Robot Motion Planning Using Neural Networks: A Modified Theory

Transcription

1 9 Robot Moton Plannng Usng Neural Networs: A Modfed Theory Subhrajt Bhattacharya, Sddharth Talapatra Department of Mechancal Engneerng, IIT Kharagpur Abstract -- A based on compettve learnng has been developed for robot moton plannng. The confguraton of a robot wth n degrees of freedom s descrbed by a pont n an n- dmensonal varable Space. The varable space has been dscretzed nto a fnte number of cells each of whch s assocated wth a neural actvty. A modfed verson of a shuntng equaton descrbng the dynamcs of the neurons has been presented. A new algorthm for updatng the neural actvtes has been suggested. The developed here has been compared to a prevous wth the ad of smulatons. Ths comparatve study clearly hghlghts how the modfed enhances the target reachng capablty of the robot. I. INTRODUCTION One of the ey ssues n moble robot systems s real tme path plannng. Autonomous navgaton nvolves obstacle avodance along wth movng towards a target. There have been several approaches towards ths ssue, encompassng both the statc and dynamc envronment problem. One approach to autonomous navgaton s the wallfollowng method [1, 2]. Here the robot navgaton s based on movng alongsde walls at a predefned dstance, whle consderng the obstacles as just another wall. Though computatonally less demandng, ths approach has only specfc applcatons (a floor cleanng robot n a long hallway). An mprovement upon ths method s the edge detecton approach [3, 4], where the postons of the vertcal edges of the obstacle are determned and the robot s steered around ether edge. However ths approach s heavly dependent upon sensor accuracy. A powerful scheme for autonomous navgaton s the Potental Felds Method, orgnally suggested by Khatb [5]. Here obstacles are consdered as centers of repulson and targets as centers of attracton (wth global effect). The robot traverses the path of least potental gradent. A shortcomng of ths method s that t assumes a nown and prescrbed world of obstacles. Later a called the Vrtual Force Feld Model [6] was developed whch employed the ntegraton of the above mentoned concept and the concept of Certanty Grd. Ths paper deals wth the use of Neural Networs n autonomous navgaton. Neural Networ approaches have been used n qute a few path plannng algorthms [7], where a Hopfeld type of networ wth contnuous neurons was employed. In ths paper we have used a loosely based path traversal to trac down a target movng n an arbtrary fas hon. Unle most approaches, ths method does not requre the global pcture, each neuron n the neural networ havng only local connectons. The constantly updatng neural actvtes generate an actvty landscape whch gudes the robot towards ts goal. All ths happens n real tme due to whch a fast movng target can be easly traced down. neuron corresponds to a target, I = -E f t s an obstacle, otherwse I = 0. E generally s a large postve quantty. [ I ] s the nhbtory nput caused by the obstacles, whle on a Compettve Learnng for real tme obstacle free Internatonal Journal of Lateral Computng IJLC, Vol.2, No.1, December 2005, ISSN X Subhrajt Bhattacharya and Sddharth Talapatra, IIT Kharagpur, Robot Moton Plannng usng Neural Networs: A Modfed Inda II. THE MODEL Our s essentally based on the bologcally nspred neural networs developed by Yang and Meng [8]. A bref dscusson on the orgnal wll be frst presented. It wll be followed by the modfcatons made as part of our present wor. Based on the physcal problem, a Neural Networ archtecture [8] s decded upon. The N-Dmensonal Neural Networ archtecture corresponds to an N-Dmensonal robot confguraton space ζ. The correspondng N-dmensonal Neural actvty landscape bascally represents the dynamcally varyng envronment of the confguraton space. The neurons are spatally arranged n the dscretzed N- dmensonal landscape representng the confguraton space ζ. Wth the th neuron s attached a quantty called Neural Actvty x and an external nput I. The shuntng equaton [9, 10] determnng the dynamc neural actvty of the th neuron s: dx dt = Ax + B x I + w x ( D+ x)[ I ] + + ( ) [ ] j[ j] 1 (1) Parameters A, B and D ndcate the passve decay rate, the upper and lower bounds of the neural actvty, respectvely. Varable x s the neural actvty of the th neuron, a contnuous varable [ D, B]. The nput I = E f the th

2 [ I] + + wj[ xj] + s the exctatory nput resultng from j = 1 the target and the local lateral connecton among the neurons. The nonlnear functon [a] + and [a] - are defned as, [a] + = max{a, 0}, and [a ] - = max{-a, 0}. q and q j represent the poston vectors to the th and j th neurons respectvely. The connecton weght w j from the jth to the th neuron s defned as w j = f( d j ), where dj = q qj s the dstance between the th and q th neurons n ζ. f( a ) s a monotoncally decreasng functon. The functon used by Yang and Meng [8] was, f( a) µ for 0< a< r a 0 for all other a 0 = (2) µ and a are postve. Ths functonal form ensures that a neuron has local connectons n a small regon around t of radus r 0 neurons. The shuntng equaton s such that the target has global mpact as the neural actvty of a target propagates whle that of an obstacle doesn t. So effect of obstacles s only local. Robot moton s decded by choosng a wnnng neuron based on the neural actvty landscape. For a gven present locaton n S, denoted by qp, the next locaton qn (also called command locaton ) s obtaned by, q x = max( x, = 1,2, L, ) (3) n qn The neghborng neurons of the neuron correspondng to the present locaton of the robot n the confguraton space are consdered, and the robot moves to the poston of one of the neg hborng neurons (ncludng the neuron correspondng to the current poston) havng the maxmum neural actvty. The values chosen by Yang and Meng [8] for the dfferent parameters n ther smulatons were, A=10, B=D=1, µ=1, r 0 =2 and E=100. Before dscussng the achevements of the present wor, some observatons on equatons (1), (2) and (3) ought to be made, whch formed the bass of our development.. Accordng to (3), a neuron wth hgher value of x has greater potental for attractng the robot, whle the one wth a lower value somewhat repels t.. The frst term n (1) always tres to brng the value of x closer to 0. For the second term, f x becomes greater than B, the term tres to reduce the value of x, else t tres to ncrease t. And as far as the thrd term s concerned, t tres to decrease the value of x tll x s greater than D. That s how B and D form the upper and lower bounds of x.. The second term uses the [ ] + operator, hence tang nto account the nfluence of the postve values (.e. potental targets or attractors) of I and xj. Whle the thrd term usng the [ ] - operator taes nto account the nfluence of the negatve values (.e. potental obstacles or repellers) of I. v. It s only n the second term that the nfluence of the nearby neurons s consdered. It s of nterest to note that whle tang nto account the nfluence of the neghborng neurons, the quantty [x j ] + has been chosen. Ths ensures that nearby neurons wth negatve actvty have no nfluence on the actvty of the th neuron. Thus t mples that although there s a way n whch nformaton about a target may propagate from one neuron to ts neghbor, the nformaton about an obstacle can t propagate n smlar fashon f we use the shuntng equaton (1). Ths somewhat gve preference of beng attracted towards a target over beng repelled from an obstacle. In fact, when placed n a complex maze wth a dstant target, the robot often stumbled over an obstacle or cut through t! Such examples demonstrated the mentoned observaton. v. Though the summaton n (1) was mentoned to be done over all the neghborng neurons, ncludng tself, from (2) we get w? 8, snce d = 0. Wth the present form of (2) there are scopes of dvergence. Hence (2) can be modfed. v. Accordng to (3), as the new poston of the robot s to be determned by fndng the neghborng neuron wth hghest actvty, ncludng tself, t s hghly probable that the robot may trace bac ts path when t encounters a complex maze ahead. Or n the worst case, the robot may eep on oscllatng wthn a few postons. Hence t s logcal to gve lower preference to those neurons whose postons have already been traversed by the robot. Keepng n vew the above mentoned observatons, the was modfed as follows: In lght of observaton v., (1) was modfed as, dx dt = Ax + B x I + w x ( ) [ ] α j[ j] 1 ( D+ x) [ I] + β wj[ xj] 1 (4) where, 0=a =1 and 0= ß=1. Note that a=1, ß=0 gves bac the orgnal Yang and Meng s. But through experments t was Internatonal Journal of Lateral Computng IJLC, Vol.2, No.1, December 2005, ISSN X Subhrajt Bhattacharya and Sddharth Talapatra, IIT Kharagpur, Robot Moton Plannng usng Neural Networs: A Modfed Inda

3 observed that a small postve value of ß maes the more effcent n complex mazes. 11 In lght of observaton v., (2) was modfed as, µ for 0< a < r0 f( a) = a for all other a (5) To solve the problem mentoned n observaton v., an nnovatve technque was devsed. In order to nsure that the robot has a lower tendency to go bac to a poston t has recently vsted, the external nputs I of those neurons are temporarly gven a small negatve value. For example (as used n our smulatons), the poston vsted n the last tme step s gven a value E/8, the one vsted two tme steps before s enforced wth a value of E/16 and the one vsted three tme steps before s enforced wth a value of -E/32. All stll prevously vsted postons are returned to ther orgnal values of I. And t was also made sure that the robot doesn t get stuc n ts own poston by mang the comparson stated n (3) among the neghborng neurons except that of the presently occuped poston. Hence the robot s forced to mae a move, preferably to a new poston, n every tme step. Fg.1a. The maze Fg.1b. Yang-Meng Fg.1c. The modfed Maze II III. RESULTS AND COMPARISONS In ths secton a comparatve study wll be presented on the performances of the orgnal of Yang and Meng and our modfed, under varous cases of complex mazes and target locaton. The parameters used wth the orgnal (Yang and Meng) have already been mentoned. The correspondng parameters used by us n the modfed are: E = 100, A = 10, B = 5, D = 1, r0 = 1.41, alpha = 0.9 and beta = 0.2. Followng are some smulaton results, where the dotted lne shows the path t raversed by the robot. The problem presently under study conssts of two dmensonal mazes and uses a neural networ actvty landscape wth same topology as physcal space. Fg.2a. The maze Maze I Internatonal Journal of Lateral Computng IJLC, Vol.2, No.1, December 2005, ISSN X Subhrajt Bhattacharya and Sddharth Talapatra, IIT Kharagpur, Robot Moton Plannng usng Neural Networs: A Modfed Inda

4 12 Fg.2b. Yang-Meng Fg.2c. The modfed Fg.4a. The maze Maze III Fg.4b. Yang-Meng Fg. 4c. The modfed Fg.3a. The maze Fg.3b Yang-Meng Fg.3c. The modfed Maze IV In the frst llustraton, we have consdered a maze whch has a well guded path leadng to the target, the sort of examples that were consdered n [8]. Here we see that both s perform equally well. In the second case, the maze consdered prevously has been slghtly modfed so as to mae the target more naccessble. The orgnal gudes the robot to a pont separated from the target by a sngle cell of the obstacle. The robot s trapped and hence unsuccessful n ts msson. The modfed however manages to gude the robot out ths trap. Ths can be attrbuted to the combned effect of two modfcatons: excludng the current neuron as a possble canddate for the wnner neuron, and mang the robot reluctant to traverse a path already covered. The last behavor s due to the thrd modfcaton shown n the last secton. The thrd maze s a smple case whch hghlghts the postve effects of the modfcatons made. In the orgnal the robot stumbles over an obstacle due to the total lac of propagaton of obstacle nformaton. The small non-zero beta n the new term ntroduced n ths (see equaton (4)) removes ths defcency. Hence we fnd that n our, the robot manages to fnd ts way around the obstacle and reach ts goal. The fourth maze s the most challengng of the lot. Once agan the proposed by Yang and Meng fals and gets stuc due to ts nablty to effectvely represent obstacles n the neural actvty map. In the modfed, the robot s strongly repelled by the obstacles and after consderng dfferent paths, t manages to fnd the correct path. Internatonal Journal of Lateral Computng IJLC, Vol.2, No.1, December 2005, ISSN X Subhrajt Bhattacharya and Sddharth Talapatra, IIT Kharagpur, Robot Moton Plannng usng Neural Networs: A Modfed Inda

5 13 IV. CONCLUSION In ths paper a for autonomous robotc moton gudance for obstacle avodance has been developed. It s loosely based on a compettve learnng. It was seen that n the neural actvty map, each neuron had only local connectons and the global pcture was not requred durng the path plannng. The neural actvtes are not allowed to attan values outsde the prescrbed range. The stablty and convergence of the orgnal shuntng equaton has been proved rgorously usng the Lyapunov stablty theory (Grossberg, 1988). Hence the system s stable. The developed n ths paper has been shown to perform better n complcated mazes, especally when the path to the target s not well guded. Smulatons have been made depctng the relatve performances of the two s n four dfferent mazes. It has been clearly shown how the modfcatons ncorporated by us have affected the performance of the robot. Unle the prevous, propagaton of neural actvty due to obstacles taes place, characterzed by the parameter ß. Also the robot s reluctant to trace a path already traversed due to a dfferent neural actvty updatng algorthm. Our algorthm forces the robot to move on to a dfferent cell n every teraton. All ths prevents the robot from gettng stuc at one place and snce the robot has a greater tendency to move to new places, ts ablty to reach ts goal s enhanced. However there s a shortcomng to ths algorthm. In certan cases, especally n those where the Yang-Meng fals, the robot taes sub optmal routes. A judcous choce of the parameters can fx ths problem, and ths can be acheved by optmzng the parameter values for shortest possble path. ACKNOWLEDGMENT We would le to than Prof. S.K. Bara, Deptt. of Cvl Engg., IIT Kharagpur for hs nvaluable gudance and suggestons whch helped us n completng the present wor. REFERENCES [1] Gralt, G., "Moble Robots." NATO ASI Seres, Vol. F11, Robotcs and Artfcal Intellgence, Sprnger-Verlag, 1984, pp [2] Ijma, J., Yuta, S., and Kanayama, Y., "Elementary Functons of a Self-Contaned Robot "YAMABICO 3.1." Proc. of the 11th Int. Symp. on Industral Robots, Toyo, 1983, pp [3] Borensten, J., "The Nursng Robot System." Ph. D. Thess, Technon, Hafa, Israel, [4] Borensten, J. and Koren, Y., "Obstacle Avodance Wth Ultrasonc Sensors." IEEE Journal of Robotcs and Automaton., Vol. RA-4, No. 2, 1988, pp [5] Khatb, O., "Real-Tme Obstacle Avodance for Manpulators and Moble Robots." 1985 IEEE Internatonal Conference on Robotcs and Automaton, March 25-28, 1985, St. Lous, pp [6] Borensten, J. and Koren, Y., 1989, "Real-tme Obstacle Avodance for Fast Moble Robots." IEEE Transactons on Systems, Man, and Cybernetcs, Vol. 19, No. 5, Sept./Oct., pp [7] J.J.Hopfeld, Neural networs and physcal systems wth emergent collectve computatonal abltes, Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, 79:2554, [8] Smon X. Yang, Max Meng, An effcent neural networ approach to dynamc robot moton plannng, Neural Networs, 13 (2000), [9] Hodgn, A. L., & Huxley, A. F. (1952). A quanttatve descrpton of membrane current and ts applcaton to conducton and exctaton n nerve. Journal of Physology (London), 117, [10] Grossberg, S. (1988). Nonlnear neural networs: prncples, mechansms, and archtecture. Neural Networs, 1, Internatonal Journal of Lateral Computng IJLC, Vol.2, No.1, December 2005, ISSN X Subhrajt Bhattacharya and Sddharth Talapatra, IIT Kharagpur, Robot Moton Plannng usng Neural Networs: A Modfed Inda