Complex Structure Workspace Sweeping Using Semiautonomous Vacuum Cleaners

Transcription

1 Complex Structure Workspace Sweepng Usng Semautonomous Vacuum Cleaners KRZYSZTOF SKRZYPCZYK Department of Automatc Control Slesan Unversty of Technology Akademcka 6, Glwce POLAND AGNIESZKA PIERONCZYK School of Engneerng, Desgn and Technology Unversty of Bradford Rchmond Road, Bradford BD7 DP UNITED KINGDOM Abstract: The paper presents the study of desgn and synthess of the system dedcated for sweepng large and complex spaces usng a team of cleanng robots. The problems of coordnaton of actvtes of ndvdual robotc-unt s dscussed. Applcaton of the Game Theory for creaton of coordnaton algorthms s proposed. The paper deals also wth desgn and mplementaton of algorthms of movement of autonomous (or semautonomous) vacuum cleaner, that would be able to work nsde of a flat, but not free of obstacles workspaces. In the paper several proposed algorthms of surface coverng are dscussed. The two algorthms were selected as the most approprate and presented n ths work. Expermental valdaton of effcency of the proposed approach were done and the results of the valdaton are dscussed n the paper. Key-Words: - Moble robotcs, game theory, mult-robot, semautonomous systems, vacuum cleaners, surface coverng Introducton Domestc robots are robots whch are used n homesteads. Ths group of robots s ntended to perform tasks such as housekeepng as well as, servng as educatonal and entertanment robots. Despte of the fact, that the area of domestc robotcs s much less funded than for example the mltary robotcs t has stll been growng and developng rapdly, smultaneously ncreasng the level of human comfort and lfe. Wth reference to human knd hstory, people have always been lookng for smplfcatons and mprovements that would made ther lfe easer. Housekeepng devces, especally domestc robots generate wde range of possbltes to acheve ths goal. The ssue s relatvely up to date matter, specally nowadays when robotcs s so popular and creaton of ts applcatons became less expensve than any tme before. The status that housekeepng robots can obtan n the future, can be compared to the status of all others domestc devces (e.g. oven, frdge, hardryer) whch are nowadays omnpresent at our homes, and create bascs of each home s equpment. That can be done only f ther performance, utlty and prces would be optmzed to accomplsh a satsfactory trade off. Of course n order to desgn a properly workng robotccleaner a number of problems related to moble robotcs must be solved. That s enough to menton path plannng, data representaton etc. [3,5,8,9]. Another aspect of applcaton of robots to the task of ndoor cleanng s usng multple cleanng devces. It s well known fact that larger number of smpler devces can perform complex tasks. When we consder the work of a team of robots, that are ntended to perform some complex task (workspace cleanng for nstance) the addtonal problem of coordnatng actons (tasks) of ndvdual robots needs to be taken nto account. Wrong coordnaton may lead to neffectve task executon or even to nablty of completon the task. The problem of coordnaton of multple robots can be stated as a conflct stuaton between ndvdual robotcagents and can be modeled as a decson makng problem. The Game Theory [,4] seems to be a convenent tool for modelng problems of conflct nature. In ths work the approach to coordnaton of multple cleanng robots operatng n a complex structured ndoor envronment s presented. The ISSN: Issue 9, Volume 5, September 200

2 problem of coordnaton of elementary tasks s modeled as a N-person, noncooperatve team decson problem. The hybrd archtecture of a system that s desgned to control the work of agents dscussed. Apart from the problem of team coordnaton there exst another, mportant ssue to solve whch s the problem of effectve and robust surface coverng by ndvdual devces []. Therefore n the paper the aspect of desgn and mplementaton of an algorthm of movement of autonomous (or semautonomous) vacuum cleaner, that would be able to work nsde of a flat, but not free of obstacles workspaces s also consdered. The hstory of all robots whch applcaton was based on the vacuum cleaners cannot be fully ordered. The cause of t s that most robots were created by the commercal companes, whch offcal secrets have not allowed to catalog all created prototypes. Fortunately most of hoverng robots, whch have been sold on market tll now, had been somehow descrbed or at least mentoned n wde range of artcles and webstes. That gave a possblty to make the hstorcal outlne of those robots. Frst mentoned, was a prototype of the frst sold on market robotc vacuum cleaner called trlobte [], manufactured by the Swedsh corporaton Electrolux was avalable for purchase n 200. One year later, n 2002 the frst generaton of the most wde known, the Roomba [6] robotc floor cleaners has been on sale. Nevertheless, to date second and thrd generatons of that robot have been produced. At once, wth each generaton new mprovements were obtaned. To begn wth better brushes and larger dust bns, fnshng wth new algorthms of cleanng. As was prevously stated, desgn and mplementaton of a robotc vacuum cleaner s the very present ssue. What s more, accordng to documentaton and artcles whch addresses ths group of devces ther algorthms should stll be enhanced n order to obtan more mmune for envronment s dsturbances and more effcent system. Addtonally constructon of the multple vehcle, cleanng system consdered n ths paper has steel been an open ssue. 2 System overvew The problem consdered n the paper s two fold one. The frst stage of research ncludes the study of constructon of multple robot system and coordnaton mechansms. The second one s focused on desgn and evaluaton of surface coverng algorthms. A general structure of the control system s presented n fg.. The system can be splt nto two layers. The frst one that s ntended to be mplemented on moble platform (cleanng and collson avodng algorthms) and the second, that provdes nformaton of robots locaton and communcaton between robots. The frst one that s consdered n ths work has typcal hybrd Fg. Overvew of the mult devce cleanng system structure. It conssts n behavor-based moton controller that s responsble for executng elementary navgatonal tasks (modeled further by operators) and non-cooperatve game based planner. The functon of the planner s to choose from admssble actons the one that provde executon of a part of prmary msson (clearng operaton). The workspace model Robots are ntended to operate nsde a well structured, complex human made workspace. In order to smplfy the navgaton problem a partal knowledge about the envronment s ntroduced to the system. In fg. 2a an exemplary offce envronment plan s presented. An overall workspace s dvded nto regons named sectors. Each sector represents an area occuped by a room, corrdor or a part of a corrdor. Moreover passages between rooms are dstngushed and ntroduces to the model as door-objects called further door for smplcty. The workspace model s stored onto two layers: topologcal and geometrcal. The frst one s gven by weghted graph: ISSN: Issue 9, Volume 5, September 200

3 (, ) {,,... }, M = V W V = v v v W V V () 2 M whch nodes represent objects of the envronment: sectors V S and doors V D where V = V S V D. On the topologcal level each object s descrbed by a real number that s an "clutterng coeffcent" c n case the object s a sector and by a probablty of beng opened o j when the object s a door. The frst coeffcent reflects the number of objects placed nsde of the gven sector. On a geometrcal level -th sector s represented t t x, y, bottom by coordnates of ts top left corner ( ) rght corner ( b b x, y ), and a center pont ( p p, ) x y. Smlarly the j-th door-object s descrbed by a crcle of p p x, y. The edges radus r j and a center of the crcle ( j j ) of the graph defne relatons of neghborhood between envronmental objects related to vertces of the graph. Weghtng factors fxed to the edges of the graph are related to some costs of movng robot from the -th to the j-th vertex (object It s worth of notcng that the model descrbes only nvarable features of the workspace. The layout of objects (furnture, equpment) placed nsde sectors s not known. Moreover t can undergo dynamc changes. Fg. 2 An Exemplary Workspace Layout (a) and ts Topologcal Model (b) The planner From the perspectve of ths work the planner s the core of the system. It conssts of three modules: local coordnator, decson process modelng module and the soluton computaton one. The work of the planner can be brefly descrbed n a followng way. The local coordnator receves nformaton of locaton of all the cleanng devces. Moreover t s provded wth a world model and a prmary task data. Dependng on a type of the task, locaton of all of teammates and a state of the task executon, a model of a decson process s bult. The model s n fact the cost functon that depends on actons made by ndvdual agents and a state of task completon. Next the problem s solved and the soluton computed. The soluton of the problem determnes an elementary acton whch s optmal for a gven agent from the pont of vew of prmary task executon. Detaled descrpton of the process of buldng the model (takng the cleanng task as an example) and the methods of soluton shall be presented n further sectons. Task executon module The role of ths module s to execute elementary navgatonal tasks. It s desgned usng the behavorbased dea of control. It s composed of sx behavors that process the state and sensory nformaton nto proper set-ponts for moton controller whch are values of lnear and angular velocty. The coordnaton of behavors actvtes s made by fxed prorty arbter. For the purpose of ths artcle that s enough to consder the module as the one whch s able to execute four dfferent elementary tasks, represented by followng operators: FndDoor(D) - the task of movng the robot nsde the area of door-object D; TraverseDoor(D,S) -the task of gong through the door-object D to the sector S; Wat() - the smple command that stop the robot; GoTo(S,S2) - the task of movng robot from the sector S to S2; Clean(S) - the task of cleanng the sector S by the robot; 3. Cleanng coordnaton strategy ISSN: Issue 9, Volume 5, September 200

4 3. The problem formulaton The task of complex structured workspace sweepng s smlar to the exploraton problem [2]. Ths task can be generally stated as a problem of vstng a gven part of the workspace by teammates wth a cost as low as possble.. In terms of ths work the cleanng task s defned as vstng a part MV VS of the workspace M n a number of steps as small as possble. Here n ths work, we model the problem as a sequence of one stage, non zero sum games n a normal form. 3.2 Game theory based approach Let us frst ntroduce a notaton that wll be used hereafter. The state of a team of robots s denoted by a set: X = { x } =,2,... N, x V (2) what s equvalent to the fact that there s the -th robot nsde the area descrbed by the vertex x. The set of all possble actons of the robot descrbed by operators s gven by the set: { } A= a, a2,... a M 3) where M s a number of all operators (n ths work M=5). A set of possble actons of the -th robot n the state x s defned by : A x = a, a,... a (4) ( ) { } 2 K and t s determned by precondton lsts of ndvdual operators. In our case they are as follows: FndDoor(D) precondtons = { x VS, wx, D } TraverseDoor(D,S) precondtons = { x = D V, w, } Wat() precondtons = φ GoTo(S,S2) D D x precondtons = { x = S VS, w S,S2 } Clean(S) precondtons = { x = S} acton a k A for k =,2,...,4 s a mappng: a : x x x X x V a A (5) n n+ n n+ k n n where x k s the current state of the -th robot, and x + k s a state the robot wll be n as a result of the acton a k. The prmary task of the team of robots s to vst and clean all objects defned by a set M G VS. We ntroduce an auxlary set defnng objects that have already been cleaned and we denote t by MV M G. Usng ths notaton we can precsely formulate a goal of the team as satsfyng the equalty: M V = M G. The task of the plannng algorthm s to choose for each robot one of the possble acton, that appled to the robot wll result n performng a part of the prmary task. The problem of selecton of proper acton s n ths work perceved as a game between ndvdual robotc-players. The result of the game related to the defned task depends on decsons made by ndvdual game partcpants. Moreover the task of exploraton has a specfc nature that can be classfed as a team-work problem where all of the players (robots) want to optmze one performance ndex. Although the envronment s n prncple dynamc we model the problem of acton plannng as a sequence of statc N- person games n a normal form. Therefore we need to defne for each stage of the plannng process the sngle cost functon value of whch depends on actons made by all of teammates and on the task completon state. We propose to defne the cost functon as a sum of three components: (,...,,... ) I a a a = I + I + I (6) N R E D The frst one s related to a some value of reward that s gven to robots for cleanng part of the workspace and t s gven by: (,..,... ) = ( ) I a a a R a R N = N R> 0 f x M x M R = k 0 otherwse n+ n+ G V where R s a postve number that denotes the reward value. The k s a number of robots vstng the same object as a result of ther actons. The value of the second component I E s dependent on an amount of energy necessary to make an acton a whch s proportonal to a cost of transton of robots between envronmental objects defned by the model M: (7) In the terms of the decson makng process model of an ISSN: Issue 9, Volume 5, September 200

5 I a a a = w x x + (8) N n n (,..,... ) (, ) E N = Thrd component denotes cost of movng the robot to the nearest (n the sense of costs defned by W) unexplored object. Let us frst denote a path of mnmal cost between an object n and m as: ( ) { } pmn n, m = vn,... vk,... vm V (9) and let the set of unexplored objects s gven as M = u = M M. Then the cost I D s gven by: { } \ U l G V (,...,,..., ) I a a a = D D N mn, = n+ ( ) D = mn p x, u mn, mn l l N (0) where D mn, s the cost of movng the -th robot from the state x + to the nearest unexplored object n 4. Surface coverng problem 4. The vacuum cleaner model As a testbed for valdaton of proposed soluton the mnature, laboratory moble robot Khepera II [7] was used. The Khepera s presented n fg. 3. As a matter of fact, that mpled also an usage of The MATLAB software, whch was requred to program the robot. Detaled specfcaton of the Khepera II and s usage n the MATLAB can be found on the web ste. Nevertheless, the robot used n experments s a complex devce and t seems reasonable to hghlght ts man features. Thus, hardware and software descrpton s presented below n order to ntroduce some essental knowledge, requred for full apprehenson of developed algorthms, that are afterwards presented. Fg.3 The Khepera II robot as a model of vacuum cleaner [7] The Khepera II s a small, dfferental wheeled moble robot, desgned as a scentfc research and teachng tool at Federal Insttute of Technology Lausanne (EPFL) n Swtzerland. Frst of all, the robot s well suted for project s purposes because t renders rapd dea generaton, prototypng and evaluaton. In ts User Manual can be found, that the robot allows also confrontaton to the real world of algorthms developed n smulaton for trajectory executon, obstacle avodance, preprocessng of sensory nformaton, hypothess on behavors processng [7]. 4.2 The problem formulaton Another am of the work s creaton of surface coverng algorthms for hoverng robot that would be a part of dscussed multple devce cleanng system. The algorthm should enable the unt to work nsde of a flat, but not free of obstacles workspaces. 4.3 Proposed soluton In ths secton detals on two desgned surface coverng algorthms are presented end dscussed. The frst s the smple random-walk based one. The second one s more complcated hybrd concept that merges two dfferent approaches random movement and spral moton. Random walk based algorthm The frst algorthm has been based on the dea of random movement (fg. 4). The robot s movng n forward drecton tll an obstacle s sensed, then t stops. Next, by comparng sensor readngs decdes n whch drecton to turn left or rght. Fnally, by generatng a random number decdes how much to turn. The overrdng aspect of ths algorthm s a dstance between the robot and obstacles. It has been done, by settng nto ts source code, a dstance threshold value. Smply, when the dstance s lower than the gven threshold value, the robot takes an acton n order to omt obstacles. As a result of ths, the manner n whch robot s movng depends strongly on ts surroundng. Despte the fact, that the algorthm assumes ongong turns from the obstacle (.e. f obstacle s sensed on the left, t turns rght) what may cause stackng n corners, the random turnng banshes the ssue. All n all, the algorthm fulflls the am of the work, allowng the robot to move around a room avodng obstacles. However, ts effcency should be compared wth more sophstcated movement manner n order to examne ts utlty. ISSN: Issue 9, Volume 5, September 200

6 Fg. 4 The vsualzaton of 8 teratons of robot s movement accordng to random walk based algorthm. Ths workng mode s presented below n a form of a flow chart (fg.5). The second algorthm has been based on the dea of a spral moton coupled wth random turnng (fg. 6). Frst of all, the robot checks f there s enough place to start movng sprally. If yes, the robot convolutes n a RHS drecton, ncreasng a radus from centre pont, the tll an obstacle s sensed. When the obstacle s sensed, robot stops and compares sensor readngs to decde n whch drecton to turn left or rght. After the decson s made, the robot turns a gven drecton for a prevously allotted angle. Fnally, n order to allow tself to begn convolutng, t moves away from the obstacle for a gven range. Invarably, the overrdng aspect of the algorthm s a dstance between the robot and obstacles. It has been done lkewse to algorthm I, by settng nto ts source code, a dstance threshold value, thanks to whch an acton n order to omt obstacles can be taken. As a result of ths, the manner n whch robot s movng depends strongly on ts surroundng characterstcs. The prncple how the algorthm works s shown n the form of a smplfed flow chart (fg. 7) that gves a full and clear vew on propertes of the algorthm. Fg. 6 The vsualzaton of 3 teratons of robot s movement accordng to algorthm II Fg. 5 The flow chart for the random walk based algorthm. Hybrd algorthm ISSN: Issue 9, Volume 5, September 200

7 envronment The layout of the smulated workspace s shown n fg.8. Fg. 8 The layout of a workspace used for a Smulaton Fg. 7 The flow chart for the hybrd algorthm. 5. Smulaton and expermental results 5. Coordnaton process In order to show how the approach dscussed n the paper works we present a result of an exemplary smulaton. We mplemented the method usng a smulaton It s an example of a typcal offce envronment. It V = v, v,... v that represent conssts of twelve sectors { } S 2 2 rooms and parts of corrdors, and thrteen passages V = v,..., v. The area of the (doors) between rooms { } d 3 25 workspace s of a sze 5 0 [m] (wdth, heght). Varous peces of furnture and equpment are placed nsde of ndvdual rooms. The robot model used for the smulaton s descrbed n secton 4.. We consder the followng cleanng task. A team of three vacuum cleaners (N=3) s ntended to clean the workspace what s equvalent to vstng all of the sectors. Thus the task s defned as M G = VS. Intally, vacuum cleaners are nsde X = v, v, v so M V =X n. A sequence of of sectors { } nt operators that was used to perform the task s presented n the table and the llustraton of the smulated process s presented n fg. 9. The symbols FD, TD, GT, W,CL denote operators: FndDoor, TraverseDoor, GoTo, Wat and Clean. We can see that algorthm works well, provdng completon of the task stated above. One can notce that robot perform only a small part of the task, explorng only one room. But t has to be taken nto account, that the door D3,6 s modeled as the one that s almost for sure closed. That s the reason of ths strange task executon. Yet another aspect s worth of commentng. The algorthm presented n paper make only one step ahead plannng. It causes that the task executon may be not optmal. Usng other plannng algorthms we would obtan better or even optmal soluton. But such an approach would be vald f the envronment was statc as well as we assumed perfect result of each acton. ISSN: Issue 9, Volume 5, September 200

8 n Robot Robot 2 Robot 3 FD( D3,4 ) FD( D9,0 ) FD( D,2_ ) 2 TD(D3,4,S3) TD(D9,0,S0) TD(D,2_,S) 3 CL(S3) CL(S0) CL(S) 3 FD(D2,3) FD(D0,) FD(D,8) 4 TD(D2,3,S2) TD(D0,,S) TD(D,8,S8) 5 CL(S2) CL(S) CL(S8) 6 W() FD(D,2) GT(S7) 7 W() TD(D0,,S2) CL(S7) 8 W() CL(S2) GT(S6) 9 W() W() CL(S6) 0 W() W() GT(S5) W() W() CL(S5) Table A Sequence of operators that provdes completon of the workspace cleanng task Two descrbed surface coverng strategesy have been tested f they are operatng properly, what means that they fulflled am and all assumed objectves. All provded tests helped n speed and proxmty sensors calbraton. Fnally, when proper work manners for both algorthms were obtaned, ther outputs gave approprate and satsfactory results for nfnte perod of tme, the free from obstacles area of the workspace would be almost fully covered and obstacles would be avoded. However, t has to be mentoned that black or very reflectve objects would not be avoded due to sensors nsenstvty for that knds of materals. At the same tme, the promnent ssue of all objectves was algorthm effcency. The effcency was seen as area coverage n tme. Takng nto account sensors and motors constrants t had been very mportant to keep llumnaton levels and ntal condtons as smlar as possble for each type of test. That assured the most applcable results from each smulaton. Owng the fact that both algorthms nvolved random movement, ther outputs were hghly unpredctable. Nevertheless, based on numerous tests, followng effcences for operaton tme equals mnute have been acheved (table 2). Workspace dmensons: 400x500 mm; no obstacles Random Moton Spral Moton 4 cm² 430 cm² Workspace dmensons: 800x500 mm; no obstacles Random Moton Spral Moton 850 cm ² 2034 cm² Workspace dmensons: 800x500x00 mm; one obstacle: 00 cm² Random Moton Spral Moton 532 cm² 296 cm² Workspace dmensons: 800x500x00 mm; fve obstacles sum: 50 cm² Random Moton Spral Moton 870 cm² 540 cm² Table 2 Comparson of the effcency of presented surface coverng algorthms. the two Fg. 9 Stages of realzaton of the cleanng task 5.2 Surface coverng As t can be observed spral moton algorthm has better effcency coeffcent for grater workspace dmensons and smaller number of obstacles. What s more, ts effcency depends strongly from the ntal pont. The best results were obtaned for ntal pont beng n the mddle of the workspace, what allowed the robot to convolute wth greater radus. On the other hand, when ISSN: Issue 9, Volume 5, September 200

9 the ntal pont s closer to the workspace s boundares, or when the workspace area s flled wth obstacles the random moton algorthm accomplshes better effcency results than the spral moton. To sum up, both algorthms that had been developed for the robotc verson of a vacuum cleaner, have been checked and they worked properly. Regardless of the fact that ther effcences vared wthn envronmental changes, ther emprcally derved effcences were stll satsfactory. 6. Concluson In ths paper we dscussed a problem of plannng and coordnaton of tasks n a mult system. The role of the dscussed system s sweepng large and complex spaces usng a team of cleanng robots. We consdered a team of robotc-vacuum cleaners that was ntended to perform a task cleanng of a human-made workspace of complex structure. We proposed both the hybrd archtecture of a control system and method of coordnaton of multple robotc agents. In the paper we also presented an algorthm of exploraton of workspace. The core of the algorthm s the model of the process that s stated as a noncooperatve game n a normal form. We appled the Nash equlbrum concept to generate a soluton of the problem. Although the result of only one smulaton was presented, we had made a numer of smulaton experments usng both varous parameters and workspace confguratons. In all cases we obtaned correct task executon. On the bass of smulatons we carred out we can conclude that ths algorthm works well and provdes effectve exploraton of even very complex-structured envronments. However, the algorthm can not guarantee optmal task performance. It s caused the algorthm uses only one-step-ahead plannng method. But ths approach on the other hand has other advantage - t allows to track dynamcal changes of the envronment and t does not need the assumpton that a gven acton s always executed n a perfect way The second ssue dscussed n the work was the desgn and mplementaton of algorthms of movement of autonomous (or semautonomous) vacuum cleaner, that would be able to work nsde of a flat, but not free of obstacles workspaces, have yet to be compared wth the acheved outcome. Unquestonably, developed algorthms fulflled ths goal. What s more, the man objectves for the project were based on analyss of developed algorthms, ther lmtatons and effcences. The Khepera II returned approprate response wth respect to both programs, keepng a dstance to objects, as well as to the area coverage. The robot has not stacked accdently at any of the workspace recesses. As t has been mentoned before, although mechancal aspect of the project has been qualfed to consder only dfferent types of a testbed robots, envronmental percepton and physcal characterstcs have gven some addtonal constrants nfluencng smultaneously on the project s fnal output. Those constrants together wth project s output have been deeply analyzed and resulted n proposals for future developments. Despte of the fact that all the descrbed system s n the desgn phase the prelmnary results obtaned are promsng. Acknowledgements Ths work has been granted by the Polsh Mnstry of Scence and Hgher Educaton from funds for years References: [] Basar, M. and G.J. Olsder Dynamc Noncooperatve Game Theory. Mathematcs n Scence and Engeneerng, Academc Press Inc. Ltd, London. [2] Burgard, W. et al Collaboratve Mult-Robot Exploraton. Proc. IEEE Internatonal Conference on Robotcs and Automaton (ICRA), San Francsco CA [3] Chng Chang Wong et al., An Intutonal Method for Moble Robot Path Plannng n Dynamc Envronment, WSEAS Recent Advances n Computers Computng and Communcatons, ISBN: , 2002, pp [4] A. Galuszka, A. Swernak, Game Theoretc Approach to Mult Robot Plannng, WSEAS Transactons on Computers, Issue 3, Vol. 3, July 2004 [5] D. Glavsk et al, Robot Moton Plannng Usng Exact Decomposton and Potental Feld Methods, Proc. of the 9 th WSEAS Internatonal Conference on Smulaton Modellng and Optmzaton, Budapest, Hungary, September, 2009 pp [6] Robot Roomba, The offcal polsh Robot Roomba webpage Avalable: [7] K-Team, Khepera II User Manual,. verson, Swtzerland, page, 2002 [8] F.G. Martn, Robotc Exploratons A Hands-On Introducton to Engneerng, st ed. New Jersey, Unted States of Amerca: Tom Robbns, 200. ISSN: Issue 9, Volume 5, September 200

10 [9] P. J. McKerrow, Introducton to Robotcs, st ed. Sngapore: Addson-Wesley Publshng Company, 99. [0] K. Skrzypczyk, Behavour Actvty Based Maps Appled to Moble Robot Navgaton, Proc. of 3 th WSEAS Internatonal Conference On Systems, Rodos, Greece, July, 2009, pp [] Trlobte (200), Robotyka The offcal Electrolux s Trlobte webpage Avalable: 8 [2] I. R. Urlch, F. Monada, J. Ncoud, Autonomous vacuum cleaner. Avalable: ftp://lamftp.epfl.ch/khepera/papers/ulrch_ras96.. ISSN: Issue 9, Volume 5, September 200