.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 conditionｆｏｒｍｅｒ ｅｎｓｕｒｅｓ ｔｈｅ ｄｉｓｔａｎｃｅ ｅｒｒｏｒ ｂｅｔｗｅｅｎ ｔｈｅ ｄｅｓｉｒｅｄ ａｎｄ ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｃｏｎｖｅｒｇｅｓ ｔｏ ０ ［１４］ ． Ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｏｆ ｅａｃｈ ＵＡＶ ｃａｎ ｂｅ ｅａｓｉｌｙ ａｃｑｕｉｒｅｄ ａｎｄ ｔｈｅ ＵＡＶ’ｓ ｔｒａｊｅｃｔｏｒｙ ｉｓ ｄｒａｗｎ． Ｆｉｇ．１ Ｆｒａｍｅ ｏｆ ｃｏｎｓｅｎｓｕｓ ｃｏｎｔｒｏｌｌｅｒ ｉｎ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ Ｆｏｒ ｔｈｅ ｆｏｒｍａｔｉｏｎ ｐｒｏｂｌｅｍ ｉｎ ａ ｍｕｌｔｉ⁃ＵＡＶ ｓｙｓｔｅｍ， ｔｈｅ ＵＡＶｓ’ ｓｔａｔｅｓ ｓｈｏｕｌｄ ｉｎｃｌｕｄｅ ｂｏｔｈ ｐｏｓｉｔｉｏｎ ａｎｄ ｖｅｌｏｃｉｔｙ， ａｎｄ ｔｈｅ ｃｏｎｔｒｏｌ ｉｎｐｕｔｓ ａｒｅ ｔｈｅ ａｃｃｅｌｅｒａｔｉｏｎｓ ［１４］ ． Ｔｈｉｓ ｍｏｄｅｌ ｉｓ ａｌｗａｙｓ ｄｅｓｃｒｉｂｅｄ ｂｙ ｓｅｃｏｎｄ⁃ｏｒｄｅｒ ｄｉｆｆｅｒｅｎｔｉａｔｉｏｎ ｅｑｕａｔｉｏｎｓ， ａｎｄ ａｓ ａ ｒｅｓｕｌｔ， ａ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｆｏｒ ｄｏｕｂｌｅ⁃ｉｎｔｅｇｒａｔｏｒ ｄｙｎａｍｉｃｓ ｉｓ ｒｅｑｕｉｒｅｄ． Ｔｈｅ ｄｙｎａｍｉｃ ｍａｔｈ ｍｏｄｅｌ ｏｆ ｔｈｅ ｍｕｌｔｉ⁃ＵＡＶ ｓｙｓｔｅｍ ｉｓ ｇｉｖｅｎ ｂｅｌｏｗ， ξ · ｉ ＝ ζｉ ζ · ｉ ＝ μｉ ξｉ ＝ ［ｘｉ ｙｉ ｚｉ］ Ｔ ì î í ï ïï ï ï （１） ｗｈｅｒｅ ξｉ，ζｉ，μｉ∈Ｒ ３ ｒｅｐｒｅｓｅｎｔ ｔｈｅ ｐｏｓｉｔｉｏｎ， ｔｈｅ ｖｅｌｏｃｉｔｙ ａｎｄ ｔｈｅ ａｃｃｅｌｅｒａｔｉｏｎ ｏｆ ＵＡＶｉ， ｒｅｓｐｅｃｔｉｖｅｌｙ ａｎｄ ｘｉ，ｙｉ，ｚｉ ａｒｅ ｔｈｅ ３⁃ｄｉｍｅｎｓｉｏｎａｌ ｐｏｓｉｔｉｏｎｓ， ｉ∈｛１，２，…，１０｝． Ｔｈｅ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｃａｎ ｂｅ ｄｅｓｃｒｉｂｅｄ ａｓ ｉｎ Ｒｅｆ ［１４］． μｉ ＝ １ ｋｉ ∑ ９ ｊ ＝ １ ａｉｊ［ζ · ｊ － Ｋｒｉ（ξｉ － ξｊ） － Ｋｖｉ（ζｉ － ζｊ）］ ＋ １ ｋｉ ａｉ１０ ［ζ · ｒ － Ｋｒｉ（ξｉ － ξ ｒ ） － Ｋｖｉ（ζｉ － ζ ｒ ）］ （２） Ｗｈｅｒｅ ａｉｊ ｉｓ ｅｌｅｍｅｎｔ （ ｉ， ｊ） ｉｎ ｔｈｅ ａｄｊａｃｅｎｃｙ ｍａｔｒｉｘ Ａ６×６ ∈Ｒ ６×６ ， ａｎｄ ｋｉ ＝ ∑ １０ ｊ ＝ １ ａｉｊ ， Ｋｒｉ，Ｋｖｉ ｉｓ ａ ３ × ３ｓｉｚｅ ｓｙｍｍｅｔｒｉｃ ｐｏｓｉｔｉｖｅ ｄｅｆｉｎｉｔｅ ｍａｔｒｉｘ， ｉ ∈ ｛１，２，…，９｝ ， ｉｎ ｔｈｉｓ ｗｏｒｋ ｔｈｉｓ ｉｓ ｒｅｐｌａｃｅｄ ｂｙ ａ Ｐａｓｃａｌ ｍａｔｒｉｘ ｆｏｒ ｓｉｍｐｌｉｃｉｔｙ． Ｔｈｅ ａｌｇｏｒｉｔｈｍ ｈａｓ ｂｅｅｎ ｐｒｏｖｅｎ ｅｆｆｅｃｔｉｖｅ ａｎｄ ｃａｎ ｍａｋｅ ａｌｌ ｔｈｅ ＵＡＶｓ’ ｂｏｔｈ ｐｏｓｉｔｉｏｎｓ ａｎｄ ｖｅｌｏｃｉｔｉｅｓ ｃｏｎｖｅｒｇｅ ｔｏ ｔｈｅ ｓｔａｔｅ ｏｆ ｔｈｅ ｒｏｏｔ． Ｔｈｉｓ ｉｓ ｂｅｃａｕｓｅ ｔｈｅ ｇｒａｐｈ ｓａｔｉｓｆｉｅｓ ｔｈｅ ｓｕｆｆｉｃｉｅｎｔ ａｎｄ ｎｅｃｅｓｓａｒｙ ｃｏｎｄｉｔｉｏｎ ｔｈａｔ ｔｈｅ ｄｉｒｅｃｔｅｄ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｔｏｐｏｌｏｇｙ ｇｒａｐｈ ｍｕｓｔ ｉｎｃｌｕｄｅ ａ ｓｐａｎｎｉｎｇ ｔｒｅｅ ［１４］ ． Ｔｏ ａｔｔａｉｎ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ｆｏｒ ｏｎｅ ｓｉｎｇｌｅ ＵＡＶ ｉｎ ｔｈｅ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ， ｔｈｅ ｅｒｒｏｒ ｂｅｔｗｅｅｎ ｔｈｅ ｄｅｓｉｒｅｄ ａｎｄ ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｓｈｏｕｌｄ ｂｅ ｔｒｅａｔｅｄ ａｓ ａｎ ｅｘｔｒａ ｖａｒｉａｂｌｅ， ｔｈｉｓ ｃａｎ ｂｅ ｄｅｓｃｒｉｂｅｄ ｂｙ ｓｉｎｇｌｅ⁃ ｉｎｔｅｇｒａｔｏｒ ｄｙｎａｍｉｃｓ ｆｏｒ ｓｉｍｐｌｉｃｉｔｙ． Ｔｈｅ ｍａｔｈ ｍｏｄｅｌ ｃａｎ ｂｅ ｄｅｓｃｒｉｂｅｄ ａｓ ｅ · ｒｉ ＝ ｕｉ （３） ａｎｄ ｔｈｅ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｉｓ ｉｎ Ｒｅｆ ［９］． ｕｉ ＝ － ∑ ９ ｊ ＝ １ ａｉｊ（ｅｒｉ － ｅｒｊ） （４） ｒ ～ ｉ ＝ ｒｐｉ ＋ ｅｒｉ （５） Ｗｈｅｒｅ ｒ ～ ｉ ｉｓ ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｏｆ ＵＡＶｉ， ＵＡＶｉ ｉｎ ｔｈｅ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ， ｅｒｉ ｉｓ ｔｈｅ ｄｉｓｔａｎｃｅ ｅｒｒｏｒ ｂｅｔｗｅｅｎ ｔｈｅ ｒｅａｌ ａｎｄ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ｏｆ ＵＡＶｉ， ｒｐｉ ｗｈｉｃｈ ｉｓ ｓｈｏｗｎ ｉｎ Ｆｉｇ． ２， ｉｓ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ｏｆ ＵＡＶｉ ｉｎ ｔｈｅ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ， ｉ∈｛１，２，…， ９｝． Ｅｑｕａｔｉｏｎｓ （１～５） ａｂｏｖｅ ｃａｎ ｂｅ ｔｒａｎｓｆｏｒｍｅｄ ｉｎｔｏ ａ ｃｏｌｌｅｃｔｉｖｅ ｍａｔｒｉｘ ｆｏｒｍ， （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 ）ｅ ｒ ＝ ｒ ～ ＋ ξ ＝ ｒｐ ＋ ｅｒ ＋ ξ ì î í ï ï ï ï ï ï （６） ｗｈｅｒｅ Ｌ９×９ ｉｓ ｔｈｅ Ｌａｐｌａｃｅ ｍａｔｒｉｘ ｏｆ ｔｈｅ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｔｏｐｏｌｏｇｙ ｇｒａｐｈ， ｂ ∈ Ｒ ９ ｒｅｆｌｅｃｔｓ ｗｈｉｃｈ ＵＡＶ ｃａｎ ｄｉｒｅｃｔｌｙ ｒｅｃｅｉｖｅ ｉｎｆｏｒｍａｔｉｏｎ ｆｒｏｍ ＵＡＶ１０ ， ξ ｒ ｉｓ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ａｎｄ ａｌｓｏ ｔｈｅ ｐｏｓｉｔｉｏｎ ｏｆ ＵＡＶ１０ ， ｗｈｉｃｈ ｓｈｏｗｓ ｔｈｅ ｄｅｓｉｒｅｄ ｐａｔｈ ｆｏｒ ａｌｌ ｔｈｅ ＵＡＶｓ， ｆｉｎａｌｌｙ， 􀱋 ｒｅｐｒｅｓｅｎｔｓ ｔｈｅ Ｋｒｏｎｅｃｋｅｒ ｐｒｏｄｕｃｔ． Ｋｒ ＝ ｄｉａｇ（Ｋｒ１ ，Ｋｒ２ ，…，Ｋｒ９ ），Ｋｖ ＝ ｄｉａｇ （ Ｋｖ１ ，Ｋｖ２ ， …，Ｋｖ９ ），ｅｒ ＝ （ｅｒ Ｔ １ ｅｒ Ｔ ２… ｅｒ Ｔ ９ ） Ｔ ，ｒ ～ ＝ （ ｒ ～ Ｔ １ ｒ ～ Ｔ ２…ｒ ～ Ｔ ９ ） Ｔ ，ｒ ｉｓ ｔｈｅ ｖｅｃｔｏｒ ｏｆ ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｏｆ ｅａｃｈ ＵＡＶ， ξ ｉｓ ｔｈｅ ｖｅｃｔｏｒ ｏｆ ｔｈｅ ｐｏｓｉｔｉｏｎ ｏｆ ｔｈｅ ｏｒｉｇｉｎ ｏｆ ｅａｃｈ ＵＡＶ’ ｓ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ． Ｅｑｕａｔｉｏｎ （ ６ ） ｉｓ ｔｈｅ ｃｏｒｅ ｏｆ ｔｈｅ ｍｕｌｔｉ⁃ＵＡＶ ｃｏｎｔｒｏｌｌｅｒ， ａｎｄ ｒ ｐｒｏｖｉｄｅｓ ｔｈｅ ｎｅｃｅｓｓａｒｙ ｐｏｓｉｔｉｏｎａｌ ｉｎｆｏｒｍａｔｉｏｎ ｆｏｒ ａｌｌ ｔｈｅ ＵＡＶｓ ｆｏｒ ｕｓｅ ｉｎ ｔｈｅ ｎｅｘｔ ｓｔｅｐ． Ａｆｔｅｒ ａｃｑｕｉｒｉｎｇ ｔｈｅ ｔａｒｇｅｔ ｉｎｓｔｒｕｃｔｉｏｎｓ， ｔｈｅ ＵＡＶｓ ｆｌｙ ｔｏｗａｒｄ ｔｈｅｉｒ ｄｅｓｔｉｎａｔｉｏｎｓ． １．２ Ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ ｔａｓｋ Ｔｈｅ ｄｅｓｉｒｅｄ ｆｏｒｍａｔｉｏｎ ｓｈａｐｅ ｉｓ ｄｅｓｉｇｎｅｄ ａｓ ｆｏｌｌｏｗｓ： ·５７２· 智 能 系 统 学 报 第 １２ 卷