｎ．Ｉｎｄｅｔａｉｌ，ｅｖｅｒｙＵＡＶｈａｓｉｔｓｉｎｉｔｉａｌ３⁃ ｄｉｍｅｎｓｉｏｎａｌｐｏｓｉｔｉｏｎ（ｘ，ｙ，ｚ）ｉｎａｎｉｎｅｒｔｉａｌｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ，ｎａｍｅｌｙｔｈｅｇｒｏｕｎｄｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ．ＴｈｅｖｅｌｏｃｉｔｙｏｆｔｈｅｏｒｉｇｉｎａｌＵＡＶｉ ’ ｓｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｃｏｎｖｅｒｇｅｓｔｏｔｈｅｓｔａｎｄａｒｄｓｔａｔｅｏｆｔｈｅｖｉｒｔｕａｌｌｅａｄｅｒＵＡＶ１０ｂｙｃｏｎｓｅｎｓｕｓ．ＴｈｅＵＡＶ１０ ’ ｓｓｔａｔｅｃｏｎｓｉｓｔｓｏｆａｔｉｍｅ⁃ｖａｒｙｉｎｇｐｏｓｉｔｉｏｎａｎｄｐｒｅｄｅｓｉｇｎｅｄｖｅｌｏｃｉｔｙ．Ｉｎｔｈｉｓｗａｙ，ａｌｌｔｈｅＵＡＶｓ ’ ｃｏｏｒｄｉｎａｔｅｓｂｅｃｏｍｅｏｎｅｃｏｍｍｏｎｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ．ＡｆｔｅｒａｄｄｉｎｇｔｈｅｒｅｌａｔｉｖｅｐｏｓｉｔｉｏｎｏｆｅａｃｈＵＡＶｉｎｔｈｅｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ，ｔｈｅｄｅｓｉｒｅｄｔｉｍｅ⁃ｖａｒｙｉｎｇｐｏｓｉｔｉｏｎｉｓａｃｈｉｅｖｅｄｆｏｒｅａｃｈＵＡＶ．Ｗｈｅｎｖｉｅｗｅｄａｓａｍａｓｓｏｆｍｏｖｉｎｇｐｏｉｎｔｓ，ａｌｌｔｈｅｓｅＵＡＶｓｃｏｎｓｔｉｔｕｔｅａｖｉｒｔｕａｌｂｏｄｙ．Ｔｈｅｆｏｒｍａｔｉｏｎｍｅｔｈｏｄｃｏｎｓｉｓｔｓｏｆｂｏｔｈｓｉｎｇｌｅ⁃ａｎｄｄｏｕｂｌｅ⁃ｉｎｔｅｇｒａｔｏｒｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｓ．ＴｈｅｌａｔｔｅｒａｉｍｓｔｏｃｏｎｃｅｎｔｒａｔｅｔｈｅＵＡＶｓｗｉｔｈｒａｎｄｏｍｉｎｉｔｉａｌｐｏｓｉｔｉｏｎｓａｎｄｅｎｓｕｒｅｔｈａｔｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｓｏｆａｌｌｔｈｅＵＡＶｓｃｏｎｖｅｒｇｅｔｏｐｏｉｎｔｓｏｎｔｈｅｐｒｅｄｅｓｉｇｎｅｄｔｒａｊｅｃｔｏｒｙ．Ｔｈｅ ·５７１· 第４期ＺＨＡＮＧＴｉａｎｊｉｅ，ｅｔａｌ．：Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ

.572. 智能系统学报第12卷 former ensures the distance error between the desired that the directed communication topology graph must and the real position converges to 0[4).The real include a spanning tree position of each UAV can be easily acquired and the To attain the desired position for one single UAV UAV's trajectory is drawn. in the moving coordinate system,the error between the Consensus of moving desired and the real position should be treated as an coordinate system extra variable,this can be described by single- State variables of integrator dynamics for simplicity.The math model can Standard moving coordinate Communication be described as state network Errors added with relative er:=u (3) pesition in the formation and the consensus algorithm is in Ref [9]. Consensus of relative position in the moving UAV's u:=- (4) relative position coordinate system -∑ag(e,-er） in the system =rpi+er (5) JA JA JAV,-UAY Where7;is the real position of UAV:,UAV:in the moving coordinate system,er;is the distance error Fig.1 Frame of consensus controller in multi-UAV formation between the real and the desired position of UAVi,rP which is shown in Fig.2,is the desired position of For the formation problem in a multi-UAV UAV;in the moving coordinate system,i1,2,.., system,the UAVs'states should include both position 9.Equations (1~5)above can be transformed into a and velocity,and the control inputs are the collective matrix form, accelerations4.This model is always described by (L☒L)E+(b☒L)E=-K[(L⑧I)E+(b☒L)]- second-order differentiation equations,and as a result, a consensus algorithm for double-integrator dynamics is K[(L☒I3)E+(b☒13)E] required.The dynamic math model of the multi-UAV e=-(L☒I3)e system is given below, r=r+专=p+er+ (6) where Loxo is the Laplace matrix of the communication (1) topology graph,bRreflects which UAV can directly 5=[x] receive information from UAVo,is the desired where专：，i:h,∈R3 represent the position,.the velocity position and also the position of UAVo,which shows and the acceleration of UAV:,respectively and x,y, the desired path for all the UAVs,finally, are the3-dimensional positions,i∈{1,2，…，l0}.The represents the Kronecker product. consensus algorithm can be described as in Ref [14]. K,=diag(K1,K2,…,Kg）,K,=diag(Kl,K2, 三，咳-K--&G,-6]+ 1 …,Ky）,er=(erer…erg),f=(r7…fg)T,ris := the vector of the real position of each UAV,g is the 心-k-》-k-5】(2 vector of the position of the origin of each UAV's moving coordinate system. Where a;is element (i,j)in the adjacency matrix Equation (6)is the core of the multi-UAV 10 controller,and r provides the necessary positional A66∈R6x6,andk=∑ag,K.,K.isa3×3size information for all the UAVs for use in the next step. symmetric positive definite matrix,i1,2,..,9, After acquiring the target instructions,the UAVs fly in this work this is replaced by a Pascal matrix for toward their destinations. simplicity.The algorithm has been proven effective and 1.2 Multi-UAV formation task can make all the UAVs'both positions and velocities The desired formation shape is designed as converge to the state of the root.This is because the follows: graph satisfies the sufficient and necessary condition

ｆｏｒｍｅｒｅｎｓｕｒｅｓｔｈｅｄｉｓｔａｎｃｅｅｒｒｏｒｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄａｎｄｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｃｏｎｖｅｒｇｅｓｔｏ０［１４］．ＴｈｅｒｅａｌｐｏｓｉｔｉｏｎｏｆｅａｃｈＵＡＶｃａｎｂｅｅａｓｉｌｙａｃｑｕｉｒｅｄａｎｄｔｈｅＵＡＶ’ｓｔｒａｊｅｃｔｏｒｙｉｓｄｒａｗｎ．Ｆｉｇ．１Ｆｒａｍｅｏｆｃｏｎｓｅｎｓｕｓｃｏｎｔｒｏｌｌｅｒｉｎｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎＦｏｒｔｈｅｆｏｒｍａｔｉｏｎｐｒｏｂｌｅｍｉｎａｍｕｌｔｉ⁃ＵＡＶｓｙｓｔｅｍ，ｔｈｅＵＡＶｓ’ ｓｔａｔｅｓｓｈｏｕｌｄｉｎｃｌｕｄｅｂｏｔｈｐｏｓｉｔｉｏｎａｎｄｖｅｌｏｃｉｔｙ，ａｎｄｔｈｅｃｏｎｔｒｏｌｉｎｐｕｔｓａｒｅｔｈｅａｃｃｅｌｅｒａｔｉｏｎｓ［１４］．Ｔｈｉｓｍｏｄｅｌｉｓａｌｗａｙｓｄｅｓｃｒｉｂｅｄｂｙｓｅｃｏｎｄ⁃ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｅｑｕａｔｉｏｎｓ，ａｎｄａｓａｒｅｓｕｌｔ，ａｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｄｏｕｂｌｅ⁃ｉｎｔｅｇｒａｔｏｒｄｙｎａｍｉｃｓｉｓｒｅｑｕｉｒｅｄ．Ｔｈｅｄｙｎａｍｉｃｍａｔｈｍｏｄｅｌｏｆｔｈｅｍｕｌｔｉ⁃ＵＡＶｓｙｓｔｅｍｉｓｇｉｖｅｎｂｅｌｏｗ， ξ · ｉ＝ ζｉ ζ · ｉ＝ μｉ ξｉ＝［ｘｉｙｉｚｉ］Ｔ ì î í ï ïï ï ï （１）ｗｈｅｒｅ ξｉ，ζｉ，μｉ∈Ｒ３ｒｅｐｒｅｓｅｎｔｔｈｅｐｏｓｉｔｉｏｎ，ｔｈｅｖｅｌｏｃｉｔｙａｎｄｔｈｅａｃｃｅｌｅｒａｔｉｏｎｏｆＵＡＶｉ，ｒｅｓｐｅｃｔｉｖｅｌｙａｎｄｘｉ，ｙｉ，ｚｉａｒｅｔｈｅ３⁃ｄｉｍｅｎｓｉｏｎａｌｐｏｓｉｔｉｏｎｓ，ｉ∈｛１，２，…，１０｝．ＴｈｅｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｃａｎｂｅｄｅｓｃｒｉｂｅｄａｓｉｎＲｅｆ［１４］． μｉ＝１ｋｉ ∑ ９ｊ＝１ａｉｊ［ζ · ｊ－Ｋｒｉ（ξｉ－ ξｊ）－Ｋｖｉ（ζｉ－ ζｊ）］＋１ｋｉａｉ１０［ζ · ｒ－Ｋｒｉ（ξｉ－ ξ ｒ）－Ｋｖｉ（ζｉ－ ζ ｒ）］（２）Ｗｈｅｒｅａｉｊｉｓｅｌｅｍｅｎｔ（ｉ，ｊ）ｉｎｔｈｅａｄｊａｃｅｎｃｙｍａｔｒｉｘＡ６×６ ∈Ｒ６×６，ａｎｄｋｉ＝ ∑ １０ｊ＝１ａｉｊ，Ｋｒｉ，Ｋｖｉｉｓａ３ × ３ｓｉｚｅｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅｍａｔｒｉｘ，ｉ ∈ ｛１，２，…，９｝，ｉｎｔｈｉｓｗｏｒｋｔｈｉｓｉｓｒｅｐｌａｃｅｄｂｙａＰａｓｃａｌｍａｔｒｉｘｆｏｒｓｉｍｐｌｉｃｉｔｙ．ＴｈｅａｌｇｏｒｉｔｈｍｈａｓｂｅｅｎｐｒｏｖｅｎｅｆｆｅｃｔｉｖｅａｎｄｃａｎｍａｋｅａｌｌｔｈｅＵＡＶｓ’ ｂｏｔｈｐｏｓｉｔｉｏｎｓａｎｄｖｅｌｏｃｉｔｉｅｓｃｏｎｖｅｒｇｅｔｏｔｈｅｓｔａｔｅｏｆｔｈｅｒｏｏｔ．Ｔｈｉｓｉｓｂｅｃａｕｓｅｔｈｅｇｒａｐｈｓａｔｉｓｆｉｅｓｔｈｅｓｕｆｆｉｃｉｅｎｔａｎｄｎｅｃｅｓｓａｒｙｃｏｎｄｉｔｉｏｎｔｈａｔｔｈｅｄｉｒｅｃｔｅｄｃｏｍｍｕｎｉｃａｔｉｏｎｔｏｐｏｌｏｇｙｇｒａｐｈｍｕｓｔｉｎｃｌｕｄｅａｓｐａｎｎｉｎｇｔｒｅｅ［１４］．ＴｏａｔｔａｉｎｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｆｏｒｏｎｅｓｉｎｇｌｅＵＡＶｉｎｔｈｅｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ，ｔｈｅｅｒｒｏｒｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄａｎｄｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｓｈｏｕｌｄｂｅｔｒｅａｔｅｄａｓａｎｅｘｔｒａｖａｒｉａｂｌｅ，ｔｈｉｓｃａｎｂｅｄｅｓｃｒｉｂｅｄｂｙｓｉｎｇｌｅ⁃ ｉｎｔｅｇｒａｔｏｒｄｙｎａｍｉｃｓｆｏｒｓｉｍｐｌｉｃｉｔｙ．Ｔｈｅｍａｔｈｍｏｄｅｌｃａｎｂｅｄｅｓｃｒｉｂｅｄａｓｅ · ｒｉ＝ｕｉ（３）ａｎｄｔｈｅｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｉｓｉｎＲｅｆ［９］．ｕｉ＝－ ∑ ９ｊ＝１ａｉｊ（ｅｒｉ－ｅｒｊ）（４）ｒ～ｉ＝ｒｐｉ＋ｅｒｉ（５）Ｗｈｅｒｅｒ～ｉｉｓｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｏｆＵＡＶｉ，ＵＡＶｉｉｎｔｈｅｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ，ｅｒｉｉｓｔｈｅｄｉｓｔａｎｃｅｅｒｒｏｒｂｅｔｗｅｅｎｔｈｅｒｅａｌａｎｄｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｏｆＵＡＶｉ，ｒｐｉｗｈｉｃｈｉｓｓｈｏｗｎｉｎＦｉｇ．２，ｉｓｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｏｆＵＡＶｉｉｎｔｈｅｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ，ｉ∈｛１，２，…，９｝．Ｅｑｕａｔｉｏｎｓ（１～５）ａｂｏｖｅｃａｎｂｅｔｒａｎｓｆｏｒｍｅｄｉｎｔｏａｃｏｌｌｅｃｔｉｖｅｍａｔｒｉｘｆｏｒｍ，（Ｌ 􀱋Ｉ３）ξ ¨ ＋（ｂ 􀱋Ｉ３）ξ ¨ ｒ＝－Ｋｒ［（Ｌ 􀱋Ｉ３）ξ ＋（ｂ 􀱋Ｉ３）ξ ｒ］－Ｋｖ［（Ｌ 􀱋 Ｉ３）ξ · ＋（ｂ 􀱋 Ｉ３）ξ ·ｒ］ｅ · ＝－（Ｌ 􀱋 Ｉ３）ｅｒ＝ｒ～＋ ξ ＝ｒｐ＋ｅｒ＋ ξ ì î í ï ï ï ï ï ï （６）ｗｈｅｒｅＬ９×９ｉｓｔｈｅＬａｐｌａｃｅｍａｔｒｉｘｏｆｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｔｏｐｏｌｏｇｙｇｒａｐｈ，ｂ ∈ Ｒ９ｒｅｆｌｅｃｔｓｗｈｉｃｈＵＡＶｃａｎｄｉｒｅｃｔｌｙｒｅｃｅｉｖｅｉｎｆｏｒｍａｔｉｏｎｆｒｏｍＵＡＶ１０， ξ ｒｉｓｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎａｎｄａｌｓｏｔｈｅｐｏｓｉｔｉｏｎｏｆＵＡＶ１０，ｗｈｉｃｈｓｈｏｗｓｔｈｅｄｅｓｉｒｅｄｐａｔｈｆｏｒａｌｌｔｈｅＵＡＶｓ，ｆｉｎａｌｌｙ， 􀱋 ｒｅｐｒｅｓｅｎｔｓｔｈｅＫｒｏｎｅｃｋｅｒｐｒｏｄｕｃｔ．Ｋｒ＝ｄｉａｇ（Ｋｒ１，Ｋｒ２，…，Ｋｒ９），Ｋｖ＝ｄｉａｇ（Ｋｖ１，Ｋｖ２， …，Ｋｖ９），ｅｒ＝（ｅｒＴ１ｅｒＴ２… ｅｒＴ９）Ｔ，ｒ～＝（ｒ～Ｔ１ｒ～Ｔ２…ｒ～Ｔ９）Ｔ，ｒｉｓｔｈｅｖｅｃｔｏｒｏｆｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｏｆｅａｃｈＵＡＶ， ξ ｉｓｔｈｅｖｅｃｔｏｒｏｆｔｈｅｐｏｓｉｔｉｏｎｏｆｔｈｅｏｒｉｇｉｎｏｆｅａｃｈＵＡＶ’ ｓｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍ．Ｅｑｕａｔｉｏｎ（６）ｉｓｔｈｅｃｏｒｅｏｆｔｈｅｍｕｌｔｉ⁃ＵＡＶｃｏｎｔｒｏｌｌｅｒ，ａｎｄｒｐｒｏｖｉｄｅｓｔｈｅｎｅｃｅｓｓａｒｙｐｏｓｉｔｉｏｎａｌｉｎｆｏｒｍａｔｉｏｎｆｏｒａｌｌｔｈｅＵＡＶｓｆｏｒｕｓｅｉｎｔｈｅｎｅｘｔｓｔｅｐ．Ａｆｔｅｒａｃｑｕｉｒｉｎｇｔｈｅｔａｒｇｅｔｉｎｓｔｒｕｃｔｉｏｎｓ，ｔｈｅＵＡＶｓｆｌｙｔｏｗａｒｄｔｈｅｉｒｄｅｓｔｉｎａｔｉｏｎｓ．１．２Ｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｔａｓｋＴｈｅｄｅｓｉｒｅｄｆｏｒｍａｔｉｏｎｓｈａｐｅｉｓｄｅｓｉｇｎｅｄａｓｆｏｌｌｏｗｓ： ·５７２· 智能系统学报第１２卷

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 .573. Table 1 Positions of 9 UAVs under the desired formation receive information from UAV,.There is also a virtual structure m leader named UAVio,which can emit an arrow to UAV number x another UAV so that the latter knows the predesigned UAV 0 12.60 10.00 trajectory. UAV2 -1.50 10.00 10.00 The communication topology among UAVs is UAV3 1.50 10.00 described as a directed graph in Fig.3.A graph with n 10.00 vertices can be denoted by G(V,E,A),V=(v1,2, UAVa -6.50 0 0 …,tn）is the set of vertices of graph G and:isa UAVs -8.00 -2.60 0 vertex.E=(e,e2,,e)is the edge set in which e UAV. -5.00 -2.60 0 means an edge.A=[a]demonstrates the adjacency UAV] 6.50 0 0 matrix of the graph G.If there is an edge from v:to, UAVs 5.00 -2.60 0 then a>0,otherwise ag=0.Usually,a=1 if a UAVo 8.00 -2.60 0 particular value is not given.The value of a is the weight of communication amount between vi and v;, which can influence the convergence speed between v UAV. 10 UAV. UAV. and v;.In the graph G(V,E,A),if there is a set of edges,(,2),(2,3）,…(v.-1,n),then there 0 is a path from v to v.If there is a vertex from which 10 many edges reach all the remaining vertices,then this UAV UAV 4UaV。 vertex is called a root and G(V,E,A)includes a direct ym o 10 spanning tree. -10 x/m Fig.2 Formation structure for UAVs There are nine UAVs flying at two different heights,they are divided into three groups,each group containing three UAVs.In every small group,the UAVs form a triangle and these are the same size,so the UAVs fly symmetrically.The side length of the triangle and the distance between different triangles is 6 much larger than the size of an UAV,so they can fly Fig.3 Communication topology for multiple UAVs safely without danger of collision.The desired heights For consensus,the adjacency matrix and the in Table I are relative values and the height of the Laplace matrix were acquired.The Laplace matrix L origin of the moving coordinate system is their reference [l]was the same size as the adjacency matrix.In the height,so when a UAV flies at zero or even a negative mati,h,=-ay,ifi≠j,otherise=立4，： height,it is still acceptable.The blue star in Fig.2 j=1j≠i According to the sufficient and necessary represents the center of the formation.x/m means condition of multi-UAV consensus,there is a directed distance along the x-axis in meters. spanning tree with its root UAVo,in the When designing the communication plan,the UAVs were numbered from I to 9.To clarify the plan, communication topology graph,and all the state variables converge to the state of the root as time tends some background knowledge on topology and graph to infinity theory is required. A graph with only vertices and arrows can be 2 Pigeon-inspired optimization with slow adopted to show the communication relationship among diving strategy all the UAVs.Each UAV acts as a vertex,and an 2.1 Basic pigeon-inspired optimization algorithm arrow from UAV to UAV2 means that UAV2 can Qiao [is]studied pigeon behavior when seeking

Ｔａｂｌｅ１Ｐｏｓｉｔｉｏｎｓｏｆ９ＵＡＶｓｕｎｄｅｒｔｈｅｄｅｓｉｒｅｄｆｏｒｍａｔｉｏｎｓｔｒｕｃｔｕｒｅｍＵＡＶｎｕｍｂｅｒｘｙｚＵＡＶ１０１２．６０１０．００ＵＡＶ２－１．５０１０．００１０．００ＵＡＶ３１．５０１０．００１０．００ＵＡＶ４－６．５０００ＵＡＶ５－８．００－２．６００ＵＡＶ６－５．００－２．６００ＵＡＶ７６．５０００ＵＡＶ８５．００－２．６００ＵＡＶ９８．００－２．６００Ｆｉｇ．２ＦｏｒｍａｔｉｏｎｓｔｒｕｃｔｕｒｅｆｏｒＵＡＶｓＴｈｅｒｅａｒｅｎｉｎｅＵＡＶｓｆｌｙｉｎｇａｔｔｗｏｄｉｆｆｅｒｅｎｔｈｅｉｇｈｔｓ，ｔｈｅｙａｒｅｄｉｖｉｄｅｄｉｎｔｏｔｈｒｅｅｇｒｏｕｐｓ，ｅａｃｈｇｒｏｕｐｃｏｎｔａｉｎｉｎｇｔｈｒｅｅＵＡＶｓ．Ｉｎｅｖｅｒｙｓｍａｌｌｇｒｏｕｐ，ｔｈｅＵＡＶｓｆｏｒｍａｔｒｉａｎｇｌｅａｎｄｔｈｅｓｅａｒｅｔｈｅｓａｍｅｓｉｚｅ，ｓｏｔｈｅＵＡＶｓｆｌｙｓｙｍｍｅｔｒｉｃａｌｌｙ．ＴｈｅｓｉｄｅｌｅｎｇｔｈｏｆｔｈｅｔｒｉａｎｇｌｅａｎｄｔｈｅｄｉｓｔａｎｃｅｂｅｔｗｅｅｎｄｉｆｆｅｒｅｎｔｔｒｉａｎｇｌｅｓｉｓｍｕｃｈｌａｒｇｅｒｔｈａｎｔｈｅｓｉｚｅｏｆａｎＵＡＶ，ｓｏｔｈｅｙｃａｎｆｌｙｓａｆｅｌｙｗｉｔｈｏｕｔｄａｎｇｅｒｏｆｃｏｌｌｉｓｉｏｎ．ＴｈｅｄｅｓｉｒｅｄｈｅｉｇｈｔｓｉｎＴａｂｌｅ１ａｒｅｒｅｌａｔｉｖｅｖａｌｕｅｓａｎｄｔｈｅｈｅｉｇｈｔｏｆｔｈｅｏｒｉｇｉｎｏｆｔｈｅｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｉｓｔｈｅｉｒｒｅｆｅｒｅｎｃｅｈｅｉｇｈｔ，ｓｏｗｈｅｎａＵＡＶｆｌｉｅｓａｔｚｅｒｏｏｒｅｖｅｎａｎｅｇａｔｉｖｅｈｅｉｇｈｔ，ｉｔｉｓｓｔｉｌｌａｃｃｅｐｔａｂｌｅ．ＴｈｅｂｌｕｅｓｔａｒｉｎＦｉｇ．２ｒｅｐｒｅｓｅｎｔｓｔｈｅｃｅｎｔｅｒｏｆｔｈｅｆｏｒｍａｔｉｏｎ．ｘ／ｍｍｅａｎｓｄｉｓｔａｎｃｅａｌｏｎｇｔｈｅｘ⁃ａｘｉｓｉｎｍｅｔｅｒｓ．Ｗｈｅｎｄｅｓｉｇｎｉｎｇｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｐｌａｎ，ｔｈｅＵＡＶｓｗｅｒｅｎｕｍｂｅｒｅｄｆｒｏｍ１ｔｏ９．Ｔｏｃｌａｒｉｆｙｔｈｅｐｌａｎ，ｓｏｍｅｂａｃｋｇｒｏｕｎｄｋｎｏｗｌｅｄｇｅｏｎｔｏｐｏｌｏｇｙａｎｄｇｒａｐｈｔｈｅｏｒｙｉｓｒｅｑｕｉｒｅｄ．ＡｇｒａｐｈｗｉｔｈｏｎｌｙｖｅｒｔｉｃｅｓａｎｄａｒｒｏｗｓｃａｎｂｅａｄｏｐｔｅｄｔｏｓｈｏｗｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｒｅｌａｔｉｏｎｓｈｉｐａｍｏｎｇａｌｌｔｈｅＵＡＶｓ．ＥａｃｈＵＡＶａｃｔｓａｓａｖｅｒｔｅｘ，ａｎｄａｎａｒｒｏｗｆｒｏｍＵＡＶ１ｔｏＵＡＶ２ｍｅａｎｓｔｈａｔＵＡＶ２ｃａｎｒｅｃｅｉｖｅｉｎｆｏｒｍａｔｉｏｎｆｒｏｍＵＡＶ１．ＴｈｅｒｅｉｓａｌｓｏａｖｉｒｔｕａｌｌｅａｄｅｒｎａｍｅｄＵＡＶ１０，ｗｈｉｃｈｃａｎｅｍｉｔａｎａｒｒｏｗｔｏａｎｏｔｈｅｒＵＡＶｓｏｔｈａｔｔｈｅｌａｔｔｅｒｋｎｏｗｓｔｈｅｐｒｅｄｅｓｉｇｎｅｄｔｒａｊｅｃｔｏｒｙ．ＴｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｔｏｐｏｌｏｇｙａｍｏｎｇＵＡＶｓｉｓｄｅｓｃｒｉｂｅｄａｓａｄｉｒｅｃｔｅｄｇｒａｐｈｉｎＦｉｇ．３．ＡｇｒａｐｈｗｉｔｈｎｖｅｒｔｉｃｅｓｃａｎｂｅｄｅｎｏｔｅｄｂｙＧ（Ｖ，Ｅ，Ａ），Ｖ＝（ｖ１，ｖ２， …，ｖｎ）ｉｓｔｈｅｓｅｔｏｆｖｅｒｔｉｃｅｓｏｆｇｒａｐｈＧａｎｄｖｉｉｓａｖｅｒｔｅｘ．Ｅ＝（ｅ１，ｅ２，…，ｅｎ）ｉｓｔｈｅｅｄｇｅｓｅｔｉｎｗｈｉｃｈｅｉｍｅａｎｓａｎｅｄｇｅ．Ａ＝［ａｉｊ］ｄｅｍｏｎｓｔｒａｔｅｓｔｈｅａｄｊａｃｅｎｃｙｍａｔｒｉｘｏｆｔｈｅｇｒａｐｈＧ．Ｉｆｔｈｅｒｅｉｓａｎｅｄｇｅｆｒｏｍｖｉｔｏｖｊ，ｔｈｅｎａｉｊ＞０，ｏｔｈｅｒｗｉｓｅａｉｊ＝０．Ｕｓｕａｌｌｙ，ａｉｊ＝１ｉｆａｐａｒｔｉｃｕｌａｒｖａｌｕｅｉｓｎｏｔｇｉｖｅｎ．Ｔｈｅｖａｌｕｅｏｆａｉｊｉｓｔｈｅｗｅｉｇｈｔｏｆｃｏｍｍｕｎｉｃａｔｉｏｎａｍｏｕｎｔｂｅｔｗｅｅｎｖｉａｎｄｖｊ，ｗｈｉｃｈｃａｎｉｎｆｌｕｅｎｃｅｔｈｅｃｏｎｖｅｒｇｅｎｃｅｓｐｅｅｄｂｅｔｗｅｅｎｖｉａｎｄｖｊ．ＩｎｔｈｅｇｒａｐｈＧ（Ｖ，Ｅ，Ａ），ｉｆｔｈｅｒｅｉｓａｓｅｔｏｆｅｄｇｅｓ，（ｖ１，ｖ２），（ｖ２，ｖ３），… （ｖｎ－１，ｖｎ），ｔｈｅｎｔｈｅｒｅｉｓａｐａｔｈｆｒｏｍｖ１ｔｏｖｎ．Ｉｆｔｈｅｒｅｉｓａｖｅｒｔｅｘｆｒｏｍｗｈｉｃｈｍａｎｙｅｄｇｅｓｒｅａｃｈａｌｌｔｈｅｒｅｍａｉｎｉｎｇｖｅｒｔｉｃｅｓ，ｔｈｅｎｔｈｉｓｖｅｒｔｅｘｉｓｃａｌｌｅｄａｒｏｏｔａｎｄＧ（Ｖ，Ｅ，Ａ）ｉｎｃｌｕｄｅｓａｄｉｒｅｃｔｓｐａｎｎｉｎｇｔｒｅｅ．Ｆｉｇ．３ＣｏｍｍｕｎｉｃａｔｉｏｎｔｏｐｏｌｏｇｙｆｏｒｍｕｌｔｉｐｌｅＵＡＶｓＦｏｒｃｏｎｓｅｎｓｕｓ，ｔｈｅａｄｊａｃｅｎｃｙｍａｔｒｉｘａｎｄｔｈｅＬａｐｌａｃｅｍａｔｒｉｘｗｅｒｅａｃｑｕｉｒｅｄ．ＴｈｅＬａｐｌａｃｅｍａｔｒｉｘＬ＝［ｌｉｊ］ｗａｓｔｈｅｓａｍｅｓｉｚｅａｓｔｈｅａｄｊａｃｅｎｃｙｍａｔｒｉｘ．Ｉｎｔｈｅｍａｔｒｉｘ，ｌｉｊ＝－ａｉｊ，ｉｆｉ ≠ ｊ，ｏｔｈｅｒｗｉｓｅｌｉｉ＝ ∑ ｎｊ＝１，ｊ≠ｉａｉｊ．Ａｃｃｏｒｄｉｎｇｔｏｔｈｅｓｕｆｆｉｃｉｅｎｔａｎｄｎｅｃｅｓｓａｒｙｃｏｎｄｉｔｉｏｎｏｆｍｕｌｔｉ⁃ＵＡＶｃｏｎｓｅｎｓｕｓ，ｔｈｅｒｅｉｓａｄｉｒｅｃｔｅｄｓｐａｎｎｉｎｇｔｒｅｅｗｉｔｈｉｔｓｒｏｏｔＵＡＶ１０，ｉｎｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｔｏｐｏｌｏｇｙｇｒａｐｈ，ａｎｄａｌｌｔｈｅｓｔａｔｅｖａｒｉａｂｌｅｓｃｏｎｖｅｒｇｅｔｏｔｈｅｓｔａｔｅｏｆｔｈｅｒｏｏｔａｓｔｉｍｅｔｅｎｄｓｔｏｉｎｆｉｎｉｔｙ［１１］．２Ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ２．１Ｂａｓｉｃｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎａｌｇｏｒｉｔｈｍＱｉａｏ［１５］ｓｔｕｄｉｅｄｐｉｇｅｏｎｂｅｈａｖｉｏｒｗｈｅｎｓｅｅｋｉｎｇ ·５７３· 第４期ＺＨＡＮＧＴｉａｎｊｉｅ，ｅｔａｌ．：Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ

.574. 智能系统学报第12卷 their way home and proposed the two-stage algorithm. overshoot,but the result is not desirable,and the The first stage of PIO is the map and compass part.X= altitude of a UAV drops to the standard height with [XX…X…X]T is the vector of the pigeons' continual oscillation.To improve the performance of the positions,i is the number of virtual pigeons,and in algorithm,better methods are required. this paper the UAVs fly in 3-dimensional space,X=2.2 Improved method with slow diving strategy [x ]V=[VI V.V]T is the vector of the The problem in the former algorithm occurs mainly pigeons'velocities,V=[]is its element, in the altitude part,the height of the UAV decreases i∈{l,2,…,9}.Both X and V are randomly too fast and it dives sharply.To slow the process,the initialized when the program starts.The update update formula should be changed,not only to consider formulas are: the best local position but also the slowest diving (e:(t+1)=:(t)·exp(-Rt)+rand·(xe-x:) velocity.In the first part of PIO,the formula is not x:(t+1)=x,(t)+,(t+1) changed,but the method of acquiring the best local if t Tlabed (7) position is modified.The criterion is a linear In the formula above,the best local position combination of the distance error between the desired is selected from the distance between the desired and and the real positions and the diving speed.An UAV present positions.In order to find the best local that dives very slowly is imitated by the others.This is the so-called "slow diving strategy".The update position,all the distance errors are calculated and formula is: ranked,and only the position with the smallest e:(t+1)=v:(t)·exp(-Rt)+ distance is defined as the best local position.For the minimum optimization question,the fitness function is: r·[p·(xhea-x:)+(1-p）·] fitness(x (t))=x(t)-xmmdm(t)(8) x:(t+1)=x,(t)+,(t+1) After calculating the distance between the desired if t Tubd (10) and the present positions of all the pigeons,only half where serves as a regulatory factor in the range of(0, of the pigeons with lower fitness are selected for the 1);if approaches 0,the slowest pigeon will iteration.This ensures that the positions of these influence the team;if o approaches 1,best local pigeons converge to the desired position as fast as position will have much more of an effect.r(0,1)is possible. a random number. In the second part,the landmark operator is The second part of PIO should be modified in the employed.Its update formula is: following way:all the UAVs converge to the center of wu+1)=水四 the flock,which is calculated by a combination of the 2 UAV positions and the slowest diving speed.The ∑，()·fx,() update equation can be described as: xc(t+1)= ∑fx,(t) +1)=0 2 x,(t+1)=x,(t)+rand·(xc(t+1)-x:(t)) ∑x,()fx,()+10·x·f代x) ift≥Tlabel (9) xc(t+1)= ∑fx,(t)+10·xa·fx) Apparently,xic is the weighted average of all the x,(t+1)=x,(t)+rand·(xc(t+1)-x:(t)) nearby pigeons in pigeon i's communication distance. (11) It is supposed that the map and compass operator has ift≥Tlabel where x is the position of the UAV with the slowest already converged pigeons to the desired trajectory to a diving speed,this part represents the slowing diving great extent.As a result,the landmark operator only strategy. needs to speed up this process.The PIO algorithm has Fitness function Eq.(10)remains.Each UAV already been proven to be robust and reliable for an aircraft path planning problems However,for the has its own fitness value. optimization of consensus,PIO can improve the

ｔｈｅｉｒｗａｙｈｏｍｅａｎｄｐｒｏｐｏｓｅｄｔｈｅｔｗｏ⁃ｓｔａｇｅａｌｇｏｒｉｔｈｍ．ＴｈｅｆｉｒｓｔｓｔａｇｅｏｆＰＩＯｉｓｔｈｅｍａｐａｎｄｃｏｍｐａｓｓｐａｒｔ．Ｘ＝［ＸＴ１ＸＴ２ … ＸＴｉ … ＸＴｎ］Ｔｉｓｔｈｅｖｅｃｔｏｒｏｆｔｈｅｐｉｇｅｏｎｓ’ ｐｏｓｉｔｉｏｎｓ，ｉｉｓｔｈｅｎｕｍｂｅｒｏｆｖｉｒｔｕａｌｐｉｇｅｏｎｓ，ａｎｄｉｎｔｈｉｓｐａｐｅｒｔｈｅＵＡＶｓｆｌｙｉｎ３⁃ｄｉｍｅｎｓｉｏｎａｌｓｐａｃｅ，Ｘｉ＝［ｘｉｙｉｚｉ］Ｔ．Ｖ＝［ＶＴ１ＶＴ２ … ＶＴｎ］Ｔｉｓｔｈｅｖｅｃｔｏｒｏｆｔｈｅｐｉｇｅｏｎｓ’ ｖｅｌｏｃｉｔｉｅｓ，Ｖｉ＝［ｖｘｖｙｖｚ］Ｔｉｓｉｔｓｅｌｅｍｅｎｔ，ｉ ∈｛１，２，…，９｝．ＢｏｔｈＸａｎｄＶａｒｅｒａｎｄｏｍｌｙｉｎｉｔｉａｌｉｚｅｄｗｈｅｎｔｈｅｐｒｏｇｒａｍｓｔａｒｔｓ．Ｔｈｅｕｐｄａｔｅｆｏｒｍｕｌａｓａｒｅ：ｖｉ（ｔ＋１）＝ｖｉ（ｔ）·ｅｘｐ（－Ｒｔ）＋ｒａｎｄ·（ｘｌｂｅｓｔ－ｘｉ）ｘｉ（ｔ＋１）＝ｘｉ（ｔ）＋ｖｉ（ｔ＋１） { ｉｆｔ＜Ｔｌａｂｅｌ（７）Ｉｎｔｈｅｆｏｒｍｕｌａａｂｏｖｅ，ｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎｘｌｂｅｓｔｉｓｓｅｌｅｃｔｅｄｆｒｏｍｔｈｅｄｉｓｔａｎｃｅｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄａｎｄｐｒｅｓｅｎｔｐｏｓｉｔｉｏｎｓ．Ｉｎｏｒｄｅｒｔｏｆｉｎｄｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎ，ａｌｌｔｈｅｄｉｓｔａｎｃｅｅｒｒｏｒｓａｒｅｃａｌｃｕｌａｔｅｄａｎｄｒａｎｋｅｄ，ａｎｄｏｎｌｙｔｈｅｐｏｓｉｔｉｏｎｗｉｔｈｔｈｅｓｍａｌｌｅｓｔｄｉｓｔａｎｃｅｉｓｄｅｆｉｎｅｄａｓｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎ．Ｆｏｒｔｈｅｍｉｎｉｍｕｍｏｐｔｉｍｉｚａｔｉｏｎｑｕｅｓｔｉｏｎ，ｔｈｅｆｉｔｎｅｓｓｆｕｎｃｔｉｏｎｉｓ：ｆｉｔｎｅｓｓ（ｘｉ（ｔ））＝ｘｉ（ｔ）－ｘｉｓｔａｎｄａｒｄ（ｔ）（８）Ａｆｔｅｒｃａｌｃｕｌａｔｉｎｇｔｈｅｄｉｓｔａｎｃｅｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄａｎｄｔｈｅｐｒｅｓｅｎｔｐｏｓｉｔｉｏｎｓｏｆａｌｌｔｈｅｐｉｇｅｏｎｓ，ｏｎｌｙｈａｌｆｏｆｔｈｅｐｉｇｅｏｎｓｗｉｔｈｌｏｗｅｒｆｉｔｎｅｓｓａｒｅｓｅｌｅｃｔｅｄｆｏｒｔｈｅｉｔｅｒａｔｉｏｎ．Ｔｈｉｓｅｎｓｕｒｅｓｔｈａｔｔｈｅｐｏｓｉｔｉｏｎｓｏｆｔｈｅｓｅｐｉｇｅｏｎｓｃｏｎｖｅｒｇｅｔｏｔｈｅｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎａｓｆａｓｔａｓｐｏｓｓｉｂｌｅ．Ｉｎｔｈｅｓｅｃｏｎｄｐａｒｔ，ｔｈｅｌａｎｄｍａｒｋｏｐｅｒａｔｏｒｉｓｅｍｐｌｏｙｅｄ．Ｉｔｓｕｐｄａｔｅｆｏｒｍｕｌａｉｓ：Ｎｐ（ｔ＋１）＝Ｎｐ（ｔ）２ｘｉＣ（ｔ＋１）＝ ∑ｘｉ（ｔ）·ｆ（ｘｉ（ｔ）） ∑ｆ（ｘｉ（ｔ））ｘｉ（ｔ＋１）＝ｘｉ（ｔ）＋ｒａｎｄ·（ｘｉＣ（ｔ＋１）－ｘｉ（ｔ）） ì î í ï ï ïï ï ï ïï ｉｆｔ ≥ Ｔｌａｂｅｌ（９）Ａｐｐａｒｅｎｔｌｙ，ｘｉＣｉｓｔｈｅｗｅｉｇｈｔｅｄａｖｅｒａｇｅｏｆａｌｌｔｈｅｎｅａｒｂｙｐｉｇｅｏｎｓｉｎｐｉｇｅｏｎｉ’ ｓｃｏｍｍｕｎｉｃａｔｉｏｎｄｉｓｔａｎｃｅ．Ｉｔｉｓｓｕｐｐｏｓｅｄｔｈａｔｔｈｅｍａｐａｎｄｃｏｍｐａｓｓｏｐｅｒａｔｏｒｈａｓａｌｒｅａｄｙｃｏｎｖｅｒｇｅｄｐｉｇｅｏｎｓｔｏｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙｔｏａｇｒｅａｔｅｘｔｅｎｔ．Ａｓａｒｅｓｕｌｔ，ｔｈｅｌａｎｄｍａｒｋｏｐｅｒａｔｏｒｏｎｌｙｎｅｅｄｓｔｏｓｐｅｅｄｕｐｔｈｉｓｐｒｏｃｅｓｓ．ＴｈｅＰＩＯａｌｇｏｒｉｔｈｍｈａｓａｌｒｅａｄｙｂｅｅｎｐｒｏｖｅｎｔｏｂｅｒｏｂｕｓｔａｎｄｒｅｌｉａｂｌｅｆｏｒａｎａｉｒｃｒａｆｔｐａｔｈｐｌａｎｎｉｎｇｐｒｏｂｌｅｍ［１５］．Ｈｏｗｅｖｅｒ，ｆｏｒｔｈｅｏｐｔｉｍｉｚａｔｉｏｎｏｆｃｏｎｓｅｎｓｕｓ，ＰＩＯｃａｎｉｍｐｒｏｖｅｔｈｅｏｖｅｒｓｈｏｏｔ，ｂｕｔｔｈｅｒｅｓｕｌｔｉｓｎｏｔｄｅｓｉｒａｂｌｅ，ａｎｄｔｈｅａｌｔｉｔｕｄｅｏｆａＵＡＶｄｒｏｐｓｔｏｔｈｅｓｔａｎｄａｒｄｈｅｉｇｈｔｗｉｔｈｃｏｎｔｉｎｕａｌｏｓｃｉｌｌａｔｉｏｎ．Ｔｏｉｍｐｒｏｖｅｔｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅａｌｇｏｒｉｔｈｍ，ｂｅｔｔｅｒｍｅｔｈｏｄｓａｒｅｒｅｑｕｉｒｅｄ．２．２ＩｍｐｒｏｖｅｄｍｅｔｈｏｄｗｉｔｈｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙＴｈｅｐｒｏｂｌｅｍｉｎｔｈｅｆｏｒｍｅｒａｌｇｏｒｉｔｈｍｏｃｃｕｒｓｍａｉｎｌｙｉｎｔｈｅａｌｔｉｔｕｄｅｐａｒｔ，ｔｈｅｈｅｉｇｈｔｏｆｔｈｅＵＡＶｄｅｃｒｅａｓｅｓｔｏｏｆａｓｔａｎｄｉｔｄｉｖｅｓｓｈａｒｐｌｙ．Ｔｏｓｌｏｗｔｈｅｐｒｏｃｅｓｓ，ｔｈｅｕｐｄａｔｅｆｏｒｍｕｌａｓｈｏｕｌｄｂｅｃｈａｎｇｅｄ，ｎｏｔｏｎｌｙｔｏｃｏｎｓｉｄｅｒｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎｂｕｔａｌｓｏｔｈｅｓｌｏｗｅｓｔｄｉｖｉｎｇｖｅｌｏｃｉｔｙ．ＩｎｔｈｅｆｉｒｓｔｐａｒｔｏｆＰＩＯ，ｔｈｅｆｏｒｍｕｌａｉｓｎｏｔｃｈａｎｇｅｄ，ｂｕｔｔｈｅｍｅｔｈｏｄｏｆａｃｑｕｉｒｉｎｇｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎｉｓｍｏｄｉｆｉｅｄ．Ｔｈｅｃｒｉｔｅｒｉｏｎｉｓａｌｉｎｅａｒｃｏｍｂｉｎａｔｉｏｎｏｆｔｈｅｄｉｓｔａｎｃｅｅｒｒｏｒｂｅｔｗｅｅｎｔｈｅｄｅｓｉｒｅｄａｎｄｔｈｅｒｅａｌｐｏｓｉｔｉｏｎｓａｎｄｔｈｅｄｉｖｉｎｇｓｐｅｅｄ．ＡｎＵＡＶｔｈａｔｄｉｖｅｓｖｅｒｙｓｌｏｗｌｙｉｓｉｍｉｔａｔｅｄｂｙｔｈｅｏｔｈｅｒｓ．Ｔｈｉｓｉｓｔｈｅｓｏ⁃ｃａｌｌｅｄ “ ｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ ”．Ｔｈｅｕｐｄａｔｅｆｏｒｍｕｌａｉｓ：ｖｉ（ｔ＋１）＝ｖｉ（ｔ）·ｅｘｐ（－Ｒｔ）＋ｒ·［φ·（ｘｌｂｅｓｔ－ｘｉ）＋（１－ φ）·ｖｓｌｏｗ］ｘｉ（ｔ＋１）＝ｘｉ（ｔ）＋ｖｉ（ｔ＋１） ì î í ï ï ïï ｉｆｔ＜Ｔｌａｂｅｌ（１０）ｗｈｅｒｅ φ ｓｅｒｖｅｓａｓａｒｅｇｕｌａｔｏｒｙｆａｃｔｏｒｉｎｔｈｅｒａｎｇｅｏｆ（０，１）；ｉｆ φ ａｐｐｒｏａｃｈｅｓ０，ｔｈｅｓｌｏｗｅｓｔｐｉｇｅｏｎｗｉｌｌｉｎｆｌｕｅｎｃｅｔｈｅｔｅａｍ；ｉｆ φ ａｐｐｒｏａｃｈｅｓ１，ｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎｗｉｌｌｈａｖｅｍｕｃｈｍｏｒｅｏｆａｎｅｆｆｅｃｔ．ｒ ∈ （０，１）ｉｓａｒａｎｄｏｍｎｕｍｂｅｒ．ＴｈｅｓｅｃｏｎｄｐａｒｔｏｆＰＩＯｓｈｏｕｌｄｂｅｍｏｄｉｆｉｅｄｉｎｔｈｅｆｏｌｌｏｗｉｎｇｗａｙ：ａｌｌｔｈｅＵＡＶｓｃｏｎｖｅｒｇｅｔｏｔｈｅｃｅｎｔｅｒｏｆｔｈｅｆｌｏｃｋ，ｗｈｉｃｈｉｓｃａｌｃｕｌａｔｅｄｂｙａｃｏｍｂｉｎａｔｉｏｎｏｆｔｈｅＵＡＶｐｏｓｉｔｉｏｎｓａｎｄｔｈｅｓｌｏｗｅｓｔｄｉｖｉｎｇｓｐｅｅｄ．Ｔｈｅｕｐｄａｔｅｅｑｕａｔｉｏｎｃａｎｂｅｄｅｓｃｒｉｂｅｄａｓ：Ｎｐ（ｔ＋１）＝Ｎｐ（ｔ）２ｘｉＣ（ｔ＋１）＝ ∑ｘｉ（ｔ）·ｆ（ｘｉ（ｔ））＋１０·ｘｓｌｏｗ·ｆ（ｘｓｌｏｗｓ） ∑ｆ（ｘｉ（ｔ））＋１０·ｘｓｌｏｗ·ｆ（ｘｓｌｏｗ）ｘｉ（ｔ＋１）＝ｘｉ（ｔ）＋ｒａｎｄ·（ｘｉＣ（ｔ＋１）－ｘｉ（ｔ）） ì î í ï ï ïï ï ï ïï ｉｆｔ ≥ Ｔｌａｂｅｌ（１１）ｗｈｅｒｅｘｓｌｏｗｉｓｔｈｅｐｏｓｉｔｉｏｎｏｆｔｈｅＵＡＶｗｉｔｈｔｈｅｓｌｏｗｅｓｔｄｉｖｉｎｇｓｐｅｅｄ，ｔｈｉｓｐａｒｔｒｅｐｒｅｓｅｎｔｓｔｈｅｓｌｏｗｉｎｇｄｉｖｉｎｇｓｔｒａｔｅｇｙ．ＦｉｔｎｅｓｓｆｕｎｃｔｉｏｎＥｑ．（１０）ｒｅｍａｉｎｓ．ＥａｃｈＵＡＶｈａｓｉｔｓｏｗｎｆｉｔｎｅｓｓｖａｌｕｅ． ·５７４· 智能系统学报第１２卷

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 ·575. 2.3 Convergence analysis of SD-PIO based on velocities,z(i)is theith element of z(t),i1,2,..., Banach fixed-point theorem 2n} In this section,convergence analysis of SD-PIO is z(i)∈u(z(t)=[xan(d)）…xa(t)h(t)…u(t)]' performed and a sufficient condition is proposed for the z()∈z),3K∈Na.st.u(2(t)=K&(). first part of SD-PIO SD-PIO-I).The main method d(T(a),T(a2)=max|T(a)-T(2)|= 1安12山 adopted to prove the stability of iteration is the Banach max Fz+Bu(z)-Fz2 Bu(z2)= 1名i名2m fixed-point theorem,as stated below: ‖Fz1+BK11-Fz2-BK22‖。(13) A map T from a normed space (V,d())into itself There is is a contraction if there exists入∈[0，l)such that for L=F(a-2)+B.[0…xa-x%0…6-%0…0] alx,y∈V,d(T(x),T(y))≤Ad(x,y) (14) In the first part of SD-PIO,each pigeon updates s.t.L=ll Fz +BK2-Fz2 BK222ll,thus the its speed and location according to Eq.(10),which largest element of Fz+BK21-Fz2 BK222 is can be transformed as selected and denoted as supposing it is thej thelement.F.and B..are the whole j th row of F and B, exp(-Rt) respectively.Ifjn,is the j th element of K121 -K2z2 and via-v2 is the (j n)th one, (12) otherwisex-xz is the (j-n)th element and v-v when it considers all the pigeons,(t 1)= is the j th one. Fz(t)+Bu(z(t)), |L=|F.(a-2)+B.[0…k4z0…k40…0]= where z(t)=[x(t)…xn(t),(t)…vn(t)]T, |F(a1-z2）+B.[0…k1…k2…0]4z≤ [1-9 exp(-Rt) ‖F(a1-2)+BK(a1-a2)‖。≤ lF+BK‖m‖a1-22‖。=lF+BKld(z1,a2） 1-ro exp(-Rt) (15) exp(-Rt) Whee4=maxl(i)-z(i)l,k,=xa-xsl/△， 1名i62 |k2=。-2sl/△，k,k2∈[-1,0)U(0,1], -rO exp(-Rt)」「0 01 1-9 k K= ,,k2 lies in the j th column, 1- k2 B =r 1- Lo 0 The equation holds if and only ifx=v- 1-9 026=Az. u(z(t)=[xeai(t)…xuan(t)un(t）…tahn(t)] The Banach fixed-point theorem requires llF+ If iteration z(t+1)=Fz(t)+Bu(z(t))has a fixed- BKI∈[0,1) pointed',as a result,z'=Fz·+Bu(z）,then max1-or exp(-Rt)+rk o +rk2(1-), SD-PIO-I will converge to 2'and each pigeon will gr exp(-Rt)+rk rk2(1-p)<1 reach its final stable location as time tends to infinity. Denoting Z as a real number space R"and distance -仁120e1-o d(y)=yyeZ,Zhas been proven (16) as a complete metric space.There is a map T:Z-Z, Equation (16)is a sufficient condition for the T(z)=Fz+Bu(z),whereF∈R2xn,B∈R2nxn, convergence of SD-PIO-I,once it is satisfied,we have u(z)=K2x2n2,then Vz1,2∈Z,T(z1),T(a2）∈Z, d(T(x),T(y)≤IF+BKld(x,y),IF+BKIe∈ it can be proven that if d(T(x),T(y))sAd(x,y), [0,1),there exists a fixed-point z'satisfying z'= A [0,1),there exists a fixed-point which satisfies Fz·+Bu(z) the condition 2'=T(2)=Fz'Bu(z)and the x=v+x states of SD-PIO-I converge to z'.The following x=x+exp(-Ri)v'+r(imx)+(1-)vi] proves this conclusion. (17) Because the best local location and the slowest m°=0 Equation(17）→ .the states of SD-PIO-I speed are among all the pigeons'locations and

２．３ＣｏｎｖｅｒｇｅｎｃｅａｎａｌｙｓｉｓｏｆＳＤ⁃ＰＩＯｂａｓｅｄｏｎＢａｎａｃｈｆｉｘｅｄ⁃ｐｏｉｎｔｔｈｅｏｒｅｍＩｎｔｈｉｓｓｅｃｔｉｏｎ，ｃｏｎｖｅｒｇｅｎｃｅａｎａｌｙｓｉｓｏｆＳＤ⁃ＰＩＯｉｓｐｅｒｆｏｒｍｅｄａｎｄａｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｉｓｐｒｏｐｏｓｅｄｆｏｒｔｈｅｆｉｒｓｔｐａｒｔｏｆＳＤ⁃ＰＩＯ（ＳＤ⁃ＰＩＯ⁃Ｉ）．ＴｈｅｍａｉｎｍｅｔｈｏｄａｄｏｐｔｅｄｔｏｐｒｏｖｅｔｈｅｓｔａｂｉｌｉｔｙｏｆｉｔｅｒａｔｉｏｎｉｓｔｈｅＢａｎａｃｈｆｉｘｅｄ⁃ｐｏｉｎｔｔｈｅｏｒｅｍ，ａｓｓｔａｔｅｄｂｅｌｏｗ：ＡｍａｐＴｆｒｏｍａｎｏｒｍｅｄｓｐａｃｅ（Ｖ，ｄ（））ｉｎｔｏｉｔｓｅｌｆｉｓａｃｏｎｔｒａｃｔｉｏｎｉｆｔｈｅｒｅｅｘｉｓｔｓ λ ∈ ［０，１）ｓｕｃｈｔｈａｔｆｏｒａｌｌｘ，ｙ ∈ Ｖ，ｄ（Ｔ（ｘ），Ｔ（ｙ）） ≤ λｄ（ｘ，ｙ）．ＩｎｔｈｅｆｉｒｓｔｐａｒｔｏｆＳＤ⁃ＰＩＯ，ｅａｃｈｐｉｇｅｏｎｕｐｄａｔｅｓｉｔｓｓｐｅｅｄａｎｄｌｏｃａｔｉｏｎａｃｃｏｒｄｉｎｇｔｏＥｑ．（１０），ｗｈｉｃｈｃａｎｂｅｔｒａｎｓｆｏｒｍｅｄａｓｘｉ（ｔ＋１）ｖｉ（ｔ＋１） é ë ê ê ù û ú ú ＝１－ｒφ ｅｘｐ（－Ｒｔ）－ｒφ ｅｘｐ（－Ｒｔ） é ë ê ê ù û ú ú ｘｉ（ｔ）ｖｉ（ｔ） é ë ê ê ù û ú ú ＋ｒ φ １－ φ φ １－ φ é ë ê ê ù û ú ú ｘｌｂｅｓｔｖｓｌｏｗ é ë ê ê ù û ú ú （１２）ｗｈｅｎｉｔｃｏｎｓｉｄｅｒｓａｌｌｔｈｅｐｉｇｅｏｎｓ，ｚ（ｔ＋１）＝Ｆｚ（ｔ）＋Ｂｕ（ｚ（ｔ）），ｗｈｅｒｅｚ（ｔ）＝ｘ１（ｔ） … ｘｎ（ｔ）ｖ１（ｔ） … ｖ [ ｎ（ｔ） ] Ｔ，Ｆ＝１－ｒφ ｅｘｐ（－Ｒｔ） ⋱ ⋱ １－ｒφ ｅｘｐ（－Ｒｔ）－ｒφ ｅｘｐ（－Ｒｔ） ⋱ ⋱ －ｒφ ｅｘｐ（－Ｒｔ） é ë ê ê ê ê ê ê ê ù û ú ú ú ú ú ú ú Ｂ＝ｒ φ １－ φ ⋱ ⋱ φ １－ φ φ １－ φ ⋱ ⋱ φ １－ φ é ë ê ê ê ê ê ê ê ù û ú ú ú ú ú ú ú ｕ（ｚ（ｔ））＝ｘｌｂｅｓｔ１（ｔ） … ｘｌｂｅｓｔｎ（ｔ）ｖｓｌｏｗ１（ｔ） … ｖ [ ｓｌｏｗｎ（ｔ）] Ｔ．Ｉｆｉｔｅｒａｔｉｏｎｚ（ｔ＋１）＝Ｆｚ（ｔ）＋Ｂｕ（ｚ（ｔ））ｈａｓａｆｉｘｅｄ⁃ ｐｏｉｎｔｅｄｚ ∗ ，ａｓａｒｅｓｕｌｔ，ｚ ∗ ＝Ｆｚ ∗ ＋Ｂｕ（ｚ ∗ ），ｔｈｅｎＳＤ⁃ＰＩＯ⁃Ｉｗｉｌｌｃｏｎｖｅｒｇｅｔｏｚ ∗ ａｎｄｅａｃｈｐｉｇｅｏｎｗｉｌｌｒｅａｃｈｉｔｓｆｉｎａｌｓｔａｂｌｅｌｏｃａｔｉｏｎａｓｔｉｍｅｔｅｎｄｓｔｏｉｎｆｉｎｉｔｙ．ＤｅｎｏｔｉｎｇＺａｓａｒｅａｌｎｕｍｂｅｒｓｐａｃｅＲｎａｎｄｄｉｓｔａｎｃｅｄ（ｘ，ｙ）＝ｍａｘ１≤ｉ≤２ｎｘｉ－ｙｉ，ｘ，ｙ ∈ Ｚ，Ｚｈａｓｂｅｅｎｐｒｏｖｅｎａｓａｃｏｍｐｌｅｔｅｍｅｔｒｉｃｓｐａｃｅ．ＴｈｅｒｅｉｓａｍａｐＴ：Ｚ → Ｚ，Ｔ（ｚ）＝Ｆｚ＋Ｂｕ（ｚ），ｗｈｅｒｅＦ ∈ Ｒ２ｎ×２ｎ，Ｂ ∈ Ｒ２ｎ×２ｎ，ｕ（ｚ）＝Ｋ２ｎ×２ｎｚ，ｔｈｅｎ ∀ｚ１，ｚ２ ∈Ｚ，Ｔ（ｚ１），Ｔ（ｚ２） ∈Ｚ，ｉｔｃａｎｂｅｐｒｏｖｅｎｔｈａｔｉｆｄ（Ｔ（ｘ），Ｔ（ｙ）） ≤ λｄ（ｘ，ｙ）， λ ∈［０，１），ｔｈｅｒｅｅｘｉｓｔｓａｆｉｘｅｄ⁃ｐｏｉｎｔｗｈｉｃｈｓａｔｉｓｆｉｅｓｔｈｅｃｏｎｄｉｔｉｏｎｚ ∗ ＝Ｔ（ｚ ∗ ）＝Ｆｚ ∗ ＋Ｂｕ（ｚ ∗ ）ａｎｄｔｈｅｓｔａｔｅｓｏｆＳＤ⁃ＰＩＯ⁃Ｉｃｏｎｖｅｒｇｅｔｏｚ ∗ ．Ｔｈｅｆｏｌｌｏｗｉｎｇｐｒｏｖｅｓｔｈｉｓｃｏｎｃｌｕｓｉｏｎ．Ｂｅｃａｕｓｅｔｈｅｂｅｓｔｌｏｃａｌｌｏｃａｔｉｏｎａｎｄｔｈｅｓｌｏｗｅｓｔｓｐｅｅｄａｒｅａｍｏｎｇａｌｌｔｈｅｐｉｇｅｏｎｓ ’ ｌｏｃａｔｉｏｎｓａｎｄｖｅｌｏｃｉｔｉｅｓ，ｚ（ｉ）ｉｓｔｈｅｉｔｈｅｌｅｍｅｎｔｏｆｚ（ｔ），ｉ ∈ ｛１，２，…，２ｎ｝ ∀ｚ（ｉ）∈ｕ（ｚ（ｔ））＝ｘｌｂｅｓｔ１（ｔ） … ｘｌｂｅｓｔｎ（ｔ）ｖｓｌｏｗ１（ｔ） … ｖ [ ｓｌｏｗｎ（ｔ）] Ｔｚ（ｉ） ∈ｚ（ｔ），∃Ｋ ∈Ｎ２ｎ×２ｎ，ｓ．ｔ．ｕ（ｚ（ｔ））＝Ｋｚ（ｔ）．ｄ（Ｔ（ｚ１），Ｔ（ｚ２））＝ｍａｘ１≤ｉ≤２ｎＴ（ｚ１）－Ｔ（ｚ２）＝ｍａｘ１≤ｉ≤２ｎＦｚ１＋Ｂｕ（ｚ１）－Ｆｚ２－Ｂｕ（ｚ２）＝ ‖Ｆｚ１＋ＢＫ１ｚ１－Ｆｚ２－ＢＫ２ｚ２‖¥ （１３）ＴｈｅｒｅｉｓＬｊ＝Ｆ·ｊ（ｚ１－ｚ２）＋Ｂ·ｊ［０ … ｘ１ａ－ｘ２ｂ０ … ｖ１ａ－ｖ２ｂ０ … ０］Ｔ（１４）ｓ．ｔ．Ｌｊ＝‖ Ｆｚ１＋ＢＫ１ｚ１－Ｆｚ２－ＢＫ２ｚ２‖¥，ｔｈｕｓｔｈｅｌａｒｇｅｓｔｅｌｅｍｅｎｔｏｆＦｚ１＋ＢＫ１ｚ１－Ｆｚ２－ＢＫ２ｚ２ｉｓｓｅｌｅｃｔｅｄａｎｄｄｅｎｏｔｅｄａｓＬｊｓｕｐｐｏｓｉｎｇｉｔｉｓｔｈｅｊｔｈｅｌｅｍｅｎｔ．Ｆ·ｊａｎｄＢ·ｊａｒｅｔｈｅｗｈｏｌｅｊｔｈｒｏｗｏｆＦａｎｄＢ，ｒｅｓｐｅｃｔｉｖｅｌｙ．Ｉｆｊ ≤ ｎ，ｘ１ａ－ｘ２ｂｉｓｔｈｅｊｔｈｅｌｅｍｅｎｔｏｆＫ１ｚ１－Ｋ２ｚ２ａｎｄｖ１ａ－ｖ２ｂｉｓｔｈｅ（ｊ＋ｎ）ｔｈｏｎｅ，ｏｔｈｅｒｗｉｓｅｘ１ａ－ｘ２ｂｉｓｔｈｅ（ｊ－ｎ）ｔｈｅｌｅｍｅｎｔａｎｄｖ１ａ－ｖ２ｂｉｓｔｈｅｊｔｈｏｎｅ．Ｌｊ＝Ｆ·ｊ（ｚ１－ｚ２）＋Ｂ·ｊ［０ … ｋ１Δｚ０ … ｋ２Δｚ０ … ０］Ｔ＝Ｆ·ｊ（ｚ１－ｚ２）＋Ｂ·ｊ［０…ｋ１…ｋ２…０］ＴΔｚ ≤ ‖Ｆ（ｚ１－ｚ２）＋ＢＫ（ｚ１－ｚ２）‖¥ ≤ ‖Ｆ＋ＢＫ‖¥‖ｚ１－ｚ２‖¥ ＝ ‖Ｆ＋ＢＫ‖¥ｄ（ｚ１，ｚ２）（１５）Ｗｈｅｒｅ Δｚ＝ｍａｘ１≤ｉ≤２ｎｚ１（ｉ）－ｚ２（ｉ），ｋ１＝ｘ１ａ－ｘ２ｂ／ Δｚ，ｋ２＝ｖ１ａ－ｖ２ｂ／ Δｚ，ｋ１，ｋ２ ∈ ［－１，０） ∪ （０，１］，Ｋ＝００ ⋱ ｋ１ ︙ ｋ２００ é ë ê ê ê ê ê ê ù û ú ú ú ú ú ú ，ｋ１，ｋ２ｌｉｅｓｉｎｔｈｅｊｔｈｃｏｌｕｍｎ，Ｔｈｅｅｑｕａｔｉｏｎｈｏｌｄｓｉｆａｎｄｏｎｌｙｉｆｘ１ａ－ｘ２ｂ＝ｖ１ａ－ｖ２ｂ＝ Δｚ．ＴｈｅＢａｎａｃｈｆｉｘｅｄ⁃ｐｏｉｎｔｔｈｅｏｒｅｍｒｅｑｕｉｒｅｓ ‖Ｆ＋ＢＫ‖¥ ∈ ［０，１） ⇔ｍａｘ｛１－ φｒ＋ｅｘｐ（－Ｒｔ）＋ｒｋ１φ ＋ｒｋ２（１－ φ），－ φｒ＋ｅｘｐ（－Ｒｔ）＋ｒｋ１φ ＋ｒｋ２（１－ φ）｝＜１ ⇔ －１＜－ φｒ＋ｅｘｐ（－Ｒｔ）＋ｒｋ１φ ＋ｒｋ２（１－ φ）＜０－１ ≤ ｋ１，ｋ２ ≤ １，ｋ１ｋ２ ≠ ０ { （１６）Ｅｑｕａｔｉｏｎ（１６）ｉｓａｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｆｏｒｔｈｅｃｏｎｖｅｒｇｅｎｃｅｏｆＳＤ⁃ＰＩＯ⁃Ｉ，ｏｎｃｅｉｔｉｓｓａｔｉｓｆｉｅｄ，ｗｅｈａｖｅｄ（Ｔ（ｘ），Ｔ（ｙ））≤‖Ｆ＋ＢＫ‖¥ｄ（ｘ，ｙ），‖Ｆ＋ＢＫ‖¥ ∈ ［０，１），ｔｈｅｒｅｅｘｉｓｔｓａｆｉｘｅｄ⁃ｐｏｉｎｔｚ ∗ ｓａｔｉｓｆｙｉｎｇｚ ∗ ＝Ｆｚ ∗ ＋Ｂｕ（ｚ ∗ ）ｘ ∗＝ｖ ∗＋ｘ ∗ ｘ ∗＝ｘ ∗＋ｅｘｐ（－Ｒｔ）ｖ ∗＋ｒ［φ（ｘ ∗ ｌｂｅｓｔ－ｘ ∗ ）＋（１－ φ）ｖ ∗ ｓｌｏｗ］ { （１７）Ｅｑｕａｔｉｏｎ（１７） ⇒ ｖ ∗ ＝０ｘ ∗ ＝ｘ ∗ ｌｂｅｓｔ { ，ｔｈｅｓｔａｔｅｓｏｆＳＤ⁃ＰＩＯ⁃Ｉ ·５７５· 第４期ＺＨＡＮＧＴｉａｎｊｉｅ，ｅｔａｌ．：Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ

,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) 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 formation

ｃａｎｏｎｌｙｃｏｎｖｅｒｇｅｔｏａｌｏｃａｌｏｐｔｉｍａｌｓｏｌｕｔｉｏｎｂｅｃａｕｓｅｅａｃｈｐｉｇｅｏｎｃａｎｏｎｌｙｒｅｃｅｉｖｅｉｔｓｎｅｉｇｈｂｏｒ’ｓｉｎｆｏｒｍａｔｉｏｎｉｎｓｔｅａｄｏｆｇｌｏｂａｌｉｎｆｏｒｍａｔｉｏｎ．ＦｏｒｔｈｅｓｅｃｏｎｄｐａｒｔｏｆＳＤ⁃ＰＩＯ，ａｌｌｔｈｅｐｉｇｅｏｎｓｃｏｎｖｅｒｇｅｔｏｔｈｅｆｌｏｃｋ’ｓｃｅｎｔｅｒ．Ｔｈｅｉｒｔｒａｊｅｃｔｏｒｉｅｓｃａｎｏｎｌｙｃｏｎｔｒａｃｔａｎｄａｐｐｒｏａｃｈｅａｃｈｏｔｈｅｒ．Ｉｎｆｏｒｍａｔｉｏｎｐｒｏｃｅｓｓｅｓｔｈｉｓｐａｒｔｓｅｒｖｅｓｏｎｌｙａｓａｔｒａｎｓｉｔｉｏｎ，ａｎｄａｓｔｈｅｔｒａｃｅｓｄｏｎｏｔｄｉｖｅｒｇｅｔｈｉｓｓｅｃｏｎｄｐａｒｔｓａｔｉｓｆｉｅｓｔｈｅｄｅｍａｎｄｓ．３Ｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｗｉｔｈａＳＤ⁃ ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍＴｈｅｆｉｒｓｔｔｈｒｅｅｓｅｃｔｉｏｎｓｉｎｔｒｏｄｕｃｅｄｃｏｎｓｅｎｓｕｓａｎｄＳＤ⁃ＰＩＯ，ｎｏｗｔｈｉｓｓｅｃｔｉｏｎｕｓｅｓｔｈｅｍｔｏｏｐｔｉｍｉｚｅｔｈｅｔｒａｊｅｃｔｏｒｙｏｆａｎＵＡＶ．ＷｈｅｎｃｏｎｓｅｎｓｕｓｉｓｏｎｌｙａｐｐｌｉｅｄｔｏｔｈｅｐｌａｎｎｅｄｒｏｕｔｅｓｏｆａｌｌｔｈｅＵＡＶｓ，ｔｈｅｒｅｉｓｕｎａｖｏｉｄａｂｌｅｏｖｅｒｓｈｏｏｔｂｅｃａｕｓｅｏｆｔｈｅｄｉｓｔｒｉｂｕｔｅｄｃｏｎｔｒｏｌｐｌａｎｉｔｓｅｌｆ．ＥｖｅｒｙＵＡＶｈａｓｉｔｓｏｗｎｌｉｍｉｔｅｄｃｏｍｍｕｎｉｃａｔｉｏｎｒｅｌａｔｉｏｎｓｈｉｐ，ｗｈｉｃｈｍｅａｎｓｉｔｃａｎｎｏｔａｌｗａｙｓｒｅｃｅｉｖｅｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙ．Ｆｏｒｅｘａｍｐｌｅ，ＵＡＶ１ｃａｎｎｅｖｅｒａｃｑｕｉｒｅｔｈｅｐｌａｎｎｅｄｔｒａｊｅｃｔｏｒｙｄｉｒｅｃｔｌｙ，ｉｔｒｅｌｉｅｓｏｎＵＡＶ３ｔｏｐａｓｓｏｎｔｈｅｉｎｆｏｒｍａｔｉｏｎ．ＩｆＵＡＶ３ ’ ｓｉｎｉｔｉａｌａｌｔｉｔｕｄｅｉｓｍｕｃｈｔｏｏｈｉｇｈ，ｉｔｍｕｓｔｄｉｖｅｔｏｍｅｅｔｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙ．Ｂｅｃａｕｓｅｏｆｉｎｅｒｔｉａａｎｄａｌａｃｋｏｆｆｏｒｅｃａｓｔｉｎｇ，ＵＡＶ３ｃａｎｏｎｌｙｐｒｏｄｕｃｅｐｏｓｉｔｉｖｅａｄｄｉｔｉｏｎｓｔｏｉｔｓｈｅｉｇｈｔｗｈｅｎｉｔｇｏｅｓｂｅｌｏｗｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙａｎｄｔｈｉｓｉｓｔｈｅｒｅａｓｏｎｆｏｒｏｖｅｒｓｈｏｏｔ．Ｈｏｗｅｖｅｒ，ａｓｆｏｒＵＡＶ１，ｏｎｌｙｗｈｅｎｉｔｓｌｅａｄｅｒＵＡＶ３ｆｌｉｅｓｂｅｌｏｗｔｈｅｐｒｏｐｏｓｅｄｈｅｉｇｈｔｄｏｅｓｉｔｂｅｇｉｎｔｏｒｅｃｏｇｎｉｚｅｔｈａｔｉｔｓｈｏｕｌｄｆｌｙａｂｉｔｈｉｇｈｅｒ，ｔｈｅｎｉｔｓｖｅｌｏｃｉｔｙｉｓｃｈａｎｇｅｄｂｙＥｑ．（７），ｆｏｌｌｏｗｅｄｂｙｉｔｓｐｏｓｉｔｉｏｎ．Ｔｈｉｓｐｒｏｃｅｓｓｉｓｍｕｃｈｓｌｏｗｅｒｔｈａｎａｃｈａｎｇｅｉｎｔｈｅｄｅｓｉｒｅｄｒｏｕｔｅ．Ａｓａｒｅｓｕｌｔ，ｔｈｅＵＡＶｓ’ ｔｒａｊｅｃｔｏｒｉｅｓｄｉｖｅｓｈａｒｐｌｙａｎｄａｐｐｅａｒｔｏｏｖｅｒｓｈｏｏｔ，ｔｈｅｎｔｈｅｙｃｏｎｖｅｒｇｅｔｏｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙ．ＳＤ⁃ＰＩＯｐｒｏｖｉｄｅｓａｇｅｎｔｌｅａｎｄｓｌｏｗｌｙｃｈａｎｇｉｎｇｔｒａｊｅｃｔｏｒｙｆｏｒｅａｃｈＵＡＶａｔｔｈｅｓｔａｒｔ，ｔｈｅｎ，ａｓｓｏｏｎａｓｔｈｅＵＡＶｓｆｌｙｄｏｗｎａｎｄｍｅｅｔｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙ，ｃｏｎｓｅｎｓｕｓｓｔａｒｔｓｔｏｏｐｅｒａｔｅ．Ｔｈｅｆｏｒｍｕｌａｏｆｔｈｅｃｏｍｂｉｎａｔｉｏｎｏｆｔｈｅｓｅｔｗｏａｌｇｏｒｉｔｈｍｓｃａｎｂｅｄｅｓｃｒｉｂｅｄａｓ：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）ｗｈｅｒｅＴｃｈｇｉｓｔｈｅｍｏｍｅｎｔｗｈｅｎｃｏｎｓｅｎｓｕｓｒｅｐｌａｃｅｓＳＤ⁃ ＰＩＯ， ± １ π × ａｒｃｔａｎ（Ｒ × （ｔ－Ｔｃｈｇ））＋０．５，ｔｈｅｉｎｖｅｒｓｅｔｒｉｇｏｎｏｍｅｔｒｉｃｆｕｎｃｔｉｏｎ，ｉｓｕｔｉｌｉｚｅｄｔｏｍｏｄｉｆｙｔｈｅｏｕｔｐｕｔｏｆｔｈｅＳＤ⁃ＰＩＯａｎｄｃｏｎｓｅｎｓｕｓ，ｉｔｗｏｒｋｓａｓａｓｗｉｔｃｈｋｅｙｔｏｓｍｏｏｔｈｌｙｃｏｎｎｅｃｔｔｈｅｔｗｏｐａｒｔｓｏｆｔｈｅｔｒａｊｅｃｔｏｒｙ．ｆ（ｘ（ｔ），ｖ（ｔ））ｉｓｔｈｅｏｕｔｐｕｔｖｅｌｏｃｉｔｙｏｆＳＤ⁃ ＰＩＯｆｒｏｍＥｑｓ．（７）ａｎｄ（９）．ＴｈｅｉｍｐｌｅｍｅｎｔａｔｉｏｎｐｒｏｃｅｄｕｒｅｏｆｔｈｅＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｉｓｇｉｖｅｎａｓｆｏｌｌｏｗｓ：１）ＩｎｐｕｔｔｈｅｎｕｍｂｅｒｏｆＵＡＶｓｉｎｔｈｅｆｏｒｍａｔｉｏｎｔａｓｋ，ｇｉｖｅｔｈｅｅｘｐｅｃｔａｔｉｏｎｆｏｒｍａｔｉｏｎａｎｄｃｏｎｓｉｄｅｒｔｈｅｃｏｍｍｕｎｉｃａｔｉｏｎｒｅｓｔｒｉｃｔｉｏｎ．２）ＤｅｓｉｇｎａｓｕｉｔａｂｌｅｃｏｍｍｕｎｉｃａｔｉｏｎｎｅｔｗｏｒｋｔｏｐｏｌｏｇｙａｎｄａｃｑｕｉｒｅｉｔｓａｄｊａｃｅｎｃｙｍａｔｒｉｘａｎｄｔｈｅＬａｐｌａｃｅｍａｔｒｉｘ．３）ＲａｎｄｏｍｌｙｉｎｉｔｉａｌｉｚｅｔｈｅｐｏｓｉｔｉｏｎｓａｎｄｖｅｌｏｃｉｔｉｅｓｏｆａｌｌｔｈｅＵＡＶｓａｌｏｎｇｗｉｔｈｔｈｅｐａｒａｍｅｔｅｒｓｉｎｖｏｌｖｅｄｉｎｔｈｅｅｑｕａｔｉｏｎｓｕｓｅｄｉｎＥｑ．（１８）．４）Ｃａｌｃｕｌａｔｅｔｈｅａｃｃｅｌｅｒａｔｉｏｎｉｎｐｕｔ，ｃｏｎｓｅｎｓｕｓ，ａｎｄｔｈｅｖｅｌｏｃｉｔｙｉｎｐｕｔｕｓｉｎｇＥｑ．（１８）５）ＵｐｄａｔｅｅａｃｈＵＡＶ’ｓｐｏｓｉｔｉｏｎａｎｄｖｅｌｏｃｉｔｙｕｓｉｎｇＥｑ．（１８）６）ＤｒａｗｔｈｅＵＡＶ’ ｓｔｒａｊｅｃｔｏｒｙａｃｃｏｒｄｉｎｇｔｏｉｔｓｐｏｓｉｔｉｏｎｉｎｅａｃｈｉｔｅｒａｔｉｏｎ．Ｔｈｅｐｒｏｃｅｓｓｉｓｓｕｍｍａｒｉｚｅｄｉｎｔｈｅｆｏｌｌｏｗｉｎｇｐｒｏｃｅｓｓｆｌｏｗｃｈａｒｔ：Ｆｉｇ．４ＴｈｅｐｒｏｃｅｓｓｏｆｔｈｅＳＤ⁃ＰＩＯｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒａｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎ ·５７６· 智能系统学报第１２卷

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 viewed

４ＳｉｍｕｌａｔｉｏｎｒｅｓｕｌｔｓａｎｄａｎａｌｙｓｉｓＴｏｖａｌｉｄａｔｅｔｈｅＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ，ｎｕｍｅｒｉｃａｌｓｉｍｕｌａｔｉｏｎｓｗｅｒｅｃｏｎｄｕｃｔｅｄｕｓｉｎｇＭＡＴＬＡＢ２０１２ａｏｎａＰＣｗｉｔｈａｎｉ５⁃４８０ｍ，２．６７ＧＨｚＣＰＵａｎｄＷｉｎｄｏｗｓ１０ｏｐｅｒａｔｉｎｇｓｙｓｔｅｍ．ＳＤ⁃ＰＩＯｗａｓｃｏｍｐａｒｅｄｗｉｔｈｃｏｎｓｅｎｓｕｓｉｔｓｅｌｆａｎｄｃｏｎｓｅｎｓｕｓｗｉｔｈｔｗｏｏｔｈｅｒｏｐｔｉｍｉｚａｔｉｏｎｓ，ｐａｒｔｉｃｌｅｓｗａｒｍｏｐｔｉｍｉｚａｔｉｏｎ（ＰＳＯ）ａｎｄＰＩＯ．ＴｈｅｒｅｓｕｌｔｓｐｒｏｖｅｔｈｅｅｆｆｅｃｔｉｖｅｎｅｓｓｏｆＳＤ⁃ＰＩＯ．ＴｈｅｍａｉｎｐａｒａｍｅｔｅｒｓｉｎＥｑｓ．（７），（９），ａｎｄ（１８）ａｒｅｌｉｓｔｅｄｉｎＴａｂｌｅ２ｂｅｌｏｗ．ＴｈｅｔｉｍｅｓｔｅｐｆｏｒｔｈｅｓｉｍｕｌａｔｉｏｎｉｓＴ＝０．０５ｓ．Ｔａｂｌｅ２Ｓｉｍｕｌａｔｉｏｎｐａｒａｍｅｔｅｒｓ γ ＲＴｌａｂｅｌ／ｓＴｃｈｇ／ｓ５５０４６ＮｉｎｅＵＡＶｓｗｅｒｅｉｎｉｔｉａｌｉｚｅｄｗｉｔｈｒａｎｄｏｍｐｏｓｉｔｉｏｎｓｉｎａ３⁃ｄｉｍｅｎｓｉｏｎａｌｅｎｖｉｒｏｎｍｅｎｔａｓｓｈｏｗｎｉｎＴａｂｌｅ３．Ｔａｂｌｅ３Ｉｎｉｔｉａｌｐｏｓｉｔｉｏｎｓｏｆ９ＵＡＶｓｍＵＡＶｎｕｍｂｅｒｘｙｚＵＡＶ１２０．８０４３０．９７１３０．４７７ＵＡＶ２１２．６８５３７．６３３３１．２５２ＵＡＶ３２２．２６０３８．６２２２４．６６１ＵＡＶ４０．３１８２７．９２９９．７０３ＵＡＶ５２１．８５９１５．８４６６．０７６ＵＡＶ６１６．９４２１３．９９９６．６４４ＵＡＶ７１９．２９３２０．７２３１７．６６２ＵＡＶ８２４．８０６２３．０６１９．３１７ＵＡＶ９３７．２９８１２．６８８７．８６１Ｔｈｅｓｅｈｅｉｇｈｔｓ，ｔｈｅｎｕｍｂｅｒｓｏｎｔｈｅｚ⁃ａｘｉｓ，ａｒｅｒｅｌａｔｉｖｅｐｏｓｉｔｉｏｎｓｃｏｒｒｅｓｐｏｎｄｉｎｇｔｏｔｈｅｈｅｉｇｈｔｏｆｏｒｉｇｉｎｏｆｔｈｅｃｏｍｍｏｎｍｏｖｉｎｇｃｏｏｒｄｉｎａｔｅｓｙｓｔｅｍｉｎＦｉｇ．２．ＥａｃｈＵＡＶｈａｄｔｈｅｓａｍｅｉｎｉｔｉａｌｐｏｓｉｔｉｏｎｉｎｔｈｅｆｏｕｒｓｉｍｕｌａｔｉｏｎｓ，ｉ．ｅ．，ｃｏｎｓｅｎｓｕｓ，ＰＳＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ，ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ，ａｎｄＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ．ＡｓｋｎｏｗｎｆｒｏｍＳｅｃｔｉｏｎｓ２ａｎｄ３，ｔｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｃｏｎｓｅｎｓｕｓｃａｎｎｏｔｓａｔｉｓｆｙｔｈｅｒｅｑｕｉｒｅｍｅｎｔｓ，ｗｈｉｃｈｉｓｃｌｅａｒｌｙｄｉｓｐｌａｙｅｄｉｎＦｉｇ．５．Ｂｅｃａｕｓｅｏｆｃｏｍｍｕｎｉｃａｔｉｏｎｒｅｓｔｒｉｃｔｉｏｎｓａｎｄｌａｃｋｏｆｆｏｒｅｃａｓｔｉｎｇ，ａｌｌｔｈｅＵＡＶｓｔｅｎｄｔｏｄｉｖｅｓｈａｒｐｌｙａｎｄｃｌｉｍｂｑｕｉｃｋｌｙ．Ｔｈｉｓｃａｕｓｅｓｇｒｅａｔｏｖｅｒｌｏａｄａｔｔｈｅｔｕｒｎｉｎｇｐｏｉｎｔ，ｗｈｉｃｈｍａｙｄａｍａｇｅｔｈｅＵＡＶ’ ｓｓｔｒｕｃｔｕｒｅａｎｄｒｅｓｕｌｔｉｎｄｉｆｆｉｃｕｌｔｉｅｓｆｏｒｆｌｉｇｈｔｃｏｎｔｒｏｌ．Ｔｈｅｉｒｒｏｕｔｅｓｓｅｅｍｔｏｔｉｅａｋｎｏｔｗｈｅｎｖｉｅｗｅｄｆｒｏｍｔｈｉｓｐｅｒｓｐｅｃｔｉｖｅ，ｈｏｗｅｖｅｒ，ｉｔｉｓｉｍｐｏｒｔａｎｔｔｏｃｌａｒｉｆｙｔｈｅｆａｃｔｔｈａｔｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｄｏｎｏｔｃｒｏｓｓａｎｄｔｈｅＵＡＶｓｄｏｎｏｔｃｒａｓｈｉｎｔｈｅｓｋｙ．Ｆｉｇ．５Ｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙｃｏｎｓｅｎｓｕｓＴｏｄｅｍｏｎｓｔｒａｔｅｔｈｅａｄｖａｎｔａｇｅｏｆＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ，ＰＳＯ⁃ ａｎｄＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｓｗｅｒｅａｄｏｐｔｅｄｔｏｃｏｍｐａｒｅｔｈｅｉｒｐｅｒｆｏｒｍａｎｃｅｓｗｉｔｈｔｈａｔｏｆｔｈｅｍｏｄｉｆｉｅｄａｌｇｏｒｉｔｈｍ．ＩｎＦｉｇ．６，ｍｏｓｔｔｒａｊｅｃｔｏｒｉｅｓａｒｅａｐｐａｒｅｎｔｌｙｂｅｔｔｅｒｔｈａｎｔｈｏｓｅｉｎＦｉｇ．５，ａｓｔｈｅｔｒａｊｅｃｔｏｒｉｅｓａｒｅｇｅｎｔｌｅａｔｔｈｅｂｅｇｉｎｎｉｎｇａｎｄｔｕｒｎｗｉｔｈｏｕｔｄｉｖｉｎｇｓｈａｒｐｌｙ．ＵＡＶ３，ＵＡＶ５，ａｎｄＵＡＶ９ｓｅｒｖｅａｓａｓｅｔｏｆｃｏｎｖｉｎｃｉｎｇｓｉｍｕｌａｔｉｏｎｓｔｈａｔｓｕｐｐｏｒｔｔｈｅｃｏｎｃｌｕｓｉｏｎｓａｂｏｖｅ．ＩｎＦｉｇｓ．７ａｎｄ８，ＰＩＯａｎｄＳＤ⁃ＰＩＯｗｏｒｋａｎｄｆｕｒｔｈｅｒｉｍｐｒｏｖｅｔｈｅｐｅｒｆｏｒｍａｎｃｅ．ＩｎＦｉｇ．７，ｔｈｅｏｖｅｒｓｈｏｏｔａｌｍｏｓｔｄｉｓａｐｐｅａｒｓ，ｂｕｔｔｈｅｒｅｉｓｓｔｉｌｌｆｌｕｃｔｕａｔｉｏｎｉｎｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｆｒｏｍｔｈｉｓｐｅｒｓｐｅｃｔｉｖｅ，ｅｓｐｅｃｉａｌｌｙｆｏｒＵＡＶ３，ＵＡＶ４，ａｎｄＵＡＶ５．ＩｎＦｉｇ．８，ｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｂｅｃｏｍｅｍｕｃｈｓｍｏｏｔｈｅｒａｎｄｔｈｅｏｖｅｒｓｈｏｏｔｃａｎｂｅｉｇｎｏｒｅｄｃｏｍｐａｒｅｄｗｉｔｈＦｉｇｓ．５ｔｏ７．ＴｈｉｓｉｓｒｅｆｌｅｃｔｅｄｉｎｔｈｅｌｏｗｅｓｔｐａｒｔｓｏｆｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｏｆＵＡＶ３，ＵＡＶ４，ａｎｄＵＡＶ５．Ｆｉｇ．６Ｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＳＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＦｉｇ．７Ｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓ ·５７７· 第４期ＺＨＡＮＧＴｉａｎｊｉｅ，ｅｔａｌ．：Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ

.578. 智能系统学报第12卷 The performance of the x trajectories can be UAV: UAV evaluated using change of direction as the criterion. -UAV After starting,if a UAV flies along a smooth trajectory 号20 —UAVg towards the desired one without turning,it must have 0月 6005004003002001000⊙mUAV4 UAV chosen the best and straightest trajectory.An y/m undesirable trajectory leads the UAV into making frequent turns,which costs too much energy and Fig.8 Trajectories of 9 UAVs by SD-PIO-based consensus increases the probability of collision. The effect of the SD-PIO-based consensus can be The X trajectories in Figs.9 and 10 belong to the analyzed according to the x and z trajectories,but the y undesirable type.In Fig.9,the trajectories of UAV2, trajectory by consensus is smooth enough and does not UAV,and UAV,curve during flight and each of them need improvement.In Figs.9 and 10,oscillation is turns twice or even three times.UAV,flies in the more obvious than the counterparts in Figs.11 and 12. direction in which x becomes larger,however at nearly The undesirable trajectories in Fig.11 partly explain the t=1 s,x begins to decrease,then at nearly t=3.5 s,x shortcomings of PIO. increases again and meets the desired value at t=5 s. 40 30 This conflict process can be explained by its topology. 30 In Fig.3,UAV,is deeply influenced by UAV,and 10 UAV3.Unfortunately,all its neighbors start at positions further along the x-axis,which means that UAV2 has to -10 15 20 catch up with its neighbor by consensus without Fig.9 X trajectory of 9 UAVs by consensus reaching its desired positions.Only when its leader realizes they have deviated from the right position,do they begin to correct it.This is a result of a lack of 304 forecasting.However,if UAV2 can directly receive 票20 information from UAVo perhaps it can perform better. UAV UAV This is the limit of communication.These two UAV; 10 20 shortcomings illustrate the necessity for improvement. s In Fig.10,the situation is not any better.UAV3, Fig.10 X trajectories of 9 UAVs by PSO-based consensus UAV,UAVs,and UAVs have the same problem.For 40 UAV example,UAVs starts at x=24.806 m at t=0,then x= 30 UAV UAV 21.5 m at nearly t=2 s,then x=25.1 m at nearly t=6 UAV: UAV s and decreases to x =18.3 m at t=8 s;the real UAV UAV trajectory goes up and down frequently.This implies UAV UAV PSO cannot satisfy the demand of performance 1 20 t/s improvement. Fig.11 X trajectories of 9 UAVs by PlO-based consensus In Figs.11 and 12,fluctuation weakens and the 40 trajectories become much smoother.Most UAVs can fly UAV UAV 30 UAV directly towards their destinations at t=7 s.In Fig.11, UAV 层20 -UAV the x position of UAVs decreases quickly from the UAV beginning to t=5 s,this fast slide slip spoils the 10 UAV UAV performance as it causes a sliding movement of a flying 15 20 UAV and makes it harder to control. Fig.12 X trajectories of 9 UAVs by SD-PIO-based consensus In Fig.12,most trajectories are pleasant and

Ｆｉｇ．８Ｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＴｈｅｅｆｆｅｃｔｏｆｔｈｅＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓｃａｎｂｅａｎａｌｙｚｅｄａｃｃｏｒｄｉｎｇｔｏｔｈｅｘａｎｄｚｔｒａｊｅｃｔｏｒｉｅｓ，ｂｕｔｔｈｅｙｔｒａｊｅｃｔｏｒｙｂｙｃｏｎｓｅｎｓｕｓｉｓｓｍｏｏｔｈｅｎｏｕｇｈａｎｄｄｏｅｓｎｏｔｎｅｅｄｉｍｐｒｏｖｅｍｅｎｔ．ＩｎＦｉｇｓ．９ａｎｄ１０，ｏｓｃｉｌｌａｔｉｏｎｉｓｍｏｒｅｏｂｖｉｏｕｓｔｈａｎｔｈｅｃｏｕｎｔｅｒｐａｒｔｓｉｎＦｉｇｓ．１１ａｎｄ１２．ＴｈｅｕｎｄｅｓｉｒａｂｌｅｔｒａｊｅｃｔｏｒｉｅｓｉｎＦｉｇ．１１ｐａｒｔｌｙｅｘｐｌａｉｎｔｈｅｓｈｏｒｔｃｏｍｉｎｇｓｏｆＰＩＯ．Ｆｉｇ．９Ｘｔｒａｊｅｃｔｏｒｙｏｆ９ＵＡＶｓｂｙｃｏｎｓｅｎｓｕｓＦｉｇ．１０Ｘｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＳＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＦｉｇ．１１Ｘｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＦｉｇ．１２Ｘｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＴｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｘｔｒａｊｅｃｔｏｒｉｅｓｃａｎｂｅｅｖａｌｕａｔｅｄｕｓｉｎｇｃｈａｎｇｅｏｆｄｉｒｅｃｔｉｏｎａｓｔｈｅｃｒｉｔｅｒｉｏｎ．Ａｆｔｅｒｓｔａｒｔｉｎｇ，ｉｆａＵＡＶｆｌｉｅｓａｌｏｎｇａｓｍｏｏｔｈｔｒａｊｅｃｔｏｒｙｔｏｗａｒｄｓｔｈｅｄｅｓｉｒｅｄｏｎｅｗｉｔｈｏｕｔｔｕｒｎｉｎｇ，ｉｔｍｕｓｔｈａｖｅｃｈｏｓｅｎｔｈｅｂｅｓｔａｎｄｓｔｒａｉｇｈｔｅｓｔｔｒａｊｅｃｔｏｒｙ．ＡｎｕｎｄｅｓｉｒａｂｌｅｔｒａｊｅｃｔｏｒｙｌｅａｄｓｔｈｅＵＡＶｉｎｔｏｍａｋｉｎｇｆｒｅｑｕｅｎｔｔｕｒｎｓ，ｗｈｉｃｈｃｏｓｔｓｔｏｏｍｕｃｈｅｎｅｒｇｙａｎｄｉｎｃｒｅａｓｅｓｔｈｅｐｒｏｂａｂｉｌｉｔｙｏｆｃｏｌｌｉｓｉｏｎ．ＴｈｅＸｔｒａｊｅｃｔｏｒｉｅｓｉｎＦｉｇｓ．９ａｎｄ１０ｂｅｌｏｎｇｔｏｔｈｅｕｎｄｅｓｉｒａｂｌｅｔｙｐｅ．ＩｎＦｉｇ．９，ｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｏｆＵＡＶ２，ＵＡＶ４，ａｎｄＵＡＶ７ｃｕｒｖｅｄｕｒｉｎｇｆｌｉｇｈｔａｎｄｅａｃｈｏｆｔｈｅｍｔｕｒｎｓｔｗｉｃｅｏｒｅｖｅｎｔｈｒｅｅｔｉｍｅｓ．ＵＡＶ２ｆｌｉｅｓｉｎｔｈｅｄｉｒｅｃｔｉｏｎｉｎｗｈｉｃｈｘｂｅｃｏｍｅｓｌａｒｇｅｒ，ｈｏｗｅｖｅｒａｔｎｅａｒｌｙｔ＝１ｓ，ｘｂｅｇｉｎｓｔｏｄｅｃｒｅａｓｅ，ｔｈｅｎａｔｎｅａｒｌｙｔ＝３．５ｓ，ｘｉｎｃｒｅａｓｅｓａｇａｉｎａｎｄｍｅｅｔｓｔｈｅｄｅｓｉｒｅｄｖａｌｕｅａｔｔ＝５ｓ．Ｔｈｉｓｃｏｎｆｌｉｃｔｐｒｏｃｅｓｓｃａｎｂｅｅｘｐｌａｉｎｅｄｂｙｉｔｓｔｏｐｏｌｏｇｙ．ＩｎＦｉｇ．３，ＵＡＶ２ｉｓｄｅｅｐｌｙｉｎｆｌｕｅｎｃｅｄｂｙＵＡＶ１ａｎｄＵＡＶ３．Ｕｎｆｏｒｔｕｎａｔｅｌｙ，ａｌｌｉｔｓｎｅｉｇｈｂｏｒｓｓｔａｒｔａｔｐｏｓｉｔｉｏｎｓｆｕｒｔｈｅｒａｌｏｎｇｔｈｅｘ⁃ａｘｉｓ，ｗｈｉｃｈｍｅａｎｓｔｈａｔＵＡＶ２ｈａｓｔｏｃａｔｃｈｕｐｗｉｔｈｉｔｓｎｅｉｇｈｂｏｒｂｙｃｏｎｓｅｎｓｕｓｗｉｔｈｏｕｔｒｅａｃｈｉｎｇｉｔｓｄｅｓｉｒｅｄｐｏｓｉｔｉｏｎｓ．Ｏｎｌｙｗｈｅｎｉｔｓｌｅａｄｅｒｒｅａｌｉｚｅｓｔｈｅｙｈａｖｅｄｅｖｉａｔｅｄｆｒｏｍｔｈｅｒｉｇｈｔｐｏｓｉｔｉｏｎ，ｄｏｔｈｅｙｂｅｇｉｎｔｏｃｏｒｒｅｃｔｉｔ．Ｔｈｉｓｉｓａｒｅｓｕｌｔｏｆａｌａｃｋｏｆｆｏｒｅｃａｓｔｉｎｇ．Ｈｏｗｅｖｅｒ，ｉｆＵＡＶ２ｃａｎｄｉｒｅｃｔｌｙｒｅｃｅｉｖｅｉｎｆｏｒｍａｔｉｏｎｆｒｏｍＵＡＶ１０，ｐｅｒｈａｐｓｉｔｃａｎｐｅｒｆｏｒｍｂｅｔｔｅｒ．Ｔｈｉｓｉｓｔｈｅｌｉｍｉｔｏｆｃｏｍｍｕｎｉｃａｔｉｏｎ．Ｔｈｅｓｅｔｗｏｓｈｏｒｔｃｏｍｉｎｇｓｉｌｌｕｓｔｒａｔｅｔｈｅｎｅｃｅｓｓｉｔｙｆｏｒｉｍｐｒｏｖｅｍｅｎｔ．ＩｎＦｉｇ．１０，ｔｈｅｓｉｔｕａｔｉｏｎｉｓｎｏｔａｎｙｂｅｔｔｅｒ．ＵＡＶ３，ＵＡＶ４，ＵＡＶ５，ａｎｄＵＡＶ８ｈａｖｅｔｈｅｓａｍｅｐｒｏｂｌｅｍ．Ｆｏｒｅｘａｍｐｌｅ，ＵＡＶ８ｓｔａｒｔｓａｔｘ＝２４．８０６ｍａｔｔ＝０，ｔｈｅｎｘ＝２１．５ｍａｔｎｅａｒｌｙｔ＝２ｓ，ｔｈｅｎｘ＝２５．１ｍａｔｎｅａｒｌｙｔ＝６ｓａｎｄｄｅｃｒｅａｓｅｓｔｏｘ＝１８．３ｍａｔｔ＝８ｓ；ｔｈｅｒｅａｌｔｒａｊｅｃｔｏｒｙｇｏｅｓｕｐａｎｄｄｏｗｎｆｒｅｑｕｅｎｔｌｙ．ＴｈｉｓｉｍｐｌｉｅｓＰＳＯｃａｎｎｏｔｓａｔｉｓｆｙｔｈｅｄｅｍａｎｄｏｆｐｅｒｆｏｒｍａｎｃｅｉｍｐｒｏｖｅｍｅｎｔ．ＩｎＦｉｇｓ．１１ａｎｄ１２，ｆｌｕｃｔｕａｔｉｏｎｗｅａｋｅｎｓａｎｄｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｂｅｃｏｍｅｍｕｃｈｓｍｏｏｔｈｅｒ．ＭｏｓｔＵＡＶｓｃａｎｆｌｙｄｉｒｅｃｔｌｙｔｏｗａｒｄｓｔｈｅｉｒｄｅｓｔｉｎａｔｉｏｎｓａｔｔ＝７ｓ．ＩｎＦｉｇ．１１，ｔｈｅｘｐｏｓｉｔｉｏｎｏｆＵＡＶ５ｄｅｃｒｅａｓｅｓｑｕｉｃｋｌｙｆｒｏｍｔｈｅｂｅｇｉｎｎｉｎｇｔｏｔ＝５ｓ，ｔｈｉｓｆａｓｔｓｌｉｄｅｓｌｉｐｓｐｏｉｌｓｔｈｅｐｅｒｆｏｒｍａｎｃｅａｓｉｔｃａｕｓｅｓａｓｌｉｄｉｎｇｍｏｖｅｍｅｎｔｏｆａｆｌｙｉｎｇＵＡＶａｎｄｍａｋｅｓｉｔｈａｒｄｅｒｔｏｃｏｎｔｒｏｌ．ＩｎＦｉｇ．１２，ｍｏｓｔｔｒａｊｅｃｔｏｒｉｅｓａｒｅｐｌｅａｓａｎｔａｎｄ ·５７８· 智能系统学报第１２卷

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 value

ｓｕｉｔａｂｌｅａｓｃｏｍｍａｎｄｓｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓ．Ｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｚｔｒａｊｅｃｔｏｒｉｅｓｃａｎｂｅｅｖａｌｕａｔｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｃｒｉｔｅｒｉｏｎ：ｄｉｖｉｎｇｓｌｏｗｌｙａｎｄｎｅｖｅｒｆｌｙｉｎｇｕｐａｎｄｄｏｗｎ．ＩｎＦｉｇ．１３，ａｌｌｔｈｅｔｒａｊｅｃｔｏｒｉｅｓｑｕｉｃｋｌｙｇｏｄｏｗｎｂｅｆｏｒｅｔ＝４ｓ，ｔｈｉｓｍｅａｎｓｔｈａｔｔｈｅＵＡＶｓｄｉｖｅｓｈａｒｐｌｙｉｎｔｈｅｂｅｇｉｎｎｉｎｇ，ａｓｓｈｏｗｎｉｎＦｉｇ．５．Ｔｈｅａｉｍｏｆｔｈｉｓｗｏｒｋｉｓｔｏｏｖｅｒｃｏｍｅｔｈｉｓｓｈｏｒｔｃｏｍｉｎｇ．ＴｈｅｔｒａｊｅｃｔｏｒｉｅｓｉｎＦｉｇ．１４ｃｌａｒｉｆｙｔｈｅａｄｖａｎｔａｇｅｏｆＰＳＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓｂｅｃａｕｓｅｔｈｅｍｏｄｉｆｉｅｄａｌｇｏｒｉｔｈｍｃａｎｅｆｆｅｃｔｉｖｅｌｙｒｅｄｕｃｅｏｓｃｉｌｌａｔｉｏｎ．ＵＡＶ２ａｎｄＵＡＶ８ｃｌａｒｉｆｙＵＡＶ８ｔｈｉｓｖｉｅｗａｓｔｈｅｙｂｏｔｈｓｌｏｗｌｙｄｅｃｒｅａｓｅｔｈｅｉｒｈｅｉｇｈｔｆｏｌｌｏｗｉｎｇｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙｉｎｓｔｅａｄｏｆｄｉｖｉｎｇｄｏｗｎｓｗｉｆｔｌｙ．Ｆｉｇ．１３Ｚｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙｃｏｎｓｅｎｓｕｓＦｉｇ．１４Ｚｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＳＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＨｏｗｅｖｅｒ，ｒｅｄｕｐｌｉｃａｔｉｖｅｃｌｉｍｂａｎｄｄｉｖｅｓｔｉｌｌｅｘｉｓｔ，ｗｈｉｃｈｍａｙｉｎｄｕｃｅｏｓｃｉｌｌａｔｉｏｎ．ＴｈｅｄａｎｇｅｒｏｕｓｔｒａｊｅｃｔｏｒｉｅｓｏｆＵＡＶ３ａｎｄＵＡＶ５ａｔＵＡＶ５ｔ∈［０，１０］ｓｉｎＦｉｇ．１４ｒｅｖｅａｌｔｈａｔｔｈｅｙａｒｅｎｏｔｔｈｅｄｅｓｉｒｅｄｏｎｅｓｔｈａｔｄｅｓｃｅｎｄｓｌｏｗｌｙａｔｔｈｅｂｅｇｉｎｎｉｎｇ，ｔｈｅｎｃｏｎｖｅｒｇｅｔｏｔｈｅｄｅｓｉｒｅｄｔｒａｊｅｃｔｏｒｙ．Ｔｈｉｓｉｓｗｈｙｔｈｅｒｅｉｓｇｒｅａｔｄｅｍａｎｄｆｏｒａｎｏｔｈｅｒｏｐｔｉｍｉｚｅｒ．ＩｎＦｉｇ．１５，ＰＩＯｉｓｕｓｅｄｔｏａｐｐｌｙａｇｌｉｄｉｎｇｐｒｏｃｅｄｕｒｅ．Ｂｅｆｏｒｅ４ｓ，ｔｈｅｍａｐａｎｄｃｏｍｐａｓｓｏｐｅｒａｔｏｒｔａｋｅｓｅｆｆｅｃｔａｎｄａｌｌｔｈｅＵＡＶｓｆｌｙｔｏｗａｒｄｓｔｈｅｂｅｓｔｌｏｃａｌｐｏｓｉｔｉｏｎ．Ｉｎｔｈｅｔｉｍｅｒａｎｇｅ，４ｔｏ６ｓ，ｔｈｅｌａｎｄｍａｒｋｏｐｅｒａｔｏｒｄｒｉｖｅｓｔｈｅＵＡＶｓｎｅａｒｂｙｔｏｔｈｅｓａｍｅｈｅｉｇｈｔ．Ｗｈｅｎａｌｌｏｆｔｈｅｍｆｌｙｔｏｔｈｅｓｔａｎｄａｒｄｈｅｉｇｈｔ，ｔｈｅＰＩＯｐａｒｔｓｔｏｐｓｗｏｒｋｉｎｇａｎｄｃｏｎｓｅｎｓｕｓｃｏｎｔｉｎｕｅｓｃｏｎｃｅｎｔｒａｔｉｎｇｔｈｅＵＡＶｓａｎｄｇｕｉｄｉｎｇｔｈｅｍｔｏｆｌｙａｌｏｎｇｔｈｅｄｅｓｉｇｎｅｄｔｒａｊｅｃｔｏｒｙ．Ｆｉｇ．１５Ｚｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＦｉｇ．１６Ｚｔｒａｊｅｃｔｏｒｉｅｓｏｆ９ＵＡＶｓｂｙＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓＴｈｅＳＤ⁃ＰＩＯ⁃ｂａｓｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍａｌｓｏｆｏｌｌｏｗｓｔｈｅｓｔｅｐｓａｂｏｖｅ，ｂｕｔｔｈｅｄｉｆｆｅｒｅｎｃｅｏｃｃｕｒｓｉｎｔｈｅｆｉｒｓｔ８ｓａｓｔｈｅＵＡＶｓｄｅｓｃｅｎｄｍｏｒｅｓｌｏｗｌｙａｎｄｓｍｏｏｔｈｌｙ．ＵＡＶ１，ＵＡＶ２ａｎｄＵＡＶ３ｃａｎｓｕｐｐｏｒｔｔｈｉｓｏｐｉｎｉｏｎ，ａｓｔｈｅｙａｔｔｈｅｓａｍｅｒｅｓｐｅｃｔｉｖｅｈｅｉｇｈｔ，ｂｕｔａｔｔ＝５ｓ，ＵＡＶ２ａｎｄＵＡＶ３ｂｙＳＤ⁃ＰＩＯａｒｅａｂｏｖｅ２０ｍｗｈｉｌｅｔｈｅｉｒｃｏｕｎｔｅｒｐａｒｔｓａｒｅｃｌｅａｒｌｙｌｏｗｅｒ．Ａｔｔ＝７ｓ，ｔｈｅｈｅｉｇｈｔｏｆＵＡＶ１ｂｙＳＤ⁃ＰＩＯｉｓ１６．０４ｍａｎｄｔｈｅｏｕｔｃｏｍｅｂｙＰＩＯｉｓｏｎｌｙ１５．４７ｍ．ＦｒｏｍｔｈｅｆｉｔｎｅｓｓｆｕｎｃｔｉｏｎＥｑ．（８），ｔｈｅｃｌｏｓｅｒｔｈｅＵＡＶｉａｐｐｒｏａｃｈｅｓｉｔｓｄｅｓｉｒｅｄｌｏｃａｔｉｏｎ，ｔｈｅｓｍａｌｌｅｒｔｈｅｆｉｔｎｅｓｓｖａｌｕｅ．ＩｎＦｉｇ．１７，ａｓｉｔｅｒａｔｉｏｎＮｃｉｎｃｒｅａｓｅｓ，ａｖｅｒａｇｅｆｉｔｎｅｓｓｖａｌｕｅｓｄｅｃｒｅａｓｅａｎｄｆｉｎａｌｌｙａｐｐｒｏａｃｈｚｅｒｏ，ａｎｄｄｉｆｆｅｒｅｎｔｏｐｔｉｍｉｚｅｒｓｐｏｓｓｅｓｓｄｉｆｆｅｒｅｎｔｃｏｎｖｅｒｇｅｎｃｅｒａｔｅｓ．ＩｎＦｉｇ．１７（ａ），ＰＩＯｓｈｏｗｓｔｈｅｆａｓｔｅｓｔｓｐｅｅｄｉｎｔｈｅｂｅｇｉｎｎｉｎｇ，ＳＤ⁃ＰＩＯｉｓａｂｉｔｓｌｏｗｅｒａｎｄＰＳＯｓｈｏｗｓｔｈｅｗｅａｋｅｓｔｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｔｈｒｅｅ．Ｈｏｗｅｖｅｒ，ａｆｔｅｒＮｃ＝８０，ＰＩＯｂｅｃｏｍｅｓｓｌｏｗａｎｄｉｓｆｉｎａｌｌｙｏｖｅｒｔａｋｅｎｂｙＳＤ⁃ＰＩＯａｔＮｃ＝９３．ＴｈｅｐｏｏｒｅｆｆｉｃｉｅｎｃｙｏｆＰＩＯｉｓｄｕｅｔｏｔｈｅｉｎａｄｅｑｕａｔｅｌｏｃａｌｏｐｔｉｍｕｍ．（ａ）Ｘｆｉｔｎｅｓｓｖａｌｕｅ ·５７９· 第４期ＺＨＡＮＧＴｉａｎｊｉｅ，ｅｔａｌ．：Ａｍｏｄｉｆｉｅｄｃｏｎｓｅｎｓｕｓａｌｇｏｒｉｔｈｍｆｏｒｍｕｌｔｉ⁃ＵＡＶｆｏｒｍａｔｉｏｎｓｂａｓｅｄｏｎｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄｏｐｔｉｍｉｚａｔｉｏｎｗｉｔｈａｓｌｏｗｄｉｖｉｎｇｓｔｒａｔｅｇｙ

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