Key-Words: - Automatic guided vehicles, Robot navigation, genetic algorithms, potential fields

Transcription

1 Autonomous Robot Navgaton usng Genetc Algorthms F. ARAMBULA COSIO, M. A. PADILLA CASTAÑEDA Lab. de Imágenes y Vsón Centro de Instrumentos, UNAM Méxco, D.F., 451 MEXICO Abstract: - In ths paper s presented a navgaton scheme, based on a genetc algorthm, for autonomous robot navgaton. Potental felds are used to ract the robot by the goal poston and reject t by the obstacles. In the scheme presented here obstacles are automatcally detected by smulated sonar sensors. The confguraton of the optmum potental feld s determned by the genetc algorthm. Intermedate goal ponts are used for global path plannng. Smulaton results show that the scheme orted has a good performance n unknown envronments wth hgh obstacle denstes. Key-Words: - Automatc guded vehcles, Robot navgaton, genetc algorthms, potental felds 1 Introducton The two man approaches for the mplementaton of robot navgaton algorthms are artfcal potental felds, and artfcal ntellgence methods. Khatb [1] ntroduced the use of potental felds for autonomous navgaton. The man dea s to generate racton and ulson forces to gude the robot to ts goal. The goal pont has an ractve nfluence on the robot and each obstacle tends to push away the robot, n order to avod collsons. Potental feld methods provde an elegant soluton to the path fndng problem. Snce the path s the result of the nteracton of approprate force felds, the path fndng problem becomes a search for optmum feld confguratons nstead of the drect constructon (e.g. usng rules) of an optmum path. Dfferent approaches have been taken to calculate approprate feld confguratons. Vadakkepat et al.[] ort the development of a GA for autonomous robot navgaton based on artfcal potental felds. Repulson forces are assgned to obstacles n the envronment and racton forces are assgned to the goal pont. The GA adjusts the constants n the force functons. Multobjectve optmsaton s performed on 3 functons whch measure each: error to the goal pont, number of collsons along a canddate path, and total path length. Ths scheme requres a prory knowledge of the obstacle postons n order to evaluate the number of collsons through each canddate path. Kun Hsang et. al [3] ort the development of an autonomous robot navgaton scheme based on potental felds and the chamfer dstance transform for global path plannng n a known envronment, and a local fuzzy logc controller to avod trap stuatons. Smulaton and expermental results on a real AGV are orted for a smple (4 obstacles) and known envronment. McFetrdge and Ibrahm [4] ort the development of a robot navgaton scheme based on artfcal potental felds and fuzzy rules. The man contrbuton of the work conssts n the use a varable for the evaluaton of the mportance of each obstacle n the path of the robot. Smulaton results on a very smple envronment (one obstacle) show that use of the mportance varable produces smoother and shorter trajectores. In ths work s presented an adaptve navgaton scheme based on artfcal potental felds whch are automatcally adjusted to avod obstacles, usng a genetc algorthm. Auxlary racton ponts have been used n order to allow the robot to navgate around large obstacles. Each chromosome n the populaton of the GA resents the set of constants used to calculate the ractve and ulsve forces. Smulated sonar sensors are used to detect obstacles and the room walls. Intermedate goal ponts (the door postons of each room n a buldng) consttute the only a prory knowledge used for global path plannng. The problem of fndng a feasble robot path s approached as an teratve search for an optmum confguraton of the potental feld forces. In the followng secton we descrbe the artfcal potental (scalar) feld and the correspondng (vector) force feld functons used n ths work. In secton 3 we present the constructon of the GA for optmsaton of the force feld durng autonomous robot navgaton. In secton 4 we present the results of 3 smulaton experments performed on 3 dfferent obstacle confguratons. In secton 5 are presented the conclusons drawn from the work orted.

2 Potental Feld and Force Feld Functons The robot s resented as a partcle under the nfluence of an artfcal potental feld U, defned as: U = U + U (1) where: U and U are the ractve and ulsve potentals respectvely. The racton nfluence tends to pull the robot towards the target poston, whle ulson tends to push the robot away from the obstacles. In a two dmensonal map, the vector feld of artfcal forces F(q) s gven by the gradent of U : F (q) = U () where: U s the gradent vector of U at q(x,y) robot poston. In ths manner, F s defned as the sum of two vectors F (q) = U and F (q) =, as shown n eq. 3. U F (q) = U (3) U.1. Attractve and Repulsve Forces For stablsaton purposes[5], the potental feld U s defned as a parabolc functon, both, U and U are also defned n ths way. The artfcal force F s obtaned from the dervaton of U, both F and F are dervatves of parabolc functons. The followng artfcal racton force F s used n ths work for the goal pont, and for each of the 4 auxlary racton ponts: 1 F ( q) = ξ ( q q a ) (4) q q where: q s the current poston of the robot; q a s the poston of an racton pont; ξ s a postve weght adjusted by the genetc algorthm. Eq. 4 s normalsed to produce an racton force ndependent of the dstance between the robot and the goal pont. The artfcal ulson force F s defned as: F ( q) = η * sqrt * d d a ( q q o ) ( d o. o ) 4 f d< d (5) F (q) = : f d > d where: q s the robot poston; q o s the obstacle poston; d= q qo d s the nfluence dstance; η s a postve weght adjusted by the genetc algorthm. As the robot gets closer to an obstacle, the ulson force of the closest obstacle cells grows n the opposte drecton of the robot trajectory. In the event of a collson wth an obstacle, the value of F s bounded by settng the mnmum value of d to.5. On the other hand f the robot dstance to an obstacle cell s hgher than d, that obstacle cell has no effect on the robot. 3 Robot Navgaton Approach The robot s resented as a partcle R that moves n the confguraton spacec, modelled as a two dmensonal grd, where each cell U nsdec can be occuped by the robot, the goal or the obstacles. There s also an assocated obstacle map M of the same sze of C. The obstacle map s ntally empty, and t s flled at the postons of the obstacles detected by the robot, as t moves nsde C. The goal cell, and 4 auxlary racton cells exert an ractve force on R gven by Eq. 4, whle each of the detected obstacle cells exerts ulson forces gven by Eq. 5. For obstacle detecton, a 5x 5 grd smulates the robot sensors. When R moves, the postons of the sensors n the mask are updated and used to calculate the dstance d mn to the closest detected obstacle (Fg.1). A predefned dstance of 5 s assgned to obstacles outsde of the detecton mask. Fg.1, Obstacle detecton

3 f In order to avod trap stuatons or oscllatons n the presence of large or closely spaced obstacles [5], 4 auxlary ractve cells have been placed around the goal cell (Fg. ). Each ractve force F of the cell c, depends on the correspondng value of ξ, whch s automatcally adjusted by the genetc algorthm descrbed n secton 3.3. collson) s severely penalsed n order to avod the selecton of the correspondng chromosome n the next generaton. In Fg. 3 s shown the plot of Eq.6 for: <=E<=44,.1<= d mn <=5, and F = 1. r Fg., Attracton feld composed of 5 racton cells wth adjustable force ntensty. 4 3 dmn 1 1 E Objectve functon for robot navgaton An adhoc objectve functon was constructed to evaluate force feld confguratons whch correspond to an optmum robot poston (.e. postons closer to the goal cell whch also avod obstacles). The objectve functon value of each canddate force feld confguraton s evaluated wth three crtera: mnmsaton of the error dstance E between the robot and the goal cell; maxmsaton of the dstance d mn to the closest obstacle cell; and mnmsaton of the magntude of the resultant ulson force vector F r. Eq. 6 shows a functon whch produces optmum (mnmum) values for mnmum E, maxmum d mn and mnmum magntude of F r. f ( q) = E / ( q) = e d mn F r :f d mn > (6) f :f d mn = where: E = qrx qgx + qry qgy q r s a canddate cell for the new robot poston; q g s the goal cell; F r s the resultant ulson force vector. The constructon of the objectve functon (f ) favours robot paths that run away from the obstacles and result n decreasng dstance to the goal cell. The case where d mn = (whch corresponds to a Fg. 3, Plot of Eq.6 for: <=E<=44,.1<= d mn <=5, and F = 1 ; r 3.3. The genetc algorthm The prncples of operaton of a GA are presented n [6]. In ths work a GA s used to optmse the values of the weghts ( ξ ) of the 5 racton cells and the values of the weghts ( η j ) of up to 155 obstacle cells. Each varable has a range of {, 1} and s bnary coded wth bts of resoluton n order to mantan a large number of values for the ulson and racton forces. A chromosome s formed by concatenaton of the 16 bnary coded varables. The GA searches for optmum values of ξ and η j n a gven bnary strng (chromosome) whch move the robot to a poston such that f (Eq 6) has a mnmum value. Only those η j whch have been detected by the robot are used to calculate the force felds gven by Eq. 5, the rest of the ulson weghts n the strng s gnored. At each generaton of the GA, every chromosome n the current populaton s evaluated usng Eqs. 4, 5, and 6. The selecton probablty (P s ) of a gven chromosome s determned wth a rankng method [7]. Each chromosome n the current populaton s assgned a number of copes wth probablty P s usng stochastc unversal samplng [8]. Sngle pont crossover s appled wth a probablty of.6, mutaton s appled to each strng wth a probablty of.1 per bt. Fnally, the next generaton of the

4 GA s formed usng ftness based renserton wth a generaton gap of.8. Ths process contnues untl the robot reaches the goal cell or generatons are completed. 4 Experments and results The algorthm was mplemented n Matlab usng the GA toolbox developed at the Unversty of Sheffeld [7]. For evaluaton we used a cell map of 4x4 cells smulatng a 5-room floor. Shown n Fg. 4a counter clockwse from the top-left corner: the frst room smulates an storage room, the next room smulates a hosptal bed, the next room smulates a meetng room, the next room s empty wth random obstacles, and the last room s a rotated hosptal bed. Three dfferent random obstacle dstrbutons of dfferent obstacle denstes were used ( Fgs. 4a, 4b, 4c). Ten experments were performed for each obstacle dstrbuton, the start and goal postons for each experment are shown n table 1, the orgn s placed at the top-left corner of the cell map. Two ntermedate goal ponts have been used to gude the robot through the corrdor corner as well as through the door of the approprate room. The postons of the ntermedate goal ponts are also shown n table 1. The robot travels from the start poston to each successve ntermedate goal pont and to the fnal goal pont. Exp. (start)-(goal) ntermedate ntermedat No goal 1 e goal 1 (34, 9)-(11, 3) (,1) (15,1) (34, 9)-(13, 14) (,1) (15,1) 3 (34, 9)-(3, 6) (,1) (15,38) 4 (34, 9)-(36, 7) (,1) (5,38) 5 (34, 9)-(37, 14) (,1) (5,) 6 (34, 4)-(3, 6) (,1) (15,1) 7 (34, 4)-(3, 14) (,1) (15,1) 8 (34, 4)-(1, 6) (,1) (15,38) 9 (34, 4)-(3, 3) (,1) (5,38) 1 (34, 4)-(38, ) (,1) (5,) Table 1. Start-goal and ntermedate goal postons of each experment In table are shown the results of the 3 experments performed, the frst column shows the obstacle dstrbuton used (Fg.4), and the (start)-(goal) poston number as prevously shown n table 1. Columns two and three show respectvely, the total dstance travelled by the robot (measured n cells), and the devaton (as a percentage) from the optmum shortest path. Exp.No. Total dstance (cells) Devaton from optmum 1 (%) a a a-3 collson collson a a a a a a a b b b-3 collson collson b b b b-7 collson collson b-8 collson collson b b c c c-3 collson collson c c c c c c c Average:.7% Table. Experment results: Total dstance 1, and Devaton from optmum 1 obtaned wth auxlary cells placed at fxed dstance (15 cells) from the goal From the results shown n table the success rate (.e. the percentage of paths completed wthout collson) s 83.3%. The average devaton from the optmum path length s 1%. In fgure 5 are shown 5 paths produced by the navgaton approach for obstacle confguraton (c) (as shown n Fg.4), whch has the hghest obstacle densty. The average tme for path completon on a Pentum III PC at 75MHz s 115s wth an average path length of 56 cells (.e..5 s/step).

5 (a) low densty (c) hgh densty (b) medum densty Fg.4, Obstacle dstrbutons tested c-1 c- c-3 c-4 c-5 Fg. 5, Paths produced by the navgaton algorthm, for maxmum obstacle densty. 5 Conclusons An autonomous robot navgaton algorthm has been developed. The scheme would enable a moble robot, equpped wth sonar sensors, to navgate through unknown obstacle dstrbutons. Intermedate goal ponts have been used to gude the robot through corrdor corners and through the door of the approprate room n a smulated floor plan. Ths knowledge s avalable from the floor plan of any buldng. Gven an start and end poston, ntermedate goal postons, can be easly calculated (e.g. usng rules). The potental felds approach has been used and modfed to allow avodance of large or closely spaced obstacles, through the use of auxlary racton cells wth adjustable force strength, 4 auxlary cells have been used n ths work provdng good smulaton results. A genetc algorthm has been used for onlne optmsaton of the force ntensty parameters of the ulson and racton cells, as well as the poston parameter of the auxlary racton cells. The scheme has been able to complete ten dfferent paths n three dfferent unknown obstacle confguratons wth a success rate of 83.3%. The total estmated processng tme (.e. ultrasound scannng plus force feld optmsaton) of an mplementaton n C of the scheme orted s.36 s/step, ths value would allow for a maxmum robot speed of.7 m/s (wth a cell sze of.1 x.1m). Ths s wthn acceptable robot speed values for real applcatons. References: [1] Khatb O. Real-Tme Obstacle Avodance for Manpulators and Moble Robots. In: Autonomous Robot Vehcles, I.J. Cox and G.T. Wlfong, Eds. Sprnger-Verlag 199; [] Vadakkepat, P., Kay Chen Tan, Wang Mng- Lang. Evolutonary artfcal potental felds and ther applcaton n real tme robot path plannng. In: Proceedngs of the Congress on Evolutonary Computaton ; July: [3] Kun-Hsang Wu, Chn-Hsng Chen, and Jung- Mng Ko. Path plannng and prototype desgn of an AGV. Mathematcal and Computer Modellng 1999; 3: [4] McFetrdge L. and Yousef Ibrahm M. New technque of moble robot navgaton usng a hybrd adaptve fuzzy-potental feld approach. Computers nd. Engng 1998; 35 (3-4): [5] Koren Y., Borensten J. Potental feld methods and ther nherent lmtatons for moble robot navgaton. In: Proceedngs of the IEEE Int. Conf. on Robotcs and Automaton 1991;

6 [6] Goldberg D.E.: Genetc Algorthms n Search, Optmsaton, and Machne Learnng. Addson- Wesley, MA, (1989). [7] Chpperfeld, P. Flemng, H. Pohlhem, C. Fonseca.: Genetc Algorthm Toolbox for use wth Matlab. User's Gude. Automatc Control and Systems Engneerng, Unversty of Sheffeld, UK (1995). [8] Baker, J.E.: Reducng bas and neffcency n the selecton algorthm. Proc. nd Int. Conf. on Genetc Algorthms. Hllsdale, N.J., USA, 1987; 14-1.