Research Article Three-Dimensional Path Planning Method for Autonomous Underwater Vehicle Based on Modified Firefly Algorithm

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1 Mathematcal Problems n Engneerng Volume 2015, Artcle ID , 10 pages Research Artcle Three-Dmensonal Path Plannng Method for Autonomous Underwater Vehcle Based on Modfed Frefly Algorthm Chang Lu, Yuxn Zhao, Feng Gao, and Lqang Lu College of Automaton, Harbn Engneerng Unversty, Harbn, Helongjang , Chna Correspondence should be addressed to Chang Lu; luchang407@hrbeu.edu.cn Receved 22 October 2014; Accepted 23 December 2014 Academc Edtor: Xn-She Yang Copyrght 2015 Chang Lu et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Path plannng s a classc optmzaton problem whch can be solved by many optmzaton algorthms. The complexty of threedmensonal (3D) path plannng for autonomous underwater vehcles (AUVs) requres the optmzaton algorthm to have a quck convergence speed. Ths work provdes a new 3D path plannng method for AUV usng a modfed frefly algorthm. In order to solve the problem of slow convergence of the basc frefly algorthm, an mproved method was proposed. In the modfed frefly algorthm, the parameters of the algorthm and the random movement steps can be adjusted accordng to the operatng process. At the same tme, an autonomous flght strategy s ntroduced to avod nstances of nvald flght. An excludng operator was used to mprove the effect of obstacle avodance, and a contractng operator was used to enhance the convergence speed and the smoothness of the path. The performance of the modfed frefly algorthm and the effectveness of the 3D path plannng method were proved through a vared set of experments. 1. Introducton AUVsarenstrongdemandwthnbothmltaryandcvl felds. More and more research nsttutons are carryng outresearchntoauvs.pathplannngsoneofthekey technologes n autonomous decson-makng for AUVs, and t has become one of the most talked about ssues n the AUV feld. Many methods for AUV path plannng have been putforwardandmostoftheauvoperatngenvronments were assumed to be two-dmensonal. For example, some researchers proposed a path plannng method usng artfcal potental feld methods [1, 2]. Chen et al. brought forward a global path plannng method for AUV based on the sparse A search algorthm [3]. Some novel optmzaton algorthms, such as the genetc algorthm () and ant colony optmzaton (ACO), were also used to solve the problems of AUV path optmzaton [4, 5]. Although these methods were proved effectve n solvng path plannng problems, they nevtably face some problems n practcal applcatons because the AUV workng envronment s a 3D marne space. 3D path plannng s a classcal optmzaton problem. Wth the development of optmzaton algorthms, many researchers apply new optmzaton algorthms to solve the pathplannngproblemnrecentyears.swarmntellgence (SI) belongs to an artfcal ntellgence (AI) dscplne that became ncreasngly popular over the last decade, and t refers to a research feld that s concerned wth a collectve behavor wthn self-organzed and decentralzed systems. Nowadays, swarm ntellgence has formed one of the hottest topcs n the developments of new algorthms nspred by nature. And methodsofswarmntellgenceareusedtosolvetradtonal NP problems and ther excellent performance and great potental have been proved n many practcal applcatons [6]. In recent years, SI-based algorthms are also appled for solvng 3D path plannng problem. For example, a 3D path plannngmethodbasedontheacowasproposedbyluetal. [7, 8]. Partcle swarm optmzaton () was also used to solve the problem of 3D path optmzaton [9 11]. These works prove that t s feasble to solve 3D path plannng problems usng SI-based algorthm. Frefly algorthm () s a more promsng SI-based optmzaton algorthm based on frefly socal features put forward by Dr. Yang n Although ths algorthm s smlar

2 2 Mathematcal Problems n Engneerng to other bonspred algorthms, t s relatvely smple both n theory and n mplementaton. One of ts most promnent advantages s that global communcatons can run based on ndvdual exercse at the same tme as random moton [12]. The exstng research results showed that the algorthm was very effectve n dealng wth a lot of optmzaton problems,andthadgraduallybeenappledtosomeknds of optmzaton felds wth good results [13 21]. As a new optmzaton algorthm, the basc has some drawbacks n convergence speed and stablty. Some mprovements have been put forward recently. For example, Dr. Yang added Levy flght to the random part of the formulaton and Levyflght frefly algorthm (L) was constructed. Expermental result showed that the mproved algorthm was more effectve n searchng for global best values [22, 23]. Farahan et al. made some mprovements to the random moton part of to ensure the random step s larger n the ntal perod, n order to avod fallng nto local optma. The random step was reduced after several teratons to make the algorthm converge to the global mnmum rapdly. Meanwhle, the movement style of was mproved and drected movement was proposed. Expermental results showed that the mproved had better performance than the basc [24]. dos Santos Coelho and de Andrade Bernert used chaotc sequences to adjust the parameters of, and the mproved algorthm had better results n optmzng benchmark test functons of relablty and redundancy allocaton [25]. Lu et al. made some mprovements to ensure parameters of adjusted accordng to the teraton numbers. Compared to the basc, the performance and convergence speed were mproved [26]. Farahan et al. put forward a new that ncreases convergence speed usng Gaussan dstrbuton to move all frefles to global best n each teraton. The modfed algorthm was tested on fve standard functons. And expermental results showed that the modfed had better performance than the basc frefly algorthm [27]. Fster et al. proposed a comprehensve revew of the. The status of applcatons wthn varous applcaton areas was summarzed. At the same tme, the revew showed that was smple, flexble, and versatle, whch was very effcent n solvng a wde range of dverse real-world problems [20]. In ths paper, a modfed and a 3D path plannng method based on the new algorthm are proposed. The length of random steps and parameters of the algorthm are desgned to be adjusted accordng to the dstance between two frefles and the teraton tmes. The autonomous flght strategy has been ntroduced to avod nvald flght. An excludng operator and a contractng operator have been used to mprove the effect of obstacle avodance, the speed of convergence, and the smoothness of the path. The paper s organzed as follows. Secton 2 analyzes the performance of basc. Secton 3 develops a modfed. A 3D path plannng method s proposed n Secton 4. Experments n Secton 5 prove the performance of the modfed and the path plannng method. Fnally, Secton 6 contans the man conclusons and descrbes some problems that warrant further research. Begn Set parameters of the algorthm Sort the frefly populaton n accordance wth the brghtness and fnd the brghtest one Satsfy fnsh condton? No Fnsh Yes Fgure 1: The executon process of. 2. Basc Frefly Algorthm and ItsPerformanceAnalyss Update locaton of each frefly The s derved from the smplfcaton and smulaton of frefly group behavor. It has three dealzed constrants whch are derved from frefly features. (1) The frefly s not gender-specfc. And they wll fly to more attractve and larger brghtness companon regardless of ts gender. (2) Frefly attractve sze s proportonal to ts brghtness. And ts brghtness decreases wth the dstance between ndvduals. If theresnobrghterormoreattractveone,thentwllflght randomly. (3) The brghtness or attractveness of a frefly s determned by the specfed value of the objectve functon [12]. For the maxmzaton problem, the lumnous ntensty s proportonal to the value of the objectve functon. Based on the above three rules, the executon process of frefly algorthm s shown n Fgure 1. The core of s that the absolute brghtness of frefles represents the value of objectve functon and the poston of the frefles represents the soluton of the problem to be solved. The relatve brghtness s ganed by comparng two frefles and t s assocated wth attracton. One frefly s attracted by a brghter one and adjusts ts poston accordng to ths attracton. Only the brghtest frefly moves randomly. The formula of s as follows: x (t+1) = x (t) +β j (r j )( x (t) x j (t))+αε, (1) β j (r j )=β 0 e γr j 2, (2) where β 0 stands for the bggest attracton and γ s the absorpton coeffcent whch controls the change n lght ntensty and determnes the speed of convergence. α s the random coeffcent whch controls the range of movement. γ j s the artesan dstance between frefly and frefly j ε s the random vector decded by Gauss dstrbuton [12]. As descrbed n formula (1), the changng poston of a frefly depends on ts attracton to brghter frefles and ther random movement. At the ntal perod of algorthm runnng, frefles are randomly dstrbuted wthn the search area. The dstance between each of them s greater, whle β j (r j ) s small and the range of movement s small, so

3 Mathematcal Problems n Engneerng 3 the explorng ablty of s nsuffcent wth slower convergence. At the later stage of the algorthm runnng, frefles gather around the optmal nsect, the dstance between each other s smaller, and β j (r j ) s bgger. At the same tme, random moton s stll takng place whch s not useful for the convergence of the algorthm [13]. In addton, the poston updatng of one frefly s affected byotherbrghterfrefles,someofwhchmaynotbenecessarly useful for solvng the problem, and ths may make the new locaton of the frefly far away from the optmal soluton (we call ths behavor nvald flght). Although nvald flght may be mproved by the nfluence of other frefles n the followng movements, t wll slow down the search process and reduce the speed of convergence. Thus, n order to mprove the ablty of basc to solve practcal problems, some changes are necessary. 3. A Modfed Frefly Algorthm There are two mportant parameters n the formula for the poston updatng: the absorpton coeffcent γ and the random coeffcent α. The smaller the γ, the greater the attracton between two frefles and the faster the convergence speed. The range of random movement s larger and the convergence speed s slower, wth a bgger α. In practcal optmzaton problems, generally global search helps the algorthm converge to an area quckly and then a local search helps obtan a hgh precson soluton. Reference [26] proposed an adaptve n whch α and γ were adjusted wth the teraton tmes, and ts performance was better than that of the basc. Because the random movement step s 1 and the changng range of the random part s 0.5to0.5, the convergence speed s very slow n solvng problems wth a large range. In order to solve ths problem, we proposed a new n whch the random movement step was desgned to be the dstance between two frefles. The poston updatng formulaofthenewsasfollows: x (t+1) = x (t) +β j (r j ) ( x (t) β j (r j ), α(t),andγ(t) can be obtaned as follows: β j (r j )=β e γ(t)r j 2, γ(t) =γ b + t N (γ e γ b ), α(t) =α b + t N (α e α b ), x j (t))+α(t) r j ε ; (3) where x and x j stand for the spatal postons of frefles and j; N s the whole teraton tme; γ e >γ b,whereγ b and γ e are thentalandultmatevalueofγ,respectvely;α e <α b,where α b and α e are the ntal and ultmate value of α,respectvely. As seen n formula (3), the attracton between two frefles s small when the dstance between them s great. Attracton therefore does not play an mportant role n poston updatng. However, as the random movement step s bg, frefles move wthn a bg range to explore a larger search (4) space. When the dstance among frefles s small, the random movement step reduces and the random movement range s small. Attracton therefore begns to play a more mportant role n poston updatng. Frefles move to the brghtest one quckly and the algorthm comes to convergence at a great speed. In addton, n order to avod nvald flght, we ntroduced the autonomous flght strategy (AFS). The new poston of a frefly should be judged before t moves. The frefly wll move and update the poston f the new poston s useful for solvng the problem. Otherwse, the frefly wll stay at the current poston. 4. 3D Path Plannng for AUV Usng the Modfed Frefly Algorthm 4.1. Codng of. In order to use to solve 3D path plannngproblem,thecodngofshouldbedonefrst.ifa path has n route ponts, t must be consttuted by n 1path segments. We assume that each frefly represents a canddate path n the frefly group, and the dmensons of poston for each frefly correspond to the route ponts. So the number of frefles corresponds to the canddate path number. So f a frefly group contans m members and each frefly n the group has n dmensons, the ()th canddate path Path() represented by the ()th frefly can be descrbed as follows: Path() ={P,j j=1,...,n}, (5) where P,j =(x,j,y,j,z,j ) stands for the (j)th route pont and t corresponds to the (j)th dmenson of the poston for the ()th frefly. In addton, as the brghtness represents the qualty of a frefly, t corresponds to the path qualty n the path plannng problem. The brghtest frefly represents the optmal path Cost Functon. There are many crtera for evaluatng a canddate path. Generally, the smaller the path length s, the better t s. Addtonally, n order to ensure the safety of AUV,thepathshouldbeawayfromobstacles,soweusedthe path length E L and rsk value E D to evaluate the canddate path. The cost functon of path plannng can be calculated as follows: E=ω 1 E L +ω 2 E D, (6) where ω 1 +ω 2 =1. So the am of path plannng s to fnd a path whch can satsfy mn(e).thecalculatngmethodsofeverycostpartare as follows Path Length. If a path has n route ponts, ts length can be calculated by the followng formula: n 1 E L =L SP1 + =1 ΔL +L Pn E, (7) where (x,y,z ), 1 < n, stands for the coordnates of the ()th route pont, L SP1 stands for the dstance

4 4 Mathematcal Problems n Engneerng z P B P n A 1 +1 C x Fgure 3: Sketch map of contractng operator. y B Fgure 2: Sketch map of excludng operator. between the start pont and the frst route pont, ΔL = (x x +1 ) 2 +(y y +1 ) 2 +(z z +1 ) 2 represents the dstance between the ()th route pont and the ( + 1)th route pont, and L Pn E stands for the dstance between the destnaton pont and the last route pont Rsk Value. Apathholdsrsksftpassesthrough obstacles.sotherskvaluecanbejudgedbyhowmuchof a path contans obstacles. In order to calculate the rsk value easly, we selected m evaluatng ponts n every path segment. The whole rsk value can therefore be judged by calculatng how many route ponts and evaluatng ponts whch are n obstacles. For example, f there are k 1 route ponts and k 2 evaluatng ponts n obstacles, the rsk value can be calculated as follows: C A E D =C(k 1 +k 2 ), (8) where C>0stands for the cost coeffcent Optmzaton Operator Excludng Operator. As the path plannng space s 3D and contans many obstacles, t s nevtable that some ponts on the path wll come nto contact wth obstacles n the process of searchng. Sometmes these ponts would come outoftheobstaclesatlast,buttwllcostmuchtme.in order to reduce searchng tme and ncrease the success rate of obstacle avodance, t s necessary to help these ponts avod obstacles quckly. An excludng operator s used here to make ths easer. A path pont nsde an obstacle wll be excluded to the surface of obstacle s plane [11]. Fgure2 shows how the excludng operator works. In Fgure 2, ABC s a plane on the surface of one obstacle and n stands for the normal drecton of ABC. P s a path pont nsde the obstacle. In order to ensure safety, P should move out of the obstacle usng an excludng operator. s on plane ABC whch stands for the new poston of P after t s excluded from obstacles. The poston of and the dsplacement of P canbecalculatedusngthefollowng formula: =P + P, P =( P n) n Contractng Operator. The generaton of a safe path should be done through the cooperaton of every path pont. Wthout takng path knowledge nto account, the plannng process wll take a long tme, especally n path plannng wth many dmensons. A path generated by only may be unsmooth, so a contractng operator s used to manage the cooperaton of route ponts and to make the path smoother [11]. Fgure 3 shows how the contractng operator works. In Fgure 3, 1 s the route pont before P and +1 s the route pont after P. On the assumpton that 1 and P +1 are fxed, P wll be pulled by these two ponts and the drecton of the resultant force s Theoretcally, wth the pullng force, P wll move to.infact,becausetsaffected by ar resstance, P can only move to. The dsplacement of P and ts real poston can be calculated by the followng formula: =P + S P, S P =k( (9) 1 + (10) +1 ), where S P s the dsplacement vector of P wth pullng forces from 1 and P +1 and k (0, 1) s the resstance coeffcent. The path plannng flow based on the modfed s as follows: frstly, ntalzng path parameters and modelng the operatng envronment and then searchng for the optmum path usng modfed, usng an excludng operator to avod obstacles and a contractng operator to optmze the path. These steps should be repeated untl a satsfactory path s found. The detaled workflow s shown n Fgure Expermental Setup 5.1. Experments Usng Modfed. Genetc algorthm () and partcle swarm optmzaton () are two classc

5 Mathematcal Problems n Engneerng 5 Table 1: Test functons. Functon Sphere Rosenbrock Rastrgn Grewank Ackley Zakharo Range [ 5.12, 5.12] [ 2.048, 2.048] [ 5.12, 5.12] [ 8, 8] [ 2.768, 2.768] [ 5, 10] Next frefly No Yes Optmzng canddate path usng contractng operator Yes Begn Modelng envronment and ntalzng parameters Sortng the frefly populaton accordng to cost functon Calculatng the new poston of the current frefly Avodng obstacles usng excludng operator Adjustng poston accordng to AFS All the frefles fnsh flght? Satsfy fnsh condton? Fnsh No Fgure 4: The flowchart for 3D path plannng. algorthms wth extreme nfluence n optmzaton feld, and research on them s relatvely mature. has some qualtes n common wth these two algorthms. In order to verfy the performance of the modfed (AM) proposed n ths paper, we used four algorthms (basc, AM,, and ) to solve sx classc test functons. The four classc test functons are as follows. Sphere: f 1 ( x) = n =1 x2 ;thefunctonhastsglobal optmum f =0at s =(0,...,0). Rosenbrock: f 2 ( x) = n 1 =1 [100(x2 x +1) 2 +(x 1) 2 ]; the functon has global optmum f =0at s =(1,...,1). Rastrgn: f 3 ( x) = n =1 [x2 10cos(2πx ) + 10]; the functon has global optmum f =0at s = (0,..., 0). Grewank: f 4 ( x) = n 1 =1 [100(x2 x +1 ) 2 +(x 1) 2 ]; the functon has global optmum f =0at s =(0,...,0). Ackley: f 5 = 20exp[ 0.2 (1/d) d =1 x2 ] exp[(1/ d) d =1 cos(2πx )] + (20 + e); the functon has global optmum f =0at s =(0,...,0). Zakharo: f 6 = D =1 s2 + ((1/2) D =1 s ) 2 + ((1/2) D =1 s ) 4 ;thefunctonhasglobaloptmumf = 0 at s =(0,...,0). Parameters of these four algorthms are set as follows. : β 0 = 1.0, γ = 1.0,andα = 1.0. randwasthe unformly dstrbuted random number n the range of 0-1. : bnary encodng was used and the bnary dgt was 20. The roulette wheel selecton method and sngle pont crossover were used. Crossover probablty was defned as p c = 0.6, mutatonprobabltywas descrbed as p m =1/n,andn was the dmenson of the optmzaton functon. : usng parameters recommended by L et al. [10], ω = and c 1 =c 2 = The maxmum speed was set to the top of the search range, V max =X max. AM: β 0 = 1.0, γ e =2.0, γ b =1.0, α e = 0.8, and α b = 2.0.randwastheunformlydstrbutedrandom number n the range of 0-1. In order to compare the optmzaton performance of thesealgorthmsfarlyandsuffcently,theywereusedto solvethefourtestfunctonsnbothsmallandwderanges. The populaton sze of every algorthm was set to 40 and the dmenson of the testng functon was set to 30. Each algorthm was run 30 tmes ndependently, and the number of ftness evaluatons was lmted to 60, Experments n a Small Range. The range of every test functon s shown n Table 1. Table 2 shows the results of smulaton for every algorthm. In order to observe the teratve process of the algorthm, the curves of the sx test functons are shown n Fgures 5 to 10. Expermental results n Table 2 show that the precson of soluton obtaned by AM s much better than that of. Meanwhle, Fgures 5 10 show that the convergence speed of AM s qucker than that of. So we can conclude that the performance of AM s better than that of and the method proposed here s effectve n mprovng the performance of basc. The performance and convergence speed of AM are better than those of and. Although the soluton precson of Sphere obtaned by AM s not the best, t s vald. Meanwhle, the soluton of Sphere

6 6 Mathematcal Problems n Engneerng Table 2: Expermental results n a small range. Functon Term AM Best E E E 07 Sphere Mean 5.58E E E E 06 STD 1.73E E E E 06 0 tmes Best E E Rosenbrock Mean 2.38E E E STD 4.34E E tmes Best 00E E Rastrgn Mean 00E E STD 00E E tmes Best 00E E 0E E 07 Grewank Mean 3.20E E E E 02 STD 1.12E E E E 02 0 tmes Best E E E +00 Ackley Mean 1.75E E E E +00 STD 7.23E E E E 01 0 tmes Best 00E E E E +01 Zakharo Mean 1.6E E E E +02 STD 9.0E E E E tmes Sphere ([ 5.12, 5.12], dmenson =30) Number of ftness evaluatons 10 4 AM Fgure 5: Test results of Sphere Rosenbrock ([ 2.048, 2.048], dmenson =30) Number of ftness evaluatons 10 4 AM Fgure 6: Test results of Rosenbrock. obtanedbyamsfoundtobezero22tmes.wecan thereforeconcludethatamsaneffectvealgorthmfor solvng problems wth a small range Experments n a Wde Range. The range of every test functon s shown n Table 3. Table 4 shows the results of smulaton for every algorthm. In order to observe the teratve process of the algorthm, the curves of the sx test functonsareshownnfgures Expermental results n Table 4 show that the precson of soluton obtaned by AM s the most effectve for solvng all sx test functons. Addtonally, t can fnd the global optmum every tme. Fgures 11 to 16 show that the convergencespeedofamsthequckestofthefour algorthms. So we can conclude that the method proposed here s effectve n mprovng the performance of basc and AM s powerful n solvng problems wth a wde range Experments of Path Plannng. In order to smulate the real salng envronmental space of AUV, a martme space

7 Mathematcal Problems n Engneerng 7 Table 3: Test functons. Functon Sphere Rosenbrock Rastrgn Grewank Ackley Zakharo Range [ 100, 100] [ 100, 100] [ 100, 100] [ 600, 600] [ , ] [ 50, 100] Table 4: Expermental results n wde range. Functon Term AM Best 00E E E E 04 Sphere Mean 00E E E E 03 STD 00E E E E 03 0 tmes Best 00E E E E +00 Rosenbrock Mean 00E E E E +01 STD 00E E E E tmes Best 00E E E E +01 Rastrgn Mean 00E E E E +01 STD 00E E E E tmes Best 00E E E E 04 Grewank Mean 00E E E E 02 STD 00E E E E 02 0 tmes Best E E E 03 Ackley Mean 00E E E E 03 STD E E E 03 0 tmes Best 0 2.0E E E +04 Zakharo Mean 1.5E E E E +04 STD 0 7.6E E E tmes Rastrgn ([ 5.12, 5.12], dmenson =30) Number of ftness evaluatons 10 4 Grewank ([ 8, 8], dmenson =30) Number of ftness evaluatons 10 4 AM AM Fgure 7: Test results of Rastrgn. Fgure 8: Test results of Grewank. wth a horzontal area about 2 2 was selected as the experment envronment for AUV path plannng. The terran was generated by nterpolaton wth elevaton data extracted from electrc chart. In order to observe the obstacle avodance capablty of the method, the actual lengths of the horzontal and vertcal coordnates were reduced ( )/1000 tmes. The resstance coeffcent was and parameters of AM were the same as n the experments descrbed above. Because the populaton sze of AM was 40 and

8 8 Mathematcal Problems n Engneerng Number of ftness evaluatons 10 4 AM Ackley ([ 2.768, 2.768], dmenson =30) Fgure 9: Test results of Ackley Rosenbrock ([ 100, 100], dmenson =30) Number of ftness evaluatons AM Fgure 12: Test results of Rosenbrock. Rastrgn ([ 100, 100], dmenson =30) Zakharo ([ 5, 10], dmenson =30) Number of ftness evaluatons Number of ftness evaluatons 10 4 AM AM Fgure 13: Test results of Rastrgn. Fgure 10: Test results of Zakharo Grewank ([ 600, 600], dmenson =30) Sphere ([ 100, 100], dmenson =30) Number of ftness evaluatons 10 4 AM Fgure 14: Test results of Grewank Number of ftness evaluatons 10 4 AM Fgure 11: Test results of Sphere. the dmenson of every frefly was set to 30, the number of canddate paths was 40 and every path should have 30 path ponts. In order to verfy the performance of the new algorthm, two experments were desgned here. In the frst experment, depths of the startng pont and destnaton

9 Ackley ([ , ], dmenson = 30) Number of ftness evaluatons 16 de ( t. La Depth (m) Mathematcal Problems n Engneerng g) 15.5 AM eg). (d 111 Lon Fgure 17: Results of experment 1. Zakharo ([ 50, 100], dmenson = 30) eg). (d Lat 1050 Depth (m) Fgure 15: Test results of Ackley on L. (deg) Number of ftness evaluatons AM Fgure 16: Test results of Zakharo. are both 300 meters. The horzontal poston of startng pont s ( E, N), and the horzontal poston of destnaton s ( E, N). The results of the path plannng are shown n Fgure 17. In the second experment, the start poston and destnaton have dfferent depths. The horzontal poston of the startng pont s ( E, N) and the depth s 230 meters. The horzontal poston of the destnaton s ( E, N), and the depth s 450 meters. The results of the path plannng are shown n Fgure 18. Fgures 17 and 18 show that there are many obstacles n the envronmental space. The lne represents the path found by AM, the black dot s the startng pont, and the red dot stands for the destnaton n the fgures. From the results of the experments we can conclude that the path plannng method usng AM can fnd the optmal path n the complex envronmental space, and the paths n both experments are smooth and do not path through obstacles. The expermental results prove that the path plannng method proposed here s feasble and AM can be used n solvng 3D path plannng problem. Fgure 18: Results of experment Conclusons Ths paper presents a 3D path plannng method based on a modfed. The performance of the modfed and the path plannng method was verfed through smulaton. The smulaton results show that the modfed has a quck convergence speed and the path plannng method based on t can fnd an effectve path n a 3D envronment. However, as a new SI-based algorthm, the frefly algorthm stll has mperfectons from a theoretcal research pont of vew and ts mathematcal theory also needs to be studed and proved. Addtonally, t has been assumed that the operatng envronment s statc and obstacle nformaton s known before the path plannng method s begun. In realty, the marne envronment s dynamc and AUVs wll face many sudden threats whle n acton. Fnally, many constrants for AUVs such as the maxmal turnng angle and the maxmal salng depth must be taken nto account. Therefore further research s requred to conclude how to plan a practcal path for AUVs n the real dynamc marne envronment. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper.

10 10 Mathematcal Problems n Engneerng Acknowledgments Ths work has been supported by the Natonal Natural Scence Foundaton of Chna under Grant no and Foundaton of Central Unversty HEUCFX41302 and HEUCF References [1] F. Wang, L. Wan, Y. R. Xu, and Y. K. Zhang, Path plannng basedonmprovedartfcalpotentalfeldforautonomous underwater vehcles, JournalofHuazhongUnverstyofScence and Technology,vol.39,no.11,pp ,2011. [2] X. L and D. Q. Zhu, Path plannng for autonomous underwater vehcle based on artfcal potental feld method, Shangha Martme Unversty,vol.31,no.2,pp.35 39,2010. [3] S. Chen, C. W. Lu, Z. P. Huang, and G. C. Ca, Global path plannng for AUV based on sparse A search algorthm, Torpedo Technology,vol.20,no.4,pp ,2012. [4] G. Yan, L. Wang, J. Zhou, and J. Cha, Path plannng based on mproved genetc algorthm for AUV, Chongqng Unversty of Technology (Natural Scence),vol.24,no.5,pp.81 85, [5] J. H. Wang, H. X. Wu, and X. C. Sh, Global path plannng for AUV based on ACO, Shpbuldng of Chna, vol. 49, no. 2, pp , [6] I. Fster Jr., X.-S. Yang, I. Fster, J. Brest, and D. Fster, A bref revew of nature-nspred algorthms for optmzaton, Elektrotehnšk Vestnk,vol.80,no.3,pp.1 7,2013. [7]L.Q.Lu,Y.T.Da,L.H.Wang,andX.L.Gan, Research on global path plannng of underwater vehcle based on ant colony algorthm, System Smulaton, vol.19,no.18, pp , [8] L.-Q. Lu, F. Yu, and Y.-T. Da, Path plannng of underwater vehcle n 3D space based on ant colony algorthm, System Smulaton,vol.20,no.14,pp ,2008. [9] J. Sun and S. T. Wu, Route plannng of cruse mssle based on mproved partcle swarm algorthm, Bejng Unversty of Aeronautcs and Astronautcs, vol.37,no.10,pp , [10] M. L, D. B. Wang, T. T. Ba, and S. Z. Sheng, Route plannng based on partcle swarm optmzaton wth threat heurstc, Electroncs Optcs & Control,vol.18,no.12,pp.1 4,2011. [11] Z. C. Fan, Path plannng method based on the algorthm of and elastc rope for underwater vehcle n three-dmensonal space [Master thess], Harbn Engneerng Unversty, [12] X.-S. Yang, Nature-Inspred Metaheurstc Algorthms, Lunver Press, [13] X.-S. Yang, Frefly algorthms for multmodal optmzaton, n Stochastc Algorthms: Foundatons and Applcatons, vol.5792 of Lecture Notes n Computer Scences, pp , Sprnger, Berln, Germany, [14] X.-S. Yang, Frefly algorthm, stochastc test functons and desgn optmzaton, Internatonal Bo-Inspred Computaton,vol.2,no.2,pp.78 84,2010. [15] R. Dutta, R. Gangul, and V. Man, Explorng sospectral sprng-mass systems wth frefly algorthm, Proceedngs of The Royal Socety of London Seres A: Mathematcal, Physcal and Engneerng Scences,vol.467,no.2135,pp ,2011. [16] T. Mauder, C. Sandera, J. Stetna, and M. Seda, Optmzaton of the qualty of contnuously cast steel slabs usng the frefly algorthm, Materals and Technology,vol.45,no.4,pp , [17] P. Aungkulanon, P. Cha-ead, and P. Luangpaboon, Smulated manufacturng process mprovement va partcle swarm optmsaton and frefly algorthms, n Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts (IMECS 11), pp , March [18] T. Apostologpoulos and A. Vlachos, Applcaton of the frefly algorthm for solvng the economc emssons load dspatch problem, Internatonal Combnatorcs, vol. 2011, Artcle ID , 23 pages, [19] M.-H. Horng and T.-W. Jang, Multlevel mage thresholdng selecton based on the frefly algorthm, n Proceedngs of the 7th Internatonal Conference on Ubqutous Intellgence & Computng and 7th Internatonal Conference on Autonomc & Trusted Computng (UIC/ATC 10),pp.58 63,October2010. [20] I. Fster, I. Fster Jr., X.-S. Yang, and J. Brest, A comprehensve revew of frefly algorthms, Swarm and Evolutonary Computaton,vol.13,pp.34 46,2013. [21] I. Fster, X.-S. Yang, D. Fster, and I. Fster Jr., Frefly algorthm: a bref revew of the expandng lterature, n Cuckoo Search and Frefly Algorthm, vol.516ofstudes n Computatonal Intellgence,pp ,2014. [22] X.-S. Yang, Frefly algorthm, Lévy flghts and global optmzaton, n Research and Development n Intellgent Systems XXVI, pp ,Sprnger,London,UK,2010. [23] X.-S. Yang and S. Deb, Eagle strategy usng Levy walk and frefly algorthms for stochastc optmzaton, n Nature Inspred Cooperatve Strateges for Optmzaton, vol. 284, pp , Sprnger, Berln, Germany, [24] S. M. Farahan, A. A. Abshour, B. Nasr, and M. Meybod, Anmprovedfreflyalgorthmwthdrectedmovement, n Proceedngs of 4th IEEE Internatonal Conference on Computer Scence and Informaton Technology, pp , [25] L. dos Santos Coelho and D. L. de Andrade Bernert, A chaotc frefly algorthm appled to relablty-redundancy optmzaton, n Proceedngs of the IEEE Congress on Evolutonary Computaton, pp , June [26] C. Lu, Z. Gao, and W. Zhao, A new path plannng method basedonfreflyalgorthm, nproceedngs of the 5th Internatonal Jont Conference on Computatonal Scences and Optmzaton (CSO 12), pp , June [27] S. M. Farahan, A. Abshour, B. Nasr, and M. Meybod, A gaussan frefly algorthm, Internatonal Machne Learnng and Computng,vol.1,no.5,pp ,2011.

11 Advances n Operatons Research Advances n Decson Scences Appled Mathematcs Algebra Probablty and Statstcs The Scentfc World Journal Internatonal Dfferental Equatons Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Complex Analyss Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Dscrete Mathematcs Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Internatonal Stochastc Analyss Optmzaton