Obstacle Avoidance by Using Modified Hopfield Neural Network

Transcription

1 bstacle Avodance by Usng Modfed Hopfeld Neral Network Panrasee Rtthpravat Center of peraton for Feld Robotcs Development (FIB), Kng Mongkt s Unversty of Technology Thonbr. 91 Sksawas road Tongkr Bangkok Thaland. Phone (662) , Fax. (662) e-mal: pan@fbo.kmtt.ac.th Kenj Nakayama Dept. of Informaton and System Eng. Faclty of Engneerng, Kanazawa Unversty , Kodatsno, Kanazawa 920 Japan Phone (81) , Fax. (81) e-mal: nakayama@t.kanazawa-.ac.jp Abstract In ths paper, path plannng of a moble robot by sng a modfed Hopfeld neral network s stded. An area, whch excldes obstacles and allows gradally changng of actvaton level of nerons from a startng pont to a goal, s derved. Path can be constrcted n ths area by searchng the next hghest actvated neron. Even thogh asymmetrc weght matrx s sed, decreasng of system energy can be nvestgated. By comparson to a symmetrc weght network, smlaton reslts show more effectve path, whch cold be generated by ths algorthm. Smlaton reslts showed constrcted path from the startng pont to the goal, whch can avod obstacle sccessflly. Keywords a modfed Hopfeld neral network, Robot path plannng, Path generator 1. Introdcton bstacle avodance s one of the most nterested sses n moble robot feld. It has been appled on ether a sngle robot or mltple robot system. Many researchers proposed ther algorthms to solve obstacle avodance problem ether statc or dynamc envronment. Y. Ara, T. Fj, H. Asama, Y. Kataoka [1] proposed collson avodance n mltple robot system. Ther robots are eqpped wth LCISS. Each robot wll recognze whether another robot or obstacle. The rle matrx for collson avodance s predetermned. Renforcement learnng s appled to acqre adaptve behavor. Mochda, Ishgro, Aok and Uchkawa [2] presented a behavor control method nspred by lvng organsms whch mmne and emoton systems were ntrodced to cope wth dynamc changng envronment. The dea of emotonal system was sed to model frstraton fncton, whch wold be sed to determne robot behavor. Ishgro, Watanabe and Uchkawa [3] ntrodced sng an mmne system concept to cope wth dynamc changng envronment. Antbody and antgen matchng were sed to determne the robot behavor. Varos knds of robot behavor were represented as antbodes. The envronment nformaton was also set as antgens. The mmne network system selected an antbody, whch was most stable for crrent staton or an antgen. R. Mehrotra and D. M. Krase [4] presented an obstacle free path plannng by sng a qadtree codng method. M. G. Lagodaks [5] proposed dynamc path plannng and obstacle avodance by sng the Hopfeld neral network. The dea of receptve feld s presented. The external npt s added n a goal and obstacles. Sch the target s hghest actvated neron. The actvaton levels of nerons are spread decreasngly arond the target as wave propagaton. However the Hopfeld neral network cannot garantee the monotoncally decreasng from the goal becase t employs symmetrcal connecton weghts. Actvaton level can propagate both drectons. Ths easly cases local peaks of actvaton level. In ths paper, an asymmetrc weght matrx s proposed n order to form a dstngshable area of actvaton level from the startng pont to the goal. The path can be generated easly by consderng hghest actvated nerons n ths area. Next secton, the Hopfeld network and selected actvaton fncton are descrbed. Then weght matrx determnaton s presented. System model, whch compose of the Hopfeld network and a path generator, s expressed. Smlaton reslts and dscsson of the paper are provded. 2. Hopfeld Network The Hopfeld neral network was proposed n1982, by physcst John J. Hopfeld [6]. It can be sed to solve nformaton retreval or optmzaton problems. It has a recrrent featre. tpt of all another nerons, v j, are fed back, weghted and smmed to be the npt to a neron. nly one neron,, s selected randomly and pdated ts state n each tme.

2 where = n j= 1 j w j v j + θ. (1) s npt potental of neron v s otpt of neron j j w j s connecton weght from neron j to θ s bas of neron Accordng to M. G. Lagodaks [5], external bas are added n the goal and obstacle nts as: By sng the hyperbolc fncton, the actvaton fncton can be presented as: 1 e 1 + e 0 φ ( ) =. (6) 0 < 0 Fg. 1 shows actvaton fnctons n each form. The actvaton fncton sed n ths paper s shown wth a thckest lne n ths fgre. + θ ( t) = 0 t arget nt obstacle nt. (2) otherwse The effect of addng bas n Eq(2) s that the goal nt wll be maxmally actvated. The obstacle nt s deactvated ndependently of other nts. The orgnal Hopfeld network assmes zero self-coplng, w = 0, and symmetrcal weghts, w = j w j. Ths symmetrcal property garantees that the system energy fncton always converges to the basn of attracton or the eqlbrm state. Ths means no state changed anymore. Many forms of actvaton fnctons can be sed to classfy an otpt level of a neron accordance wth ts npt. In the dscrete Hopfeld network, we can se a sgn fncton to classfy otpt to be ether 1 or 0 as: 1 ( ) = 0 0 < 0 sgn. (3) Sch otpt s dscrete wth vale {1,0}. The otpt s 1 means the neron s actvated or fred. Whle the otpt s 0 means the neron s deactvated or qenched. For the analog Hopfeld network, a sgmod fncton or a hyperbolc fncton can be sed. For the sgmod fncton, the otpt s analog wth vale [0,1] as: φ. (4) 1 ( ) = 1 + e For the hyperbolc fncton, the otpt s analog wth vale [-1,1] as: 1 e ( ) = 1 + e φ. (5) In ths paper, system state s set wth vale [0,1]. Fg. 1 shows actvaton fnctons 3. Weght Matrx Determnaton M. G. Lagodaks [5] mapped the Confgraton Space, C nto neron space. Each neron corresponds to sbset C of C. Sch all nerons can be represented as overall Confgraton Space. Each neron has a receptve feld, RF whch s sbset of nts of ts neghborhood. Each nt connects to the nts n ts RF. There are no connectons otsde the receptve feld. The symmetrcal property s concerned sch f any nt s n RF j then nt j s n RF and vce versa. Weght s determned as a decreasng fncton, whch depends on Ecldean dstance ρ (, j) between the nts. Weght between nt and nt j can be represented as: w j = f ( ρ(, j)), (7) The fncton can take varos forms as: f( ρ) α γρ = e. (8) where γ s a postve real nmber

3 α s sed to span the wdth of a decreasng fncton. It may be 2, 4 or more. In ths paper, weght matrx s determned by sng the dstance between the goal and each neron. The strongest weght wll be set on a neron, whch s nearest to the goal drecton. The weght wll be decreased spread ot of the strongest weght as shown n Fg. 2 k-1 k k th 5th 4th 1st G Goal drecton Fg. 2 shows relatons of weght n each nt As shown n Fg. 2, the largest weght s the weght from a neron (k,) to a neron (k-1,+1), whch locates toward the goal drecton. The connecton weghts from the neron (k,) to (k-1,) and (k,+1) have the 2 nd largest vale. By settng the weght n ths manner, nerons, whch close to the goal drecton, have hgher actvaton level and are actvated easly. Sch the level of actvatng can be dstngshed obvosly. However the symmetrcal property cold not be conserved as shown n Fg.3. The weght from a neron (k+1,+1) to (k,) s stronger than weght from the neron (k,) to (k+1,+1). composed the modfed Hopfeld network and the path generator as shown n Fg. 4. Modfed Hopfeld Network Fg. 4 shows system model Path Generator 5. Mechansm of Path Generator In the modfed Hopfeld neral network, the startng pont and goal states are always set to be maxmally actvated nerons. Sch there are twopeaked srface n an actvated neron space. The levels of actvated nerons are propagated decreasngly from them. The obstacles are also set to be deactvated nerons and they show the valleylke. By settng the connecton weght as dscssed above, the levels of actvated nerons can be dstngshed obvosly from the startng pont to the goal. By searchng the hghest actvated neron from the startng pont to the goal, path can be easly obtaned. However n the Hopfeld neron network, the hgh nt state trends to actvate ts neghborng nerons. Some nerons are actvated mproperly. For example, the goal s set n one sde of a wall of obstacles. The startng pont s set to another sde of the wall as shown n Fg k-1 k k+1 1st Fg. 3 shows asymmetrc weght featre As expressed n Eq (8), the connecton wegh can be set accordance wth the nearness to the goal by varyng γ n each drecton. 4. System Model As dscssed above, the level of actvated neron can be separated notceably. Path can be generated from the hghest actvated neron from the startng pont to the goal. Sch system model s G 1st Fg. 5 Startng pont (S) and goal (G) are set n dfferent sde of wall of obstacles (). The startng, goal and obstacle nts are marked by S, G and, respectvely. The nerons, whch are n corner, may be hghly actvated becase they close to the goal and the startng pont. Sch selectng hghest neron from ts neghborhood s nsffcent condton for constrctng the good path. good path means there are less nmbers of steppng to move from the startng pont to the goal. Path generator shold concern next step of ts neghborng nerons. The startng neron s set to be a crrent nt. By consderng ts closet nerons, two hghest actvated nerons are searched. The level of actvatng of ther adjacent nts, excldng the crrent neron and ts neghborhood, are compared

4 and selected the hghest one. The selected neron wll be pdated to be the crrent nt. Followng ths manner, the path can be constrcted. Not only consderng the next step of ts neghborng nts bt the path generator shold also concern preventon of movng back to the prevos nt n order to avod loop constrctng. In order to compare the level of actvatng of adjacent nts, a dagram of crrent neron (k,) and ts vcnty are presented n Fg. 6. shown n Fgs Another three applcatons are smlated to show an effectve path generated by or model. The smlaton reslts are shown n Fgs k-2 k-1 k k+1 k+2 Fg. 7a shows system states and system energy for symmetrc weght matrx approach. Fg. 6 shows a neron (k,) and ts vcnty There are 8 neghborng nerons arond the neron (k,). Each neghborng nt has ts nearby nts. Neron (k-1,-1) has 5 next closest nts,.e., (k,-2), (k-1,-2), (k-2,-2), (k-2,-1), (k-2,). Neron (k,-1) has 3 next neghborng nts,.e., (k+1,-2), (k,-2), (k-1,-2). By consderng n ths way, the adjonng nts n each drecton can be obtaned. Three hghest nts n each drecton are averaged. Two of them correspondng to two hghest neghborng nts are sed n comparng for next step selecton. 6. Smlaton Reslts The smlaton reslts are composed of 2 parts,.e., comparson to a symmetrc weght matrx and applyng to another applcatons. A network composes of 10x10 nerons. The startng, goal and obstacle nts are marked by S, G and, respectvely. For the modfed Hopfeld network, the startng and goal states are fxed to be maxmally actvated nerons. The obstacle states are set to be deactvated nts. Intal state of another nerons are selected randomly and scaled n a range of [0,0.5]. The smlaton was done ntl approached to an eqlbrm state. The srface of eqlbrm states s plotted for more nderstandng. The system energy s shown to nvestgaton of the decreasng of energy. Path generator constrcts a path from the actvated nerons n an eqlbrm state. The constrcted path s mapped to a Confgraton space of robot wth 14x14 nts. By comparson to a symmetrc weght matrx, weght matrx determnaton by sng M. G. Lagodaks [5] method s sed. The envronment as shown n Fg. 5 s tested. The smlaton reslts are Fg. 7b shows constrcted path by sng symmetrc weght matrx approach for applcaton 1. Fg. 8a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 1.

5 Fg. 8b shows constrcted path by sng asymmetrc weght matrx approach for applcaton 1. Fg. 10a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 3. Fg. 9a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 2. Fg.10b shows constrcted path by sng asymmetrc weght matrx approach for applcaton 3. Fg. 9b shows constrcted path by sng asymmetrc weght matrx approach for applcaton 2. Fg. 11a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 3. (600 steps of learnng)

6 Fg.11b shows constrcted path for applcaton 3. (600 steps of learnng) Fg. 13a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 4. (500 steps of learnng) Fg. 12a shows system states and system energy by sng asymmetrc weght matrx approach for applcaton 4. Fg. 12b shows constrcted path by sng asymmetrc weght matrx approach for applcaton 4. Fg.13b shows constrcted path for applcaton 4. (500 steps of learnng) 7. Dscsson By comparng to the symmetrc method, the smlaton reslts show more effectve paths generated by the modfed Hopfeld network as shown n Fgs. 7 and 8. A symmetrc property leads to propagaton of the actvaton level arond the goal. It s qte dffclt to constrct a good path n the meanng of less nmber of steppng. In Fgs. 10 and 12, the system energy was decreased toward the mnmm pont and grown p after long learnng. Ths s a reslt of asymmetrc weght matrx. Ths decreasng of the system energy cannot garantee. By learnng ntl the mnmm of system energy, the path was constrcted more effectve than long learnng as shown n Fgs. 11 and 13. Sch path can be constrcted effectvely by leanng ntl t approachs to the mnmm energy pont. Learnng ntl t approaches to the Eqlbrm state s not necessary for or method. Path can be generated n a shorter tme than sng the orgnal Hopfeld network.

7 8. Conclson In ths research, obstacle avodance for moble robot s stded. Asymmetrc weght matrx has been ntrodced. Even thogh the decreasng of the system energy cannot be garanteed, t can be demonstrated. The modfed Hopfeld network wll be learned ntl the mnmm energy pont. Smlaton reslts showed that path generator constrcted an effectve path. Ths leads to redcng of path plannng tme. 10. Reference [1] Y. Ara, T. Fj, H. Asama, Y. Kataoka, Adaptve Behavor Acqston of Collson Avodance among Mltple Atonomos Moble Robots, Proceedng of IRS, 1997 [2] Mochda, T.; Ishgro, A.; Aok, T.; Uchkawa, Y., Behavor arbtraton for atonomos moble robots sng emoton mechansms, Internatonal Conference on Intellgent Robots and Systems, IRS, Pttsbrgh, Pennsylvana, USA,1995 [3] Ishgro, A., Watanabe, R., Uchkawa, Y., An mmnologcal approach to dynamc behavor control for atonomos moble robots, Internatonal Conference on Intellgent Robots and Systems, IRS, Pttsbrgh, Pennsylvana, USA, 1995 [4] R. Mehrotra and D. M. Krase, bstacle-free Path Plannng for Moble Robots [5] M. G. Lagodaks, Hopfeld Neral Network for Dynamc Path Plannng and bstacle Avodance, Techncal Paper, Unversty of Sothwestern Losana. [6] S. Haykn, Neral Networks: A Comprehensve Fondaton, Prentce Hall Internatonal, Inc., New Jersy