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.TheＴｈｅｒｅ ａｒｅ ｔｗｏ ｍａｉｎ ｐａｒｔｓ ｏｆ ｔｈｅ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ［１３－１４］ ． Ｆｉｒｓｔ， ｔｈｅ ｆｏｒｍａｔｉｏｎ ｉｓ ｔｒｅａｔｅｄ ａｓ ａ ｖｉｒｔｕａｌ ｒｉｇｉｄ ｂｏｄｙ， ｅｖｅｒｙ ＵＡＶ ｈａｓ ａ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ａｎｄ ｔｈｉｓ ａｃｔｓ ａｓ ａ ｐｏｉｎｔ ｏｎ ｔｈｅ ｂｏｄｙ． Ｕｓｉｎｇ ｃｏｎｓｅｎｓｕｓ， ｔｈｅ ＵＡＶｓ’ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎｓ ｃｏｎｃｅｎｔｒａｔｅ ｔｏｇｅｔｈｅｒ ａｎｄ ｃｒｅａｔｅ ｔｈｅ ｄｅｓｉｒｅｄ ｆｏｒｍａｔｉｏｎ． Ｈｏｗｅｖｅｒ， ａ ｄｉｓｔａｎｃｅ ｅｒｒｏｒ ａｌｗａｙｓ ｅｘｉｓｔｓ ｂｅｔｗｅｅｎ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ａｎｄ ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｆｏｒ ｅａｃｈ ＵＡＶ， ａｎｄ ｔｈｅ ｓｅｃｏｎｄ ｐａｒｔ ｏｆ ｔｈｅ ａｌｇｏｒｉｔｈｍ ｄｅａｌｓ ｗｉｔｈ ｔｈｉｓ ｐｒｏｂｌｅｍ ｂｙ ｃｏｎｓｉｄｅｒｉｎｇ ｔｈｉｓ ｅｒｒｏｒ ａｓ ａ ｖａｒｉａｂｌｅ ｉｎ ｃｏｎｓｅｎｓｕｓ． Ｂｙ ｄｅｖｉｓｉｎｇ ａ ｕｓｅｆｕｌ ａｌｇｏｒｉｔｈｍ， ｔｈｅ ｅｒｒｏｒ ｃｌｏｓｅｓ ｔｏ ｚｅｒｏ ａｓ ｔｉｍｅ ｔｅｎｄｓ ｔｏ ｉｎｆｉｎｉｔｙ． Ｔｈｅ ｒｅａｌ ｐｏｓｉｔｉｏｎ ｆｏｒ ｅａｃｈ ＵＡＶ ｉｓ ｔｈｅ ｓｕｍ ｏｆ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎ ａｎｄ ｔｈｅ ｅｒｒｏｒ． Ｉｎ ｔｈｉｓ ｗａｙ， ＵＡＶｓ ｗｉｔｈ ｒａｎｄｏｍ ｉｎｉｔｉａｌ ｌｏｃａｔｉｏｎｓ ａｌｗａｙｓ ｃｏｎｃｅｎｔｒａｔｅ ａｎｄ ｆｌｙ ａｌｏｎｇ ｔｈｅｉｒ ｐｒｅｄｅｓｉｇｎｅｄ ｒｏｕｔｅｓ． Ｔｈｅｓｅ ｔｗｏ ｐａｒｔｓ ｏｆ ｔｈｅ ａｌｇｏｒｉｔｈｍ ａｄｄｅｄ ｔｏ ｔｈｅ ｃｏｍｍｕｎｉｃａｔｉｏｎ ｎｅｔｗｏｒｋｓ ｏｆ ｔｈｅ ｉｎｄｉｖｉｄｕａｌ ｖｅｈｉｃｌｅｓ ｃｏｍｐｒｉｓｅ ｔｈｅ ｗｈｏｌｅ ＵＡＶ ｓｙｓｔｅｍ ［１３－１４］ ． Ｈｏｗｅｖｅｒ， ｃｏｎｓｅｎｓｕｓ ｃａｎ ｏｎｌｙ ｅｎｓｕｒｅ ｔｈａｔ ａｌｌ ｔｈｅ ｖａｒｉａｂｌｅｓ ａｒｅ ｃｌｏｓｅ ｔｏ ｔｈｅ ｇｉｖｅｎ ｓｔａｔｅ ａｎｄ ｄｏｅｓ ｎｏｔ ｋｎｏｗ ｗｈｅｔｈｅｒ ｔｈｅ ｖａｌｕｅｓ ｏｆ ｖａｒｉａｂｌｅｓ ａｔ ｅａｃｈ ｍｏｍｅｎｔ ａｒｅ ｏｐｔｉｍａｌ． Ｉｔ ｉｓ ａ ｇｏｏｄ ｉｄｅａ ｔｏ ｉｍｐｒｏｖｅ ｃｏｎｓｅｎｓｕｓ ｗｉｔｈ ｏｐｔｉｍａｌ ａｌｇｏｒｉｔｈｍｓ． Ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄ ｏｐｔｉｍｉｚａｔｉｏｎ （ ＰＩＯ） ｉｓ ａ ｎｏｖｅｌ ｓｗａｒｍ ｉｎｔｅｌｌｉｇｅｎｃｅ ｏｐｔｉｍｉｚｅｒ ｐｒｏｐｏｓｅｄ ｂｙ Ｄｕａｎ ａｎｄ Ｑｉａｏ ｉｎ ２０１４ ［１５］ ． Ｔｈｅ ｍｅｔｈｏｄ ｉｍｉｔａｔｅｓ ｐｉｇｅｏｎｓ􀆳 ｂｅｈａｖｉｏｒ ｗｈｅｎ ｓｅｅｋｉｎｇ ａ ｗａｙ ｂａｃｋ ｔｏ ｔｈｅｉｒ ｌｏｆｔ． Ｉｔ ｉｓ ｂｅｌｉｅｖｅｄ ｔｈａｔ ｐｉｇｅｏｎｓ ａｄｏｐｔ ｄｉｆｆｅｒｅｎｔ ｔｏｏｌｓ ａｔ ｄｉｆｆｅｒｅｎｔ ｓｔａｇｅｓ ｗｈｅｎ ｒｅｔｕｒｎｉｎｇ ｈｏｍｅ． Ｉｎ ｔｈｅ ｂｅｇｉｎｎｉｎｇ， ｃｏｍｐａｓｓ ａｎｄ ｍａｐ ｆａｃｔｏｒｓ ａｒｅ ａｄｏｐｔｅｄ ［１６］ ， ｅａｃｈ ｐｉｇｅｏｎ􀆳ｓ ｖｅｌｏｃｉｔｙ ｉｓ ｕｐｄａｔｅｄ ｂｙ ｔｈｅ ｌｉｎｅａｒ ｃｏｍｂｉｎａｔｉｏｎ ｏｆ ｉｔｓ ｎｅｉｇｈｂｏｒｓ’ （ ｉｎｃｌｕｄｉｎｇ ｉｔｓｅｌｆ） ｂｅｓｔ ｌｏｃａｔｉｏｎｓ ａｎｄ ｉｔｓ ｆｏｒｍｅｒ ｖｅｌｏｃｉｔｙ． Ｗｈｅｎ ｐｉｇｅｏｎｓ ａｐｐｒｏａｃｈ ｈｏｍｅ， ｓｏｍｅ ｓｍａｒｔ ｏｎｅｓ ｗｉｔｈ ｇｏｏｄ ｍｅｍｏｒｉｅｓ ｃａｎ ｒｅｃｏｇｎｉｚｅ ｆａｍｉｌｉａｒ ｌａｎｄｍａｒｋｓ ｗｈｉｃｈ ｇｕｉｄｅ ｔｈｅｍ ［１７］ ， ｔｈｅ ｏｔｈｅｒｓ ｊｕｓｔ ｎｅｅｄ ｔｏ ｆｏｌｌｏｗ ｔｈｅｓｅ ｌｅａｄｅｒｓ． Ｔｈｅ ｓｅｃｏｎｄ ｐａｒｔ ｏｆ ＰＩＯ ｉｓ ｄｅｓｉｇｎｅｄ ｔｏ ｃｏｎｃｅｎｔｒａｔｅ ｔｈｅ ａｖｅｒａｇｅ ｌｏｃａｔｉｏｎ ｏｆ ａｌｌ ｔｈｅ ｐｉｇｅｏｎｓ． Ｉｔ ｉｓ ｗｉｄｅｌｙ ｋｎｏｗｎ ｔｈａｔ ｅａｃｈ ｏｐｔｉｍｉｚｅｒ ｆｉｔｓ ｏｎｅ ｏｒ ｓｅｖｅｒａｌ ｐｒｏｂｌｅｍｓ， ａｎｄ ＰＩＯ ｗａｓ ｆｉｒｓｔ ｐｒｏｐｏｓｅｄ ｆｏｒ ａｉｒ ｒｏｂｏｔ ｐａｔｈ ｐｌａｎｎｉｎｇ ［１５，１８］ ， ｉｍａｇｅ ｐｒｏｃｅｓｓｉｎｇ ［１７，１９］ ， ａｎｄ ｏｒｂｉｔ ｆｏｒｍａｔｉｏｎｓ ［２０］ ． Ｗｈｅｎ ｉｔ ｃｏｍｅｓ ｔｏ ｔｈｅ ｏｐｔｉｍｉｚａｔｉｏｎ ｏｆ ｃｏｎｓｅｎｓｕｓ ｍｅｎｔｉｏｎｅｄ ａｂｏｖｅ， ｄｉｓａｄｖａｎｔａｇｅｓ ａｐｐｅａｒ ｓｕｃｈ ａｓ ｔｈｅ ｔｒａｊｅｃｔｏｒｙ ｃｌｏｓｅｓ ｔｏ ｔｈｅ ｄｅｓｉｒｅｄ ｔｒａｃｅ ｗｉｔｈ ｏｓｃｉｌｌａｔｉｏｎ ｔｏ ａｎ ｕｎｓａｔｉｓｆａｃｔｏｒｙ ｄｅｇｒｅｅ ［２１］ ． Ｔｈｉｓ ｍｅａｎｓ ｔｈａｔ ＰＩＯ ｎｅｅｄｓ ｔｏ ｂｅ ｍｏｄｉｆｉｅｄ ａｎｄ ａ ｐｉｇｅｏｎ⁃ｉｎｓｐｉｒｅｄ ｏｐｔｉｍｉｚａｔｉｏｎ ｗｉｔｈ ｓｌｏｗ ｄｉｖｉｎｇ ｓｔｒａｔｅｇｙ （ ＳＤ⁃ＰＩＯ） ｉｓ ｐｒｏｐｏｓｅｄ． Ｔｈｅ ｒｅｍａｉｎｄｅｒ ｏｆ ｔｈｉｓ ｐａｐｅｒ ｉｓ ｏｒｇａｎｉｚｅｄ ａｓ ｆｏｌｌｏｗｓ． Ｉｎ Ｓｅｃｔｉｏｎ ２， ｄｅｔａｉｌｓ ｏｆ ｔｈｅ ａｐｐｌｉｃａｔｉｏｎ ｏｆ ｃｏｎｓｅｎｓｕｓ ｉｎ ａ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ ａｒｅ ｇｉｖｅｎ． Ｔｈｅ ｃｏｎｓｔｒｕｃｔｉｏｎ ｏｆ ＳＤ⁃ＰＩＯ ａｎｄ ｔｈｅ ｃｏｎｖｅｒｇｅｎｃｅ ａｎａｌｙｓｉｓ ｉｓ ｃｏｎｓｉｄｅｒｅｄ ｉｎ Ｓｅｃｔｉｏｎ ３． Ｉｎ Ｓｅｃｔｉｏｎ ４， ｔｈｅ ＳＤ⁃ＰＩＯ⁃ ｂａｓｅｄ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍ ｉｓ ｐｒｏｐｏｓｅｄ ｆｏｒ ａ ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ， ｆｏｌｌｏｗｅｄ ｂｙ ｎｕｍｅｒｉｃａｌ ｓｉｍｕｌａｔｉｏｎ ａｎｄ ｒｅｓｕｌｔｓ ａｎａｌｙｓｉｓ ｉｎ Ｓｅｃｔｉｏｎ ５． Ｆｉｎａｌｌｙ， ｓｅｖｅｒａｌ ｃｏｎｃｌｕｓｉｏｎｓ ｃｏｎｃｅｒｎｉｎｇ ｔｈｅ ｕｓｅ ｏｆ ｏｐｔｉｍａｌ ａｌｇｏｒｉｔｈｍｓ ｉｎ ｃｏｎｓｅｎｓｕｓ ｔｈｅｏｒｙ ａｒｅ ｄｒａｗｎ． １ Ｃｏｎｔｒｏｌｌｅｒ ｄｅｓｉｇｎ ｂａｓｅｄ ｏｎ ｃｏｎｓｅｎｓｕｓ １．１ Ｍｕｌｔｉ⁃ＵＡＶ ｆｏｒｍａｔｉｏｎ ｃｏｎｔｒｏｌｌｅｒ ｄｅｓｉｇｎ Ｉｔ ｉｓ ｓｕｐｐｏｓｅｄ ｔｈａｔ ｅａｃｈ ＵＡＶ ｐｏｓｓｅｓｓｅｓ ａ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ ｗｉｔｈ ｔｈｒｅｅ ａｘｅｓ （ ｘ， ｙ， ａｎｄ ｚ）． Ｔｈｅ ｓｈａｐｅ ｏｆ ｔｈｅ ＵＡＶ ｉｓ ｉｇｎｏｒｅｄ ａｎｄ ａ ｍａｓｓ ｐｏｉｎｔ ｍｏｄｅｌ ｉｓ ｕｓｅｄ ｉｎｓｔｅａｄ． Ｔｈｉｓ ｉｓ ｂｅｃａｕｓｅ ｔｈｅ ｔｉｍｅ ｄｅｌａｙ ｏｆ ａｎ ａｉｒｃｒａｆｔ ｃｏｎｔｒｏｌ ｌｏｏｐ ｉｓ ｍｕｃｈ ｓｈｏｒｔｅｒ ｔｈａｎ ｔｈａｔ ｏｆ ｔｈｅ ｎａｖｉｇａｔｉｏｎ ａｎｄ ｇｕｉｄａｎｃｅ ｌｏｏｐｓ． Ｔｈｅｒｅｆｏｒｅ， ｗｈｅｎ ｄｅａｌｉｎｇ ｗｉｔｈ ｆｏｒｍａｔｉｏｎ ｃｏｎｔｒｏｌ， ｔｈｅ ＵＡＶ ｍｏｄｅｌ ｃａｎ ｂｅ ｓｉｍｐｌｉｆｉｅｄ ａｓ ａ ｐｏｉｎｔ， ｗｈｉｃｈ ｍｅａｎｓ ｔｈｅｒｅ ｉｓ ｎｏ ｎｅｅｄ ｔｏ ｃｏｎｓｉｄｅｒ ｅａｃｈ ＵＡＶ’ｓ ａｔｔｉｔｕｄｅ． Ｕｎｄｅｒ ｔｈｉｓ ａｓｓｕｍｐｔｉｏｎ， ａｌｌ ｔｈｅ ＵＡＶ’ｓ ｘ， ｙ， ａｎｄ ｚ ａｘｅｓ ａｒｅ ｐａｒａｌｌｅｌ． Ｃｏｎｓｅｎｓｕｓ ｍｅａｎｓ ｔｈａｔ ｍｕｌｔｉｐｌｅ ＵＡＶｓ ｒｅａｃｈ ａｎ ａｇｒｅｅｍｅｎｔ ｏｎ ａ ｃｏｍｍｏｎ ｓｔａｔｅ， ｅ． ｇ．， ｔｈｅ ＵＡＶ’ ｓ ｐｏｓｉｔｉｏｎ． Ｉｎ ｄｅｔａｉｌ， ｅｖｅｒｙ ＵＡＶ ｈａｓ ｉｔｓ ｉｎｉｔｉａｌ ３⁃ ｄｉｍｅｎｓｉｏｎａｌ ｐｏｓｉｔｉｏｎ （ ｘ， ｙ， ｚ ） ｉｎ ａｎ ｉｎｅｒｔｉａｌ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ， ｎａｍｅｌｙ ｔｈｅ ｇｒｏｕｎｄ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ． Ｔｈｅ ｖｅｌｏｃｉｔｙ ｏｆ ｔｈｅ ｏｒｉｇｉｎａｌ ＵＡＶｉ ’ ｓ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ ｃｏｎｖｅｒｇｅｓ ｔｏ ｔｈｅ ｓｔａｎｄａｒｄ ｓｔａｔｅ ｏｆ ｔｈｅ ｖｉｒｔｕａｌ ｌｅａｄｅｒ ＵＡＶ１０ ｂｙ ｃｏｎｓｅｎｓｕｓ． Ｔｈｅ ＵＡＶ１０ ’ ｓ ｓｔａｔｅ ｃｏｎｓｉｓｔｓ ｏｆ ａ ｔｉｍｅ⁃ｖａｒｙｉｎｇ ｐｏｓｉｔｉｏｎ ａｎｄ ｐｒｅｄｅｓｉｇｎｅｄ ｖｅｌｏｃｉｔｙ． Ｉｎ ｔｈｉｓ ｗａｙ， ａｌｌ ｔｈｅ ＵＡＶｓ ’ ｃｏｏｒｄｉｎａｔｅｓ ｂｅｃｏｍｅ ｏｎｅ ｃｏｍｍｏｎ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ． Ａｆｔｅｒ ａｄｄｉｎｇ ｔｈｅ ｒｅｌａｔｉｖｅ ｐｏｓｉｔｉｏｎ ｏｆ ｅａｃｈ ＵＡＶ ｉｎ ｔｈｅ ｍｏｖｉｎｇ ｃｏｏｒｄｉｎａｔｅ ｓｙｓｔｅｍ， ｔｈｅ ｄｅｓｉｒｅｄ ｔｉｍｅ⁃ｖａｒｙｉｎｇ ｐｏｓｉｔｉｏｎ ｉｓ ａｃｈｉｅｖｅｄ ｆｏｒ ｅａｃｈ ＵＡＶ． Ｗｈｅｎ ｖｉｅｗｅｄ ａｓ ａ ｍａｓｓ ｏｆ ｍｏｖｉｎｇ ｐｏｉｎｔｓ， ａｌｌ ｔｈｅｓｅ ＵＡＶｓ ｃｏｎｓｔｉｔｕｔｅ ａ ｖｉｒｔｕａｌ ｂｏｄｙ． Ｔｈｅ ｆｏｒｍａｔｉｏｎ ｍｅｔｈｏｄ ｃｏｎｓｉｓｔｓ ｏｆ ｂｏｔｈ ｓｉｎｇｌｅ⁃ａｎｄ ｄｏｕｂｌｅ⁃ｉｎｔｅｇｒａｔｏｒ ｃｏｎｓｅｎｓｕｓ ａｌｇｏｒｉｔｈｍｓ． Ｔｈｅ ｌａｔｔｅｒ ａｉｍｓ ｔｏ ｃｏｎｃｅｎｔｒａｔｅ ｔｈｅ ＵＡＶｓ ｗｉｔｈ ｒａｎｄｏｍ ｉｎｉｔｉａｌ ｐｏｓｉｔｉｏｎｓ ａｎｄ ｅｎｓｕｒｅ ｔｈａｔ ｔｈｅ ｄｅｓｉｒｅｄ ｐｏｓｉｔｉｏｎｓ ｏｆ ａｌｌ ｔｈｅ ＵＡＶｓ ｃｏｎｖｅｒｇｅ ｔｏ ｐｏｉｎｔｓ ｏｎ ｔｈｅ ｐｒｅｄｅｓｉｇｎｅｄ ｔｒａｊｅｃｔｏｒｙ． Ｔｈｅ ·５７１· 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