Finding Proper Configurations for Modular Robots by Using Genetic Algorithm on Different Terrains

Transcription

1 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 Fndng Proper Confguratons for Modular Robots by Usng Genetc Algorthm on Dfferent Terrans Sajad Haghzad Kldbary, Saeed Bagher Shourak, and Salman Faraj Abstract Ths paper presents a novel self-reconfgurable robotc system named ACMoD where each module can move tself ndvdually. It can also attach to other modules to buld varous confguratons and change ths confguraton adaptvely on dfferent terrans. In ths paper, we have proposed Genetc Algorthm for optmzng the path of modular robots through a statc grd of dfferent terran blocks. Each chromosome conssts of path and modular robot confguratons. Soluton of the proposed algorthm s a proper path and confguraton pattern for crossng the envronment wth mnmum effort related to a pre-defned mult-objectve functon. Fnally, for nvestgatng the effcency of the proposed algorthm, the performance of proposed algorthm s compared to Djkstra algorthm n dfferent envronments. Index Terms Djkstra algorthm, modular robots, path plannng. genetc algorthm, I. INTRODUCTION Self-reconfgurable modular robots (SRMR) refer to a class of robots whch are made of large number of dentcal and ndependent small components called modules. They can connect to each other and reconfgure nto dfferent shapes [1]. These knds of robots have the capablty to reconfgure and adapt to dfferent task, condtons and envronments. Ths ablty s the man reason brngng such robots nto consderaton n recent years. The path plannng problem has been one of the mportant ssues n moble robotcs [2]-[4]. Path plannng s an optmzaton problem [2] whch s defned to fnd a sutable collson-free path for robot from the start locaton to the goal wth dfferent evaluaton crtera [3], [5]. Path plannng generally can be dvded nto two classes that nclude path plannng n statc [6], [7] and dynamc [8] envronments. In statc path plannng, the whole nformaton of envronment s known and global path can be generated. However, n dynamc path plannng the robot respond to the envronment change whch s known as sensor based approach [6], [8]. Ths paper s focused on global path plannng n statc envronment. Generally, the process of path plannng has two man steps that nclude envronment descrpton (envronment model) and usng a proper search II. NEW MODULE DESIGN For testng the proposed path plannng algorthm, we use a set of 3-DoF modular robots called ACMoD. These modules have the capablty to reconfgure automatcally from terran to terran, as requred n our method. Each module conssts of two wheels rotatng freely compared to a central jont whch s lmted, but more powerful. Ths desgn helps to create more flexble confguratons especally for legged robots. Fg. 1 shows ACMoD wth some feasble confguratons regardng physcal lmts of selected servomotors and jonts. Manuscrpt receved December 11, 2012; revsed January 30, F. A. Sajad Haghzad Kldbary s wth Artfcal Creatures Lab, Electrcal Engneerng School, Sharf Unversty of Technology, Tehran, Iran (e-mal: haghzad@ee.sharf.edu). S.B. Saeed Bagher Shourak s head of Artfcal Creatures Lab, Electrcal Engneerng School, Sharf Unversty of Technology, Tehran, Iran (e-mal: bagher-s@sharf.edu). T.C. Salman Faraj s n Ecole Polytechnque Fédérale de Lausanne (EPFL), Swtzerland (e-mal: salman.faraj@epfl.ch). Fnd some vdeos showng the performance of the robot at DOI: /IJMMM.2013.V1.78 algorthm to fnd optmal or near optmal path. In most of path plannng methods, the envronment s lmted to two dmensons and obstacles are presented by polygon shapes [4]-[8]. So far, many methods have been ntroduced to descrbe the envronment such as vsblty graph [9], Vorono dagram [10], MAKLINK graph [11] and cell decomposton [12]. Varous search algorthms have been used such as artfcal potental feld method [13], neural networks [14], ant colony algorthm [15], partcle swarm optmzaton [16] and genetc algorthm [2]-[6], [8]. Each method has ts own advantages over others n certan aspects. In the recent years, genetc algorthms have been wdely used n the feld of path plannng for moble robots. So far, most of presented algorthms are based on fxed-structure and they have not addressed path plannng and onlne reconfgurng, smultaneously [16]. So they are not sutable path plannng methods for modular robots. In ths paper, accordng to the capablty of new desgned modular robot to change confguratons, the GA s presented to produce a proper path and confguraton pattern for crossng the envronment. Path evaluaton crtera are combned wth mnmum tme, lowest energy and shortest dstance. Chromosomes are consstng of dfferent paths and dfferent confguratons wth varable length. In our method, unlke most of earler methods, all chromosomes n ntal populaton and after applyng GA operators are feasble wthout havng collson wth obstacles. Smulaton results prove that our method can successfully plan a path and confguraton pattern for modular robots wth convncng performance, compared to fxed-structure robots. The rest of the paper s organzed as follows: n Secton II, our new module desgn s explaned n detals together wth ts local navgaton method. The proposed GA s ntroduced n Secton III. In Secton IV, Djkstra algorthm s used for modular robot path plannng. In Secton V smulaton results of GA and Djkstra algorthm n varous envronments are presented and analyzed. Fnally, the concluson and suggestons for future research are gven n Secton VI. 360

2 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 localzaton algorthm durng reconfguraton process for the purpose of ths work. Fg. 2. Navgaton algorthm used n reconfguraton process of the modules. On the rght mage (b), the yellow path shows the outcome of the ERRT algorthm. Ths path ends up n a crcle n front of the target jont n the master robot. On the left mage (a) whch shows the crcle from top vew, artfcal potental felds navgate the robot locally to the target pont where the two jonts are close enough to attach. Note the spral component added to normal potental felds rght at the destnaton whch causes the robot to rotate constantly and not beng stopped wth an undesred orentaton. Fg. 1. Schematc of the module wth some smple confguratons that are feasble regardng lmted torque of the selected servomotors actuatng the three jonts. Note that the center of mass wll be adjusted a lttle under the geometrc center of the robot defned by three dotted orange axes. Ths enables the robot to do locomoton smply by dsplacng ths pont compared to the contact pont of the wheels lke a Segway robot, but beng stable. An mportant dfference between ths desgn and other conventonal modular robots s the ablty to move ndvdually and fndng each other. Such capablty s also n [17], but the ndvdual movement of ths robot s not so robust to possble roughness n the terran. Our desgn benefts from large wheels that can perform better n dfferent envronments. Wth these wheels, the module cannot rotate by 1800 from the mddle whch s not needed also, snce most of common legged robot structures and also other wheeled robots could be bult by ths smple desgn as shown n Fg. 1. Ths ntroduces a smpler way to reconfgure for a new terran. When the robot decdes to change confguraton, t dsassembles tself frst, then all modules get far enough from each other and the new confguraton starts to buld. Indvduals are commanded by a leader among them to pursue a pass toward a target robot n order to make a new connecton. An underlyng assumpton s that they can create a rough local map of ther relatve postons and orentatons. There are lots of algorthms n lterature that help a robot go to the destnaton whereas avodng obstacles. Ths work takes advantage from the ERRT algorthm lke [18] and [19] whch s robust to envronment uncertantes. Bascally, ths algorthm navgates the slave robot nto a crcle n front of the target jont of the master robot whch wats and does not move. After the arrval of slave robot n the crcle, the navgaton contnues wth potental felds toward the exact desred locaton where the two jonts can make a connecton. We assume a maxmum detectable dstance for a jont determned by desgn so that t can be recognzed by another jont to make the connecton. Ths varable determnes the accuracy of the localzaton algorthm too. Ths scenaro s depcted n Fg. 2. Ths abstract desgn s studed well n terms of feasblty n [20] and s currently beng developed at ACL. Modules are smulated as well usng Mcrosoft DrectX and Nvda PhysX lbrares as developed n [20]. All the parameters used n smulatons are determned by specfcatons of selected off the shelf components for the robot. We also assume a perfect 361 III. PROPOSED METHOD Our path plannng algorthm for modular robots s based on Genetc algorthm (GA). It s a randomzed search technque based on the prncple of survval of the fttest n nature [4], [7]. In ths secton the proposed GA wll be dscussed n detals. A. Envronment Representaton Envronment descrpton s the frst step to plan the path. We assume that the envronment s statc and does not contan any movng objects. Another assumpton s that our approach s global,.e., we have complete knowledge about the envronment as shown n Fg. 3. We represent the envronment wth a grd of dfferent terrans to establsh a 2D work space model. Ths model s represented by orderly numbered grds as n [2], [4], [7]. Ths method s better than Cartesan coordnates [3], [5] because seral number representaton s more concse and saves memory [21]. In ths representaton each grd cell s a specfc type of terran that at least one of the modular confguratons can cross t. Fg. 4 shows our method. Unlke [2], we are not concerned about the smoothness of the path for the purpose of ths work, assumng that most of the energy consumpton s related to passng the terran. B. Chromosome Representaton One of the mportant ssues n GA s chromosome representaton [4]. In order to apply GA to path plannng, we need to encode the path nto genes. A complete set of genes form a chromosome. As t s shown n Fg. 4, unlke prevous works [2], [3], [7] that chromosomes only represent a sequence of grds, n ths paper, a vald chromosome represents a sequence of grd labels and confguratons. The frst gene always contans the start locaton of the robot and the one before the last contans the goal cell. In ths representaton the chromosome's length s varable. C. The Generaton of Intal Populaton In most of prevous proposed GA algorthms, ntal

3 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 populaton s generated randomly [4], [7], [21]. Ths s qute smple, but requres defnng addtonal operators and usng penalty terms n ftness functon to correct or dstngush between feasble and nfeasble path [4], [7]. These methods therefore ncrease computaton tme drastcally. Fg. 3. Modular robot envronment and symbols that are used for each terran n smulatons. To ncrease accuracy of the soluton, the resoluton of grds can be ncreased n case of a real envronment. 4) Step 4: Calculate the Eucldean dstance between free grds to the center of obstacle grds, then add sum of nverse of Eucldean dstances to prevous values of grds and ntal small value (B) by the followng equaton: L G B (1/ r ) (1) j 1 In above equaton, L s number of obstacles and for each grd ( G ) we calculate the equaton. 5) Step 5: Durng the movement of the robot from startng locaton to the end, n each grd cell the vector R s dvded by the values of adjacent grds (n Step 4) to form a new possblty vector. The next step of the moton s obtaned by usng Roulette wheel selecton method on the new possblty vector. 6) Step 6: Apply short-cut and Loop Remove operators to remove unnecessary cells from the path, f any. 7) Step 7: Fnally, for each grd cell of ths path assgn a confguraton randomly. In case of an obstacle-free envronment, to generate ntal populaton we do not need step 1 and 4. Also n step 5, we only use the Roulette wheel selecton. j Fg. 4. The envronment n Fg. 3 represented by orderly numbered grds referred to as robot plannng area. The black grds are obstacles. Obstacle for modular robots means that no modular confguraton can pass t. The blue grds show a vald path that connect the start cell to the goal. Chromosomes are therefore encoded by nteger numbers begnnng from start cell and endng n the goal one. For creatng a chromosome, each cell s assgned to two genes. The frst gene represents the number of grd cell and the second gene represents a confguraton for that cell. In our method the ntal populaton s generated randomly, but all paths are feasble so that to use few GA operators whch decreases the computaton tme. We assume that the robot can move n eght cardnal and nter-cardnal drectons. Generaton of the ntal populaton conssts of the followng steps: 1) Step 1: Frst assgn a small value (B) to free cells and nfnte to blocked ones. Also assgn nfnte borders (grey cells), as shown n Fg. 5. 2) Step 2: Defne R as a drecton vector whch shows the possblty of movement for the robot n eght drectons as shown n Fg. 5. 3) Step 3: Draw a lne from start cell to the goal. The most possblty s then assgned to the three drectons around ths lne (green dots n Fg. 5). Fg. 5. In ths fgure we have envronment wth 10*10 grds, B s a small value assgned to free cells and r1 and r2 are Eucldean dstances. For each grd cell the formula mentoned n fgure s evaluated. Each step of generatng the ntal populaton s shown n fgure. After generatng a vald path, for each grd cell of ths path we assgn a confguraton randomly. D. Evolutonary operators Our proposed GA uses crossover, mutaton and two customzed operators, short-cut and Loop Remove. These operators are explaned bellow: 1) Selecton: New generaton s formed by selectng the chromosomes from prevous generaton and applyng crossover, mutaton and other operators. Roulette wheel selecton used here s based on ftness functon, where chromosomes wth a small ftness functon have more chance (possblty) to survve. 2) Crossover: The crossover operaton means combnng two parents n order to exchange nformaton between them. We use sngle-pont crossover,.e., one of the common genes of parent chromosomes (odd genes) s selected randomly and two new chromosomes are generated by combnng parents from ths common gene. 362

4 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 Ths operator s llustrated n Fg. 6. 3) Mutaton: Ths operator ncreases dversty of a populaton to prevent local convergence. Mutaton alters one or more randomly selected genes of a chromosome. Our method uses sngle-pont mutaton and unlke crossover, operates on both odd and even genes as demonstrated n Fg. 6. 4) Short-cut: Ths operator ams to reduce the total dstance of a path. Short-cut deletes unnecessary ntermedate cells that are between two other cells. An example s beng shown n Fg. 6. 5) Loop Remove: Sometmes after generatng ntal populaton or applyng GA operators, some loops may appear n the path. Ths operator s used to delete these loops whch are redundant. Fg. 6 demonstrates ths operator as well. dfferent chromosomes. ET ( e, c ) and TT ( e, c ) are the energy and tme needed for the confguraton c to cross the envronment e. EC ( c, c 1) and TC ( c, c 1) represent energy and tme needed to reconfgure the confguraton c to c. a 1 and b refer to the weghts of nfluence of tme and energy on the total cost and A s a coeffcent proportonal to the length of the shortest path traveled by the robot n each grd. We know all the costs for crossng dfferent terrans and changng confguratons together. Ths nformaton s obtaned by smulaton tests. One of the energy tables s shown n Table. I and others are smlar to [22]. TABLE I : ENERGY (KJ) CONSUMPTION OF DIFFERENT CONFIGURATIONS OVER DIFFERENT TERRAINS ( E (, ) T e c ) c. e. 4-legged 3-legged Segvey Indvdual Snake Smooth Fence nf nf 4.70 Brdge nf nf nf Cobble nf 4.60 Star nf nf nf Gap nf nf 4.75 Hll nf nf 6.50 In all steps of optmzatons, the normalzed values of these tables are used to fnd the proper path. Fg. 6. Genetc operators: the crossover manpulates only odd genes (grd genes). As shown n fgure, n ths paper there are two knds of mutatons: grd mutaton (for odd genes) and confguraton mutaton (for even genes). Ths operaton has two constrants. In grd mutaton, the path should reman contnuous and n confguraton mutaton, the new confguraton should be able to cross the cell. Short-cut and Loop Remove reduce path length. In [22], the convergence of GA for each of these operators s nvestgated. E. Evaluaton (Ftness Functon) The ftness functon s so mportant for the stablty and convergence of GA [4], [14]. Each new generaton s evaluated by a ftness functon. Ftness functons are usually weghted sums of evaluaton crtera [21]. In ths paper, a proper path s evaluated accordng to the mnmum tme, mnmum energy and shortest dstance. Our cost functon s a combnaton of tme, energy and dstance: F A ( a E ( e, c ) b T ( e, c )) N 1 1 N 1 T T ( a E ( c, c ) b T ( c, c )) C 1 C 1 a b 1 (3) where N s the total length of a path whch may vary for (2) IV. DIJKSTRA ALGORITHM In order to measure the performance of our proposed algorthm, we compare t wth Djkstra algorthm. Djkstra s a determnstc optmzaton algorthm used for fndng the optmal shortest path from a start node to a goal node n a graph [3], [23]. So far, most of presented Djkstra algorthms are based on fxed-structure [10], [23]. However n ths paper, Djkstra s used for modular robot path plannng. In ths algorthm for each vertex of the grd, we put fve nodes as we have fve confguratons of our modular robot. These nodes are connected to each other by edges as shown n Fg. 7. The weght of each edge s proportonal to confguraton, type of terran and confguraton change durng the robot moton. For each edge, cost of crossng the grd (edge coeffcent) s obtaned by followng equaton: W a ( c E ( e, c ) E ( c, c )) k T C ' b ( c T ( e, c ) T ( c, c )) T C ' Here, represents grd number, the value of ' can be varable and depends on next grd number and c s 1 for horzontally and vertcally move, and s 2 for dagonal move. ET ( e, c ), TT ( e, c ), E ( c, c ' ), T ( c, c ' ), a and b are C C the same as cost functon proposed for GA. In Secton III the proposed GA was explaned and n Secton IV we ntroduced the graph that s used for Djkstra algorthm. In next secton we wll nvestgate the results of (4) 363

5 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 both algorthms for path plannng of our new modular robot. soluton [3]. Table II shows the results of proposed GA and Djkstra algorthms. In order to demonstrate the performance of our method, we calculate the average of GA results after 20 runs. Fg. 7. Part of a graph that Djkstra algorthm s appled on. For each vertex of the grd, fve nodes are lad due to fve confguratons used for modular robot. In ths graph the edge that connects vertex 1 to vertex 1 (on the other sde of the grd) tells that the confguraton 1 (4-legged) passes the grd wthout any reconfguraton. As another example, the edge connectng vertex 1 to vertex 2 shows that the confguraton 1 (4-legged) passes the grd and reconfgure to confguraton 2 (3-legged). V. SIMULATION RESULTS In ths secton, optmzaton results are compared for both Djkstra and GA algorthms, mplemented usng MATLAB. All the smulatons are done on a computer wth Intel core 5, 2.4 GHz CPU. The probablty of grd mutaton p m1 s 0.3, confguraton mutaton p m2 s 0.7, and probablty of crossover p c s 1. Eltsm strategy s also used to preserve the best chromosomes. In each generaton, 20 percent of populaton wthout applyng any genetc operators are transferred to next generaton, and then GA operators are appled to all chromosomes accordng to ther possbltes [2], [4]. Then the whole generaton s replaced by offsprngs. In all smulatons, a and b are 0.5. A. Small Envronment For nvestgatng the soluton of proposed GA, we compared proposed GA wth Djkstra algorthm. We ntalze both algorthms wth same frst cell, frst confguraton and the goal cell. The smulaton results for both GA and Djkstra algorthm are shown n Fg. 8. The process of gudng the robot from start locaton to the goal s based on a state machne. Each confguraton has ts own Central Pattern generator (CPG) that performs the locomoton. All of them can execute forward, backward and steerng commands. Therefore we navgate them from pont to pont knowng the optmum path and confguraton pattern comng from optmzatons. If a reconfguraton s requred, the modular robot dsassembles and assembles agan as descrbed n Sec. II. In Table II output of Djkstra and Genetc algorthms wth ntal populaton sze of 50 and after 50 teratons are obtaned. In terms of performance, ths cost s 94.9 present of Djkstra algorthm. We run GA wth varous populaton szes and teratons. The goal s to nvestgate the behavor of GA n each case two measures are typcally used to compare algorthms quanttatvely, frst the tme complexty of the algorthms, and second the qualty of Fg. 8. Smulaton results for both Genetc and Djkstra algorthms. An envronment wth 10*10 cells and 32 obstacles are used n ths test. "S" s the start cell and "G" stands for the goal. In smulatons we use the frst fve confguratons that are labeled n Fg. 1 whch all consst of seven modules. Seven terrans that are determned n Fg. 3 are used n smulatons. After applyng Djkstra, for each cell the shortest nternal path s calculated. In the fgure green symbols determne the fnal cells and confguratons for Djkstra algorthm and red symbols determne the cells and confguratons of the GA after fve teratons. For both algorthms robot moves on shortest path n each determned cell. TABLE II : COST OF GA AND DIJKSTRA ALGORITHMS IN AN ENVIRONMENT WITH OBSTACLES AND 10*10 GRIDS (THE FIRST COLUMN IS INITIAL POPULATION) GA Iteraton=20 Iteraton=30 Djkstra Cost Tme(S) cost Tme(S) Cost Tme(S) B. Large Envronment Here, algorthms are tested n large envronments and the results are shown n Table III. TABLE III : COST OF GA AND DIJKSTRA IN AN ENVIRONMENT WITH OBSTACLES AND VARIOUS SIZES (THE FIRST COLUMN IS NUMBER OF CELLS, INITIAL POPULATION FOR GA IS 150 [S]) Iteraton=20 Iteraton=30 Djkstra GA Cost Tme Cost Tme Cost Tme 100* * * Out of Memory Performance of these algorthms depends on parameters of algorthms and nput sze. It s clear that the envronment resoluton s the man factor n complexty and computatonal tme of algorthm. Smulaton results show that analytcal method lke Djkstra s better than heurstc algorthms lke GA n lower resolutons of envronment. Smulaton results tell us that wth small nput sze the path obtaned by GA needs more tme than Djkstra, but when the nput sze s too large Djkstra becomes neffcent. In large szes GA becomes more effectve and tres to fnd a nearly optmal soluton 364

6 Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 1, No. 4, November 2013 [11] W. Yu-Qn and Y. Xao-Peng, Research for the Robot Path Plannng Control Strategy Based on the Immune Partcle Swarm Optmzaton Algorthm, n Proc. Int.. Conf. on Intellgent System Desgn and Engneerng Applcaton, 2011, pp [12] C. Ca and S. Ferrar, Informaton-drven sensor path plannng by approxmate cell decomposton, IEEE Trans. on Systems, Man, and Cybernetcs, Part B: Cybernetcs, vol. 39, no. 3, pp , [13] E. Rmon and D. E. Kodtschek, Exact Robot Navgaton Usng Artfcal Potental Functons, IEEE Trans. on Robotcs and Automaton, vol. 8, no. 5, October 1992, pp [14] D. Xn, C. Hua-hua, and G. We-kang, Neural network and genetc algorthm based global path plannng n a statc envronment, Journal of Zhejang Unversty Scence, pp , [15] T. Guan-Zheng, H. Huan, and S. Aaron, Ant Colony System Algorthm for Real-Tme Globally Optmal Path Plannng of Moble Robots, Acta Automatca Snca, vol. 33, no. 3, pp March [16] T. Lu, C. Wu, B. L, J. Lu, The Adaptve Path Plannng Research for a Shape-shftng Robot Usng Partcle Swarm Optmzaton, n Proc. Int. Conf. on Natural Computaton, 2009, pp [17] G. G. Ryland and H. H. Cheng, Desgn of Mobot, an Intellgent Reconfgurable Moble Robot wth Novel Locomoton, ICRA 2012, pp [18] Jr. James, J. Kuffner, and S. M. LaValle. RRT-Connect: An effcent approach to sngle-query path plannng, n Proc. IEEE Int. Conf. on Robotcs and Automaton, 2000, pp [19] V. R. Desaraju and J. P. How, Decentralzed Path Plannng for Mult-Agent Teams n Complex Envronments usng Rapdly-explorng Random Trees, n Proc. ICRA 2012, pp [20] S. Faraj, Desgn and Smulaton of a new structure for moble modular robots, B.S. thess, Dept. Elect. Eng., Sharf Unv. of Technology, Tehran, Iran, [21] L. Weqang, Genetc Algorthm Based Robot Path Plannng, IEEE Int. Conf. on Intellgent Computaton Technology and Automaton, 2008, pp [22] S. H. Kldbary, Fndng Proper Modular Robots Structure by Usng Genetc Algorthm, M.S. thess, Dept. Elect. Eng., Sharf Unv. of Technology, Tehran, Iran, [23] H. Wang, Y. Yu, and Q. Yuan, Applcaton of Djkstra algorthm n robot path-plannng, n Proc. Second Int. Conf. on Mechanc Automaton and Control Engneerng, 2011, pp provded that we tune parameters to allow the algorthm search all the space. Djkstra for small nput works well and t requres less tme to fnd the optmal path. However when the nput s too large, as shown n Table III, shortage of memory occurs or t takes a lot of tme to fnd the optmal path. In GA we can tune the executon tme by changng GA parameters lke number of generatons, reducng the qualty of favorte soluton, populaton sze, mutaton and crossover possbltes. VI. CONCLUSION In ths paper we presented a new module beng able to move ndvdually. We proposed GA for modular robots global path plannng n statc envronment. In ths method chromosomes consst of dfferent paths and confguratons wth varable length. Intal populaton s generated randomly, but all paths are feasble to save computaton tme. We consder three parameters n ftness functon: mnmum tme, lowest energy and shortest dstance. MATLAB smulaton s used to verfy the proposed algorthm both n terms of qualty and runnng tme compared to Djkstra algorthm. Smulaton results show a tradeoff between qualty of path and optmzaton tme. An nterestng topc for future research would be to mprove the proposed algorthm, amng to fnd the optmal path for modular robots n a dynamc envronment. ACKNOWLEDGMENT The authors would lke to thank Ramn Halavat for hs knd dscussons. REFERENCES Z. Guanghua, D. Zhcheng, and W. We, Realzaton of a Modular Reconfgurable Robot for Rough Terran, n Proc. the IEEE Int. Conf. on Mechatroncs and Automaton, 2006, pp [2] C.-C. Tsa, H.-C. Huang, and C.-K. 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Conf. on Intellgent Networks and Intellgent Systems, 2010, pp [1] 365 Sajad Haghzad Kldbary receved the B.Sc. degree n Electrcal Engneerng n 2009 from Raz Unversty, Kermanshah, Iran, and M.Sc. degree on Dgtal electroncs from Department of Electrcal Engneerng, Sharf unversty of Technology, Tehran, Iran n Hs research nterests nclude robotcs, artfcal ntellgence, Neural Networks, Genetc and Evolutonary Algorthms and FPGA crcut desgn Saeed Bagher Shurak receved hs B.Sc. n Electrcal Engneerng and M.Sc. n Dgtal Electroncs from Sharf Unversty of Technology, Tehran, Iran, n 1985 and He joned soon to Computer Engneerng Department of Sharf Unversty of Technology as a faculty member. He receved hs Ph.D. on fuzzy control systems from Tsushn Dagaku, Tokyo, Japan, n He contnued hs actvtes n Computer Engneerng Department up to He s currently a Professor n Electrcal Engneerng Department of Sharf Unversty of Technology. Hs research nterests nclude control, robotcs, artfcal lfe, and soft computng. Salman Faraj receved B.Sc. degree from Sharf Unversty of Technology n Electrcal Engneerng and Dgtal Systems, Tehran, Iran n He s currently a M.Sc. student n Robotcs, Ecole Polytechnque Fédérale de Lausanne (EPFL), Swtzerland. Hs nterests n research nclude modular robots, dstrbuted ntellgence, legged robots locomoton and nverse dynamcs.