G3.6 The Evolutionary Planner/Navigator in a mobile robot environment

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1 Engineering G3.6 The Evlutinary Planner/Navigatr in a mbile rbt envirnment Jing Xia Abstract Based n evlutinary cmputatin cncepts, the Evlutinary Planner/Navigatr (EP/N) represents a new apprach t path planning and navigatin. The majr advantages f the EP/N include being able t achieve bth near-ptimality f paths and high planning efficiency, being able t accmmdate different ptimizatin criteria, being flexible t changes, and being rbust t uncertainties. The EP/N unifies ff-line planning and n-line planning/navigatin prcesses in the same evlutinary algrithm t deal with unknwns in an envirnment gracefully and flexibly. It prvides high safety fr the rbt withut requiring cmplete infrmatin abut the envirnment. G3.6.1 Prject verview The mtin planning prblem fr mbile rbts is typically frmulated as fllws (Yap 1987): given a rbt and a descriptin f an envirnment, plan a path f the rbt between tw specified lcatins which is cllisin free and satisfies certain ptimizatin criteria. Traditinally there are tw appraches t the prblem: ff-line planning, which assumes a perfectly knwn and stable envirnment, and n-line planning, which fcuses n dealing with uncertainties when the rbt traverses the envirnment. On-line planning is als referred t by many researchers as the navigatin prblem. (Althugh sme researchers als interpret navigatin as a lw-level cntrl prblem fr path fllwing, we d nt use such an interpretatin here.) A great deal f research has been dne in mtin planning and navigatin (see Yap 1987 and Latmbe 1991 fr surveys). Hwever, different existing methds encunter ne r many f the fllwing difficulties: high cmputatin expenses inflexibility in respnding t changes in the envirnment inflexibility in respnding t different ptimizatin gals inflexibility in respnding t uncertainties inability t cmbine advantages f glbal planning and reactive planning. The EP/N system was develped t address these difficulties; the inspiratin t use evlutinary techniques was triggered by the fllwing ideas/bservatins: Randmized search can be the mst effective in dealing with NP-hard prblems and in escaping lcal minima. Parallel search actins nt nly prvide great speed but als prvide grund fr interactins amng search actins t achieve even greater efficiency in ptimizatin. Creative applicatin f the evlutinary cmputatin cncept rather than dgmatic impsitin f a standard algrithm prves t be mre effective in slving specific types f real prblems. Intelligent behavir is the result f a cllectin f simple reactins t a cmplex wrld. A planner can be greatly simplified, much mre efficient and flexible, and increase the quality f search, if search is nt cnfined t be within a specific map structure. It is mre meaningful t equip a planner with the flexibility f changing the ptimizatin gals than the ability f finding the abslutely ptimum slutin fr a single, particular gal. c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:1

2 The Evlutinary Planner/Navigatr in a mbile rbt envirnment The EP/N embdies the abve ideas by fllwing the evlutin prgram apprach, that is, cmbining the cncept f evlutinary cmputatin with prblem specific chrmsme structures and genetic peratrs (Michalewicz 1994). With such an apprach, the EP/N pursues all the advantages as described abve. Less bvius, thugh, is that, with the unique design f chrmsme structure and genetic peratrs, the EP/N des nt need a discretized map fr search, which is usually required by ther planners. Instead, the EP/N searches the riginal and cntinuus envirnment by generating paths based n evlutinary cmputatin. The bjects in the envirnment can simply be indicated as a cllectin f straight-line walls. This representatin accmmdates bth knwn bjects and partial infrmatin f unknwn bjects btained frm sensing. Thus, there is little difference between ff-line planning and n-line navigatin fr the EP/N. In fact, the EP/N unifies ff-line planning and n-line navigatin in the same evlutinary algrithm and chrmsme structure. The structure f the EP/N is shwn in figure G3.6.1, where FEG the ff-line evlutinary algrithm, and NEG the n-line evlutinary algrithm are essentially the same evlutinary algrithm as t be described. The nly difference between FEG and NEG is in certain values f parameters (see sectin G3.6.5) ne may chse. The different parameter values are t accmmdate slightly different bjectives f FEG and NEG: FEG emphasizes the ptimality f a path while NEG emphasizes the swiftness in generating a feasible path. Nte that bth FEG and NEG perfrm glbal planning, and NEG generates an alternative subpath by glbal planning based n the updated knwledge f the envirnment btained frm sensing. Mrever, if n bject is initially knwn in the envirnment, then FEG will generate a straight-line path with just tw ndes: the start and the gal lcatins. It will slely depend n the NEG t lead the rbt twards the gal while aviding unknwn r newly emerged bstacles. Task Basic Map FEG Path Alternative Subpath Path Selectr Cntrller Actin NEG Sensing Envirnment Figure G The EP/N structure. G3.6.2 Design prcess We nw describe the evlutinary algrithm which bth FEG and NEG adpt in detail. G Chrmsmes and initializatin A path cnsists f ne r mre straight-line segments, with the starting lcatin, the gal lcatin, and (pssibly) the intersectin lcatins f tw adjacent segments defining the ndes. A feasible path cnsists f nly feasible ndes. An infeasible path has at least ne infeasible nde which is either nt cnnectable t the next nde n the path due t bstacles r lcated inside sme bstacle. Chrmsmes are represented as rdered lists f path ndes: each nde, apart frm the pinter t the next nde, cnsists f x and y crdinates f the knt pint and a state variable b, which indicates whether r nt the nde is feasible (figure G3.6.2). Each chrmsme can have a varied number f ndes, which prvides great flexibility. The methds fr checking the feasibility f a nde (i.e. lcatin validity and cnnectivity) are relatively simple and are based n algrithms described by Pavlidis (1982). The initializatin f chrmsmes is a randm prcess subject t the fllwing input parameters: a ppulatin size P and the maximum number f ndes in a chrmsme N. Fr each chrmsme, a randm number is generated within [2,N] t determine its length, that is, the number f ndes. The crdinates x and y are als created randmly fr each nde f such a chrmsme within the cnfine f the envirnment. P chrmsmes are generated in this way. c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:2

3 The Evlutinary Planner/Navigatr in a mbile rbt envirnment x y b x y b n n n Figure G A chrmsme representing a path. G Evaluatin The evaluatin functin Path Cst(p) measures the path cst f a chrmsme p. Since p can be either feasible r infeasible, we adpt tw separate evaluatin functins eval f and eval u t handle the feasible and infeasible cases respectively. Our design f the evaluatin functin Path Cst(p) has gne thrugh a lng prcess f develpment, as will be discussed in sectin The eval f and eval u t be described are the mst recent results f such develpment. It seems t be relatively easy t cmpare tw feasible paths. Intuitively, we think eval f shuld be a functin f the ttal length f a path dist, its smthness smth and the clearance clear between the path and the surrunding bstacles. There can be many ways t define the functin eval f. At present, we simply define it as the linear cmbinatin f dist, smth, and clear: eval f (p) = w d dist(p) + w s smth(p) + w c clear(p) where the cnstants w d, w s, and w c represent the weights n the ttal cst f the path s length, smthness, and clearance, respectively. We define dist, smth, and clear as the fllwing: dist(p) = n 1 i=1 d(m i,m i+1 ), the ttal length f the path, where d(m i,m i+1 ) dentes the distance between tw adjacent path ndes m i and m i+1. smth(p) = max n 1 i=2 s(m i), the maximum curvature at a knt pint, where curvature is defined as θ i s(m i ) = min{d(m i 1,m i ), d(m i,m i+1 )} and θ i [0,π] is the angle between the extensin f the line segment cnnecting ndes m i 1 and m i and the line segment cnnecting ndes m i and m i+1 (figure G3.6.3). clear(p) = max n 1 i=1 c i, where { gi τ if g c i = i τ e a(τ gi) 1 therwise g i is the smallest distance frm the segment m i m i+1 t all detected bjects, τ is a parameter defining a safe distance, and a is a cefficient. With this frmulatin, ur gal is t minimize the functin eval f. m i θ i m i-1 m i+1 Figure G θ i at each nde m i. We tk int accunt several factrs in the design f eval u : the number f intersectins f a path with bstacles, the depth f intersectin (i.e. hw deeply a path cuts thrugh bstacles), the rati between the numbers f feasible and infeasible segments, the ttal lengths f feasible and infeasible segments, and s n, and implemented tw designs fr eval u. One design f eval u is as fllws: eval u (q) = µ + η c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:3

4 The Evlutinary Planner/Navigatr in a mbile rbt envirnment where µ is the number f intersectins f a whle path with bstacles and η is the average number f intersectins per infeasible segment. With this evaluatin functin, the path csts f the three paths frm figure G3.6.4 are eval u (p) = = 4 eval u (q) = = 6 eval u (r) = = 8 which match ur intuitin: path p is the ne which will generate a feasible ffspring mst easily, and path q is much mre prmising than path r. This eval u, hwever, may nt be perfect, since q culd be cnsidered the best amng the three paths (i.e. it shuld have the lwest cst) frm a different perspective. q s r e p Figure G Three infeasible paths p, q, and r. The ther design makes eval u equal t the summatin f all the penetratin distances, where a penetratin distance D is defined as the minimum distance t mve an infeasible path segment ut f an bstacle it penetrates. This design is reasnable in almst all cases but is mre cmputatinally expensive than the first design. Assciated with this apprach f designing eval u independently f eval f is the issue f hw t cmpare feasible paths against infeasible nes. This issue requires the answer f the fllwing questin: Is any feasible slutin better than any infeasible ne? In the EP/N, we have chsen the (smewhat risky) answer yes, which makes such cmparisns relatively easy fr us and is als cnsistent with ur designs f eval u. With this chice, we add t the value f eval u f any infeasible path p a cnstant ρ (within a given generatin f the evlutinary prcess) t make the path less attractive than a feasible ne: ρ = max{0, max p F {eval f(p)} min q U {eval u(q)}} where F and U dente the sets f feasible and infeasible paths respectively. Nte that ρ measures the difference between the wrst feasible and the best infeasible paths. In actual implementatin, we d nt really cmpute ρ; instead, we simply srt the feasible paths and infeasible paths separately frm the best t the wrst based n their separate evaluatin functins. Then, we append the srted list f infeasible paths at the tail f the srted list f feasible paths. (This wrks with any ranking selectin.) c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:4

5 The Evlutinary Planner/Navigatr in a mbile rbt envirnment G Genetic peratrs The current versin f EP/N uses eight types f genetic peratr t evlve chrmsmes int pssibly better nes. These peratrs are sufficient t generate an arbitrary path, but may nt all be needed in all situatins. The applicatin f each peratr is cntrlled by a prbability. Hw t select the best cmbinatin f peratrs, that is, hw t determine thse prbabilities, very much depends n envirnmental characteristics and specific cnstraints impsed n a task. Our current versin f EP/N is able t feed back hw useful an peratr is, which helps us in determining the prbabilities. Hwever, mre research is needed (see G3.7.5). Frm ur current experience n fairly cmplex envirnments, the EP/N system perfrmed the best with all eight types f peratr present with cnsiderable prbabilities (e.g. in the range ). Nw we intrduce each type f peratr, as illustrated in figure G3.6.5: crssver: recmbines tw (parent) paths int tw new paths. The parent paths are divided randmly int tw parts respectively and recmbined: the first part f the first path with the secnd part f the secnd path, and the first part f the secnd path with the secnd part f the first path. Nte that there can be different numbers f ndes in the tw parent paths. mutatin 1: used fr fine tuning nde crdinates in a path fr shape adjustment. mutatin 2: used fr large change f nde crdinates in a path. insertin: inserts new ndes int a path. deletin: deletes ndes frm a path. swap: swaps the crdinates f selected adjacent ndes in a path. smth: smths turns f a feasible path by cutting crners, that is, fr a selected nde, the peratr inserts tw new ndes n the tw path segments cnnected t that nde respectively and deletes that selected nde. repair: repairs an infeasible segment in a path by pulling the segment arund its intersecting bstacles. crssver mutatin_1 mutatin_2 insertin deletin swap smth repair Figure G The rles f the genetic peratrs. c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:5

6 The Evlutinary Planner/Navigatr in a mbile rbt envirnment Nte that we deliberately left ut details n hw exactly ndes were selected and changed in many peratrs, since such decisins culd be made in varius ways frm purely randm t incrprating much heuristic knwledge. In the earlier EP/N, such decisins were made mstly randmly. The current EP/N is equipped with versins f peratrs using mre knwledge. Fr example, it has tw versins f mutatin 1. The first versin changes the crdinates f a nde randmly within sme bunds which decrease as evlutin prceeds; it applies t any path. The secnd versin, hwever, applies t nly a feasible path, and it changes the crdinates f a (feasible) nde randmly within sme lcal clearance f the path s that the path remains feasible afterwards. Bth versins select ndes randmly. The merits f different types and versins f peratr will be further discussed in G G Reprductin Fr the selectin prcess, a ppulatin f P chrmsmes are first srted based n their fitness values (i.e. cst values) frm the best t the wrst, and a rulette wheel f P slts is then prduced with the ith slt C2.2 sized prprtinal t the fitness value f the ith chrmsme (Michalewicz 1994; als see Sectin C2.2). By spinning the wheel, the chrmsmes which have better fitness values (i.e. lwer cst values) will have better chances t be selected fr reprductin. In rder t be mre efficient, in a later versin f the EP/N, we adpted a fixed rulette wheel f P slts with linearly decreasing slt sizes instead f generating a different rulette wheel at each generatin. In this way, the chance fr a chrmsme t be selected is nt necessarily prprtinal t its fitness value but is still better than the chance fr a wrse chrmsme t be selected. Generally, a parameter S P determines the number f chrmsmes t be selected fr reprductin. At generatin t, the selected S chrmsmes frm the ppulatin P(t) are altered by the genetic peratrs t generate S ffspring. The S ffspring plus the P S best chrmsmes in the riginal ppulatin P(t) frm the next generatin f ppulatin P(t +1). In ur latest versin f EP/N, nly ne genetic peratr is used at each generatin, and S = 1 (r 2 if the peratr chsen is crssver). The selectin f peratrs is als based n a rulette wheel with slts sized prprtinal t the prbabilities f the peratrs. Nte that in this versin the time perid fr a single generatin is the shrtest. G3.6.3 Develpment and implementatin The develpment f the EP/N is an ever-living evlutin prcess itself: different ideas have been experimented with and many imprvements have been made since the earliest versin, but there are still many new ideas and features that can be incrprated in the EP/N system (see sectin G3.6.5). Instead f seeking a cmplete prduct, we see the EP/N mre as representing a new directin, alng which there are many new hpes but als new challenges, and a new framewrk, under which these new hpes, in terms f new ideas and strategies, can be explred and tested, and the new challenges can be dealt with. Indeed, we have already discvered a mixture f hpes and challenges s far. G Develpment f fitness functin The earliest versin f the EP/N (Lin 1993, Lin et al 1994a, 1994b) can be characterized as having a single fitness criterin and a simple penalty functin. Only the shrtest distance criterin was used: the C5.2 path cst was simply the length f the path, and a path was better than anther ne if it was shrter. (This was just as in many traditinal appraches t path planning.) Infeasible paths were penalized by adding large penalty cnstants t their csts, making their lengths exceedingly lng. Such treatment hampered the ability f the EP/N t wrk well in difficult envirnments because f its many drawbacks. First, the shrtest path may nt be safe, that is, sufficiently away frm bstacles, and it may nt be mre efficient than a lnger path if it is nt smth. Fr example, in figure G3.6.6, the path q is lnger than p but is bviusly better. Hence, we changed the evaluatin (r fitness) functin t include factrs f clearance and smthness. We experimented with varius ways f defining the fitness functin and encuntered the prblem f hw t evaluate the fitness f an infeasible path, fr which clearance and smthness d nt make much sense. This investigatin deepened ur understanding f the prblems intrduced by using simple penalties t discriminate against infeasible paths. c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:6

7 The Evlutinary Planner/Navigatr in a mbile rbt envirnment s p q e Figure G The lnger path is better. q s p e Figure G Tw infeasible paths. The majr prblem with using a simple penalty functin is that it des nt prvide a reasnable basis fr cmparing tw infeasible paths, since the merit f a path is nt merely reflected by its length and what makes ne feasible path better than anther simply may nt be applicable t the cmparisn f tw infeasible paths. (In fact, as we d nt even cnsider cmparing tw infeasible paths in ur daily lives, we have much less intuitin t help us than in the case f cmparing tw feasible nes.) Fr example, in figure G3.6.7, path p has the shrtest distance (a straight line) and a perfect smthness, if smthness is cunted. The ther path q has lnger distance and wrse smthness. Thus, with the same cnstant penalty n bth paths, p will be ranked better than q, althugh it seems that q is actually better in the sense that q can be mutated int a feasible path relatively easily. One may ask what will happen if we simply eliminate infeasible paths altgether and nly evlve the feasible nes. Unfrtunately, in ur prblem, except fr cases with very simple envirnments which have nly a few bstacles, the randmly generated initial ppulatin usually cnsists f infeasible paths nly, and since the feasible slutin space is nncnvex and has a cmplex bundary depending n bstacles, it is ften mre difficult t prduce/reprduce nly feasible paths than t deal with infeasible nes. Therefre, evaluating infeasible paths is extremely imprtant and almst inevitable. Anther imprtant incentive fr evlving infeasible paths is that it can speed up the search fr the ptimum slutin by prviding shrtcuts acrss the infeasible slutin space. Hence, ur investigatin results in the current slutin f evlving feasible and infeasible paths by separate fitness functins as described in sectin G G Develpment f peratrs Initially, the first six peratrs were used in the EP/N. Different schemes and prbabilities f applying thse peratrs were experimented, and the effects f the peratrs were investigated. We later added the smth peratr t imprve feasible paths. We tested the peratrs in different envirnments and fund that fr cmplex planning tasks in certain cmplex envirnments purely randm peratrs did nt wrk very well. This led us t design peratrs using mre knwledge abut the envirnment. The repair peratr was intrduced using knwledge f bstacles. In fact, we fund that much f the knwledge c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:7

8 The Evlutinary Planner/Navigatr in a mbile rbt envirnment needed by mre intelligent peratrs had already been made available during evaluatin f path fitness. In the latest versin f EP/N, we added new versins fr mutatin 1 (as explained in sectin G abut genetic peratrs), as well as fr deletin and smth using such knwledge. Our experience shwed that repair was highly effective in generating feasible paths; smth and the mre intelligent versin f mutatin 1 were highly effective in imprving feasible paths; crssver was cnsistently effective in evlving bth infeasible and feasible paths. This was particularly imprtant since crssver was cmpletely randm. Its simplicity als seemed t speed up cnsiderably the evlutin prcess. The nly peratr that remved ndes frm a path was deletin, which thus was highly effective in keeping the EP/N system efficient (in time and space) and allwing ther peratrs t be active. It seemed that the cmbined effrt f different peratrs generally wrked better in cmplex situatins. As mentined in sectin G , hw t determine the best cmbinatin f peratrs (i.e. prbabilities) is nt a trivial issue and is definitely ne f the majr future research tpics (see sectin G3.6.5). G Implementatin The earlier versins f the EP/N prgram were run n 486 r Pentium PCs. The later versins f the EP/N were run under Unix n Sun SparcStatins. N cmmercial EA tls were used. (a) (b) (c) (d) Figure G (a) T = 0: paths are generated randmly. (b) T = 100: evlutin has taken 0.91 secnds. (c) T = 600: evlutin has taken secnds; the best path has 25 ndes and a cst f (d) T = 1000: evlutin has taken secnds; the best path has 20 ndes and a cst f G3.6.4 Results In figures G3.6.8 G3.6.11, we present sme ff-line planning results btained frm running the latest versin f the EP/N system n a Sun Sparc 20 in different envirnments with the same set f parameter values as fllws: c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:8

9 The Evlutinary Planner/Navigatr in a mbile rbt envirnment (a) (b) (c) (d) Figure G (a) T = 0: paths are generated randmly. (b) T = 150: evlutin has taken 2.13 secnds. (c) T = 300: evlutin has taken 3.92 secnds; the best path has fur ndes and a cst f (d) T = 500: evlutin has taken 7.06 secnds; the best path has five ndes and a cst f prbabilities f applicatin fr peratrs crssver, mutatin 1, mutatin 2, insertin, deletin, swap, smth, and repair are 0.6, 0.8, 0.5, 0.5, 0.5, 0.5, 0.9, and 0.8 respectively ppulatin size is 30 cefficients w d,w s,w c,a, and τ in the evaluatin functin eval f (p) are 1.0, 1.0, 1.0, 7.0, and 10 respectively. Snapshts were taken at fur different states, indicated by fur different values f generatin index T, f evlutin fr each task/envirnment, where tw-thirds f the ppulatin were displayed at states (a) and (b), and nly the best path was displayed at states (c) and (d). Despite the fact that the parameter values were chsen rather arbitrarily and the same ne size fit all values were applied t different envirnments with n individual adjustment, the EP/N system perfrmed quite well as clearly shwn by the results. Especially ntewrthy is the efficiency the EP/N demnstrated in finding feasible paths as shwn in states (b) and the near-ptimal paths as shwn in states (c). Frm states (c) t states (d), hwever, the pace f evlutin was much slwed as expected. G3.6.5 Cnclusins The EP/N represents a prmising new apprach in rbt planning which is full f ptential and a new applicatin f evlutinary cmputatin cncepts which is full f interesting challenges. The EP/N is remarkably rbust despite imperfectins in the design f evaluatin functins, the design and applicatin (i.e. prbabilities) f genetic peratrs, and the like. It cnfirms the nature and advantage f an evlutinary system. One imprtant issue in future research is hw t further use dmain knwledge (i.e. specific envirnmental knwledge) effectively in the EP/N system t imprve perfrmance. Althugh in ur latest c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:9

10 The Evlutinary Planner/Navigatr in a mbile rbt envirnment (a) (b) (c) (d) Figure G (a) T = 0: paths are generated randmly. (b) T = 100: evlutin has taken 1.60 secnds. (c) T = 350: evlutin has taken 9.88 secnds; the best path has 19 ndes and a cst f (d) T = 1000: evlutin has taken secnds; the best path has 17 ndes and a cst f versin f the EP/N we incrprated dmain knwledge in bth fitness evaluatin and genetic peratrs, there are ther cmpnents/prcesses, such as initializatin prcess and determinatin f parameter values, which may benefit frm dmain knwledge. Fr example, rather than randm initializatin, an initial ppulatin may cnsist f (i) a set f paths created by mutating r repairing the shrtest path between start and gal lcatins r (ii) sme mixture f chrmsmes having randmly generated crdinates and chrmsmes having crdinates with prblem specific knwledge as btained frm (i). It is highly desirable t make the EP/N capable f adapting its parameter values based n dmain knwledge and the states f evlutin. Currently, all peratrs f the EP/N have cnstant prbabilities f applicatin, which are fixed at the beginning f an evlutin prcess. Hwever, different peratrs may have different impacts (rles) at different stages f the evlutin prcess due t different situatins encuntered in an envirnment. Fr example, in n-line navigatin, if the rbt fllws the current best path withut running int any unexpected bstacles, the significance f mutatin 1 shuld grw, whereas the prbability f mutatin 2 shuld be kept at the minimum level. On the ther hand, if the rbt is trapped in sme lcatin f the envirnment (e.g. surrunded by previusly unknwn bstacles), the prbability f mutatin 2 shuld increase; at the same time the significance f mutatin 1 culd shrink. While the rle f repair shuld be very significant at the early stage f evlutin, the rle f smth shuld becme mre significant at the later stage when the ppulatin cnsists f mre feasible chrmsmes. Similar bservatins and cmments culd be made fr the ther parameters. Anther imprtant issue is t imprve the rganizatin f the EP/N t stress adaptability and learning fr n-line navigatin. Fr example, instead f generating a subpath fr the rbt t get arund an bstacle as is the case in the current versin, the NEG may simply generate an alternative path fr the rbt t reach its gal, where the path is based n the past experience, which can be a pl f feasible paths btained previusly. It culd als be interesting t study ther frms f memry, such as ne based c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:10

11 The Evlutinary Planner/Navigatr in a mbile rbt envirnment (a) (b) (c) (d) Figure G (a) T = 0: paths are generated randmly. (b) T = 60: evlutin has taken 1.74 secnds. (c) T = 150: evlutin has taken 6.19 secnds; the best path has seven ndes and a cst f (d) T = 400: evlutin has taken secnds; the best path has six ndes and a cst f n multichrmsme structures with a dminance functin (Gldberg 1989) r ne emplying machine learning techniques. Acknwledgement The authr wuld like t thank Zbigniew Michalewicz and Lixin Zhang fr their imprtant cntributin t the imprvement and implementatin f the latest versin f the EP/N. References Gldberg D E 1989 Genetic Algrithms in Search, Optimizatin and Machine Learning (Reading, MA: Addisn Wesley) Latmbe J C 1991 Rbt Mtin Planning (Deventer: Kluwer) Lin H-S 1993 Dynamic Path Planning fr a Mbile Rbt Using Evlutin Prgramming Master Thesis, UNCC Lin H-S, Xia J and Michalewicz Z 1994a Evlutinary navigatr fr a mbile rbt Prc. IEEE Int. Cnf. Rbtics and Autmatin (San Dieg, 1994) (Piscataway, NJ: IEEE) pp b Evlutinary algrithm fr path planning in mbile rbt rbt envirnment Prc. 1st IEEE Cnf. n Evlutinary Cmputatin (part f the IEEE Wrld Cngress n Cmputatinal Intelligence) (Orland, FL, 1994) (Piscataway, NJ: IEEE) pp Michalewicz Z 1994 Genetic Algrithms + Data Structures = Evlutin Prgrams 2nd edn (Berlin: Springer) Pavlidid T 1982 Algrithms fr Graphics and Image Prcessing (New Yrk: Cmputer Science) Yap C-K 1987 Algrithmic mtin planning Advances in Rbtics, vl 1: Algrithmic and Gemetric Aspects f Rbtics ed J T Schwartz and C-K Yap (Hillsdale, NJ: Erlbaum) pp c 1997 IOP Publishing Ltd and Oxfrd University Press Handbk f Evlutinary Cmputatin release 97/1 G3.6:11