Solving the Local Minima Problem for a Mobile Robot by Classification of Spatio-Temporal Sensory Sequences

Transcription

1 Solvng the Local Mnma Problem for a Moble Robot by Classfcaton of Spato-Temporal Sensory Sequences K. Madhava Krshna, Prem K Kalra* Department of Electrcal Engneerng Indan Insttute of Technology Kanpur, Inda e-mal: kalra@tk.ac.n Receved 3 January 2000; accepted 28 June 2000 The local mnma problem occurs when a robot navgatng past obstacles towards a desred target wth no pror knowledge of the envronment gets trapped n a loop. Ths happens especally f the envronment conssts of concave obstacles, mazes, and the lke. To come out of the loop the robot must comprehend ts repeated traversal through the same envronment, whch nvolves memorzng the envronment already seen. Ths paper proposes a new real-tme collson avodance algorthm wth the local mnma problem solved by classfyng the envronment based on the spato-temporal sensory sequences. A double layered classfcaton scheme s adopted. A fuzzy rule base does the spatal classfcaton at the frst level and at the second level Kohonen s self-organzng map and a fuzzy ART network s used for temporal classfcaton. The robot has no pror knowledge of the envronment and fuzzy rules govern ts obstacle repulsng and target attractng behavors. As the robot traverses the local envronment s modeled and stored n the form of neurons whose weghts represent the spato-temporal sequence of sensor readngs. A repetton of a smlar envronment s mapped to the same neuron n the network and ths prncple s exploted to dentfy a local mnma stuaton. Sutable steps are taken to pull the robot out of the local mnma. The method has been tested on varous complex envronments wth obstacle loops and mazes, and ts effcacy has been establshed John Wley & Sons, Inc. * To whom all correspondence should be addressed. ( ) ( ) Journal of Robotc Systems 17 10, by John Wley & Sons, Inc.

2 550 Journal of Robotc Systems INTRODUCTION Path plannng s one of the key ssues n moble robot navgaton. Autonomous moble robots are used n varous applcatons such as n automatc freeway drvng, 1 cleanng of hallways, 2 and exploraton of dangerous regons. 3 These applcatons demand robust and adaptable methods for path plannng. Path plannng s tradtonally dvded nto two categores, global path plannng and local path plannng. In global path plannng, pror knowledge of the workspace s avalable. Some of the global approaches nclude the confguraton space method, 4 potental feld method, 5 the generalzed Vorono dagram, 6 and free space representatons. 7 The plannng s done offlne and the robot has complete knowledge of ts work area and ts path when t starts. One of the man ssues n ths area of work s to reduce the tme complexty of these algorthms 8 10 and mnmze the cost of the search. Local path plannng methods use ultrasonc sensors, laser range fnders, and on-board vson systems to perceve the envronment and plannng s done onlne. The workspace for the navgaton of the moble robot s assumed to be unknown and consstng of statonary obstacles. One of the earlest mplemented strateges of ths type s the wall followng method, 11 where the robot moton s based on movng adjacent to the walls at a fnte dstance. The drawback s that the algorthm fals n cluttered envronments and f the walls are n the form of loops. Other onlne approaches to path plannng nclude edge detecton, 12, 13 vrtual force feld, 14 and fuzzy logc methods. Fuzzy rulebase technques have an advantage n that they do not requre an analytcal model of the envronment, but desgnng the rulebase s heurstc. Though fuzzy path plannng schemes work well n cluttered envronments they fal n envronments where the rules that are fred for target attractor and obstacle repulsor modules gve output actons that neutralze each other and the robot gets nto an nfnte loop or a local mnma. There are methods n lterature that tackle the local mnma problem such as the Bug algorthms of Lumelsky 18, 19 20, 21 and ther mprovements. Recently a vrtual obstacle approach 22 and a vrtual target approach 23 have been proposed as possble solutons for the local mnma problem. Wall followng s a common strategy adopted n general for surmountng the local mnma stuaton and many algorthms slp nto a wall followng mode upon encounterng the frst obstacle en route to the target. 20, 22, 23 However, the approaches do try to evaluate whether the robot s n a trapped stuaton before a sutable strategy s nvoked to overcome the mnma. Among ths the approach of Huang 20 makes an emprc guess to dentfy the robot s trapped condton whle the recent strateges 22, 23 are more robust n determnng the same. The man contrbuton of ths artcle s n the dentfcaton of the local mnma stuaton durng the robot s traversal, much akn to the way a human mght understand hs trapped state by recollectng some of the landmarks he had seen n hs last traversal of the same envronment. Ths remembrance s provded by a classfer network, whch classfes the spato-temporal sequences of sensor readngs. A two layered classfcaton scheme s employed. At the frst layer fuzzy rulebase does the spatal classfcaton. At the second layer Kohonen s SOM 24 and the fuzzy ART 25 network s used for learnng and classfyng the temporal sequences of spatal patterns classfed by the frst layer. The classfcaton scheme mparts an understandng of the robot s local envronment and correlates the same wth prevous experences of a smlar envronment. The robot s local envronment s classfed n terms of landmarks through each neuron n the SOM or ART layer whose weght vector represents a landmark. When the robot sees a smlar landmark at the same spatal locaton where t had seen t prevously t understands ts entanglement n a loop. Sutable actons are then taken to pull the robot out of ts trap. The SOM nterprets the local envronment of the robot n terms of landmarks learnt offlne and can recall prevously stored patterns but s not plastc enough to new stuatons. The onlne classfcaton property of fuzzy ART makes the classfer plastc to new patterns but ts ablty to understand the envronment s poor compared to the SOM. Hence the SOM and ART networks are employed n the second layer to aval the advantages of both. The organzaton of the artcle s as follows. A fuzzy navgaton scheme for gudng the robot past unknown obstacles s dscussed n secton 2. The learnng of the spato-temporal sequence of sensory nputs s dealt n secton 3. The robustness of the algorthm s shown through smulaton n secton 4. Secton 4 also compares the present algorthm wth an exstng approach, 20 whch s an extenson of

3 Krshna and Kalra: Local Mnma Problem for a Moble Robot 551 Lumelsky s Bug algorthm. Secton 5 presents the conclusons and future drectons. 2. REAL-TIME NAVIGATION PROBLEM The objectve of the sensor-based navgaton of a moble robot s to reach the target n any unknown workspace, cluttered wth obstacles of any shape, sze, and orentaton. The decson for the proper turn angle of the moble robot s taken based on sensory nformaton and the angular dfference between the robot s current drecton of moton and the goal orentaton wth respect to orgn of the reference frame. The nput space, U, of the moble robot can be represented as T u u u u U s df T where u s u0 u1 u2 u3 u4 u5 u 6 T, the sensor vector, denotes the range readngs obtaned by the seven sensors placed n the form of an arc on the crcumference of the robot, subtendng an angle of 105 degrees at the center. The nput udf denotes the angular dfference between the moble robot s nstantaneous drecton vector and the vector jonng the robot s center to the target, also called as the dfference angle. Ultrasonc sensors are n general used to obtan nformaton regardng the local envronment. In our smulaton we have used an array of seven ultrasonc sensors Ž They have been grouped as the left Ž sensors 0 2., center Ž 3., and rght Ž 4 6. arrays. The actual ntrcaces nvolved n the measurement of tme of flght values usng ultrasonc sensors have not been modeled n our smulaton algorthm Fuzzy Inference Engne The sensor nputs whch are n the form of pxels from the current locaton of the robot to the obstacle poston are normalzed to 0, 1. The nference engne parttons the problem nto two man modules. One module governs the target reachng acton and the other governs the obstacle avodance acton of the robot. Target Reachng Module The nput dfference angle n the range of 180, 180 s fuzzfed usng the membershp functon, whch can be represented as u df for 90 udf 0 90 Ž u. Ž 1. left df udf 90 0 otherwse u df for 0 udf 90 Ž u. 90 Ž 2. rght df 1 90 udf otherwse The defuzzfed output turn angle due to the target reachng behavor, y t, s obtaned as c Ž u. c Ž u. t l left df r rght df y Ž 3. Ž u. Ž u. left df rght df where c l, cr are constants whose values take 9 and 9, respectvely. Obstacle Avodance Module The membershp functons that determne the degree of farness or nearness to the obstacle are defned as follows. The nputs whch are dstances of the obstacles to the sensors are measured n terms of pxels and normalzed to 0, 1. near Ž u. 2.5 Ž u. near u u 1 0 otherwse Ž 4. 4 where I 0, 1,...,6 2.5 Ž u. far u 7 Ž u. far u 1 0 otherwse Ž 5. 4 where I 0, 1,...,6 The output defuzzfed turn angle due to the obstacle avodance behavor, y o, s gven by Ý Ý n Ž u. f Ž u. near far o y Ž 6. Ž u. Ž u. Ý Ý near far

4 552 Journal of Robotc Systems 2000 where 0, 6 and n, f are constants determned n a heurstc fashon. The net angle by whch the robot must turn s decded by assgnng weghts to the ndvdual modules dependng on the closeness of the obstacle to the robot. The weghts ascrbe the proporton of mportance to ether of the modules. The angle y by whch the robot shall fnally turn after assgnment of weghts can be represented as Ž. Ž. Ž. t y far ul far uc far ur y Ž Ž. Ž. Ž.. o near u l, near u c, near u r y Ž 7. where Ž A, B, C. A B C AB BC CA ABC and the factors by whch the angles due to target reachng, y t, and obstacle avodance behavor, y o, are multpled represent the weghts assgned to the modules. The robot turns by the angle y, obtaned as above, and traverses some dstance n that drecton before samplng the envronment for ts next nput. The dstance s Ž measured n pxels. the robot moves between any two samples of the envronment s obtaned as t o Ž. s w s w s where w 1, w2 are the weghts assgned n the same Ž. t o ven as n Eq. 7 and s, s are the dstances due to target reachng and obstacle avodance behavor, whch are gven by s Ž u. s Ž u. t 1 near t 2 far t s Ž 9. Ž u. Ž u. near t far t where ut represents the Eucldean dstance of the target from the current poston of the robot s center and s1 1, s2 3. s max Ž Ž u.. s mn Ž Ž u.. n near f far I I s o max Ž Ž u.. mn Ž Ž u.. I near far I Ž where I 0, 1,...,6. Though the above approach works well when the envronment s hghly cluttered t fals when the obstacles are n form of loops and bends due to the contradctng actons mposed by the target attractng and goal repulsng modules. The robot consequently s trapped n an nfnte loop. Ths s because a purely fuzzy approach fals to provde some knd of a memory or remembrance of the envronment the robot has already traversed, whch s needed for the robot to come out of such loops. Hence there s the need for ncorporatng memory n the navgaton algorthm, the topc of the next secton. 3. SPATIO-TEMPORAL CLASSIFIER Artfcal neural networks have proved through the last decade ther powerful classfcaton propertes as a result of ther nherent abstracton and generalzaton capabltes. Kohonen s self-organzng maps Ž SOM. and ART networks are one of the most popular unsupervsed learnng mechansms. The SOM has a specal property of effectvely creatng spatally organzed nternal representatons of varous features of nput sgnals and ther abstractons 24 whle the advantage of fuzzy ART s that t s very stable to prevously stored nputs and plastc to new nputs. Both the schemes have been used n varous applcatons such as speech recognton, 26 vector quantzaton, 27 and robotcs. 28, 29 A bref revew regardng the theory and learnng of a SOM network s gven n Appendx A and the fuzzy ART s gven n Appendx B Spatal Classfcaton The need for an ntal spatal classfer can be understood as follows. At any nstant t when the robot samples ts envronment t obtans the sensory vector u Ž t. u Ž t. u Ž t. u Ž t. u Ž t. u Ž t. u Ž t. s u Ž. 6 t T. At each nstant t a vector of dmenson seven s present as the nput and a sequence of such vectors must be learnt and stored by the classfer. In other words the classfer network must learn varous combnatons of sequences u Ž t., u Ž s s t 1,...,. u Ž t n 1. s. The complexty of such a learn- ng algorthm ncreases tremendously n space and tme. The classfer would need a network of over 20,000 neurons to store the varous combnatons wthout loss of sgnfcant data and the learnng tme nvolved s huge. In order to avod ths each of the seven dmensonal patterns at a gven nstant s mapped nto a partcular class wthout losng essental data. Ths can be consdered as a vector quantzaton problem and a fuzzy classfcaton scheme s employed for ths. Then the SOM or the ART can be traned on the sequence of quantzed vectors or classes rather than on a sequence of the nput vectors tself. The classfcaton s done as follows. In-

5 Krshna and Kalra: Local Mnma Problem for a Moble Robot 553 tally the sensory vector u Ž. s t s represented as u Ž. Ž. Ž. Ž. s t ul t uc t ur t T where l, c, and r are nterpreted as left, center, and rght sensor readngs. The sensors 0, 1, and 2 are grouped together as left and ther mnmum readng s taken as the readng of the left sensor;.e., u Ž t. mnu Ž t., u Ž t., u Ž t.4 l Smlarly u Ž t. mnu Ž t., u Ž t., u Ž t.4 and u Ž. r c t u Ž. 3 t. Now based on these left, center, and rght readngs a fuzzy classfcaton scheme s employed as shown n the table below. The left, center, and rght sensors readngs are gven to the fuzzy rule bases Ž descrbed n secton 2. and a readng s classfed as very near f the nearness degree of the readng s more than 0.9, as near f the nearness degree s n 0.3, 0.9, and as far f nearness degree s less than 0.3. Thus each nput vector ut Ž. s classfed nto one of the nne classes Ž as shown n Table I. and the SOM s traned over a sequence of such classes. The next ssue n consderaton s the upper bound on n, the number of classes n the sequence over whch the second layer of the classfer shall be traned. To obtan nformaton regardng the envronment t s essental to keep track of changes n the classes of a sequence. Often the changes n the sequence are more mportant than smply the sequence tself. It has been observed through experence wth varous smulaton envronments that all the mportant landmarks of an envronment can be represented by one or two changes of a sequence. For example, n Fgure 1Ž. a a robot meetng a dead end whle gong through a narrow corrdor s represented as over 15 tme nstants whereas n Fgure 1Ž b. the robot traverses to a dead corrdor of shorter length and hence the sequence obtaned s wthn nne nstants. Hence a robot passng through a corrdor to ht a dead end s represented by the followng temporal Table I. Spatal classfcaton of the sensory nput space by fuzzy rule base. Fgure 1. Ž. a Robot passng through a long corrdor wth a dead end. Ž b. A smlar shorter coorder. sequence T. The landmark s represented by two changes n the sequence vz. 3 1, 1 0. Hence the bound on the number of classes n the temporal sequence s fxed to be 3. In general a two or three dmensonal vector of classes can represent a landmark. The structure of the double layered classfer network s shown n Fg. 2. The structure represents the process of extractng the temporal order of the classes before gvng t to the second layer. As dscussed n the prevous paragraph any two consecutve classes n the fnal sequence to be classfed must be dfferent. The comparator Ž Fg. 2. whch compares the present class Ct Ž. wth the class of the prevous nstant Ct 1 Ž. does ths. When they are dfferent then Ct 1 Ž. s extracted to form the frst class n the sequence to be classfed by the second layer. The process s repeated and when three such classes are extracted the sequence, y, y, y Ž see Fg. 2. s nput to the second layer of the classfer Temporal Classfer The second layer as mentoned earler classfes the sequence of classes of dmenson three. The fundamental dfference n the way the SOM and ART networks learn and classfy gves rse to dfferent results. The SOM models the local envronment of the robot nto well defned landmarks such as corners Ž meetng of two walls., mazes, and blnd ends that are encountered n a typcal envronment. Each neuron n the SOM lattce codes a partcular landmark. Ths requres that the SOM be traned for Rght sensor Center sensor Left sensor Class Very near Very near Very near 0 Near Near Near 1 Near Near Far 2 Near Far Near 3 Near Far Far 4 Far Near Near 5 Far Near Far 6 Far Far Near 7 Far Far Far 8 Structure of the Spato-temporal Classfer Net- Fgure 2. work.

6 554 Journal of Robotc Systems 2000 varous knds of landmarks offlne before usng the traned lattce durng real tme. Durng real-tme navgaton the robot can encounter a temporal sequence of classes for whch the SOM was not traned. Ths can under some exceptonal stuatons stll gve rse to an nfnte loop stuaton, whch s dscussed n secton 4. One way to crcumvent ths problem s by tranng the SOM offlne for an exhaustve combnaton of sequences. Snce each sequence s of dmenson three, no two consecutve classes n a sequence are same, and the total number of classes as classfed by the spatal classfer s nne, the SOM shall be a map of 648 Ž 9X8X9. neurons. But most of the neurons represent combnatons of sequences, whch have no possblty of occurng n any real-tme envronment. As a matter of fact experments suggest 70% of such a lattce or 454 neurons do not wn at all n varous real-tme envronments. Another soluton for ths s to ncorporate an onlne learnng network n the second layer. Fuzzy ART wth a complement codng scheme serves as a good alternatve. Fuzzy ART s capable of learnng new patterns and smultaneously remembers the earler patterns to a reasonable accuracy. It elmnates the need for mantanng a map of large neurons, most of whch are dle. The ART network can dynamcally add new patterns accordng to the local envronment of the robot and can afford to forget patterns that have not come wthn a recent tme nterval, say n the last 100 samples of the envronment, thus memorzng the envronment to the extent needed. But the SOM s not entrely useless for t can correlate the local envronment seen at real-tme n terms of landmarks learnt offlne. Ths knd of classfcaton by the SOM provdes the algorthm wth an mmedate memory Ž IM. and s dscussed n secton 4. The ART, however, cannot provde for ths, as t does not understand a classfed pattern to represent a partcular landmark. It just learns a pattern, but s not sure what t represents n the real world. It should be noted that the terms mmedate and dstant memory used n ths artcle have nothng to do wth the long term and short term memores of the standard ART archtectures. The classfer network has been ncorporated wth both the SOM and ART network n ts second layer to aval the advantages of both. The SOM s a lattce of 49 neurons wth ther weghts representng some of the common landmarks, whle the ART s used as a knd of backup when the robot encounters a new pattern not learnt by the SOM. It has been found that the ART needs to mantan not more than 15 neurons whose weghts represent the most recent patterns not dentfed by the SOM. Thus ART also serves to reduce the space requred by the algorthm from 648 vectors, each of dmenson 3, to 64 vectors Ž 49 from SOM 15 due to ART. of the same dmenson Learnng Landmarks and the Vector of Lower Bounds Snce the SOM dentfes a real-tme sequence to a predefned landmark the mnmum number of tmes Ž the lower bound. each class must occur n a sequence s to be known. Ths prevents msdentfcaton of spurous sequences as landmarks. For example, n landmarks such as a narrow corrdor leadng to a dead end Ž Fg. 1., the frst class n the sequence, 3, must occur at least four tmes, the second class must occur at least twce, and the last should occur once. The mnmum number of tmes each class occurs n a sequence s termed the vector of lower bounds Ž Fg. 3.. A smulaton envronment conssts of varous landmarks, some of whch are shown n Fgure 4, consstng of corners Ž meetng of two walls. of varous orentatons and shapes, a long wall wth a narrow slt, half open doors, narrow corrdors wth dead ends, etc. These are landmarks that can be Fgure 3. Lattce neurons and the correspondng vector of lower bounds. Fgure 4. A smulaton envronment wth some of the landmarks.

7 Krshna and Kalra: Local Mnma Problem for a Moble Robot 555 expected n any typcal envronment. The robot s made to navgate by placng startng and target locatons across the landmarks. The robot navgates usng the fuzzy algorthm and the temporal sequences formed as descrbed n Secton 3.1 are stored as an array of vectors. Each vector n the array s a vector of classes. An example s shown n Fgure 5, where the start and target locatons are gven across a landmark n the form of a rght bend. The SOM s ntalzed wth a lattce of 7 7 Kohonen neurons wth normalzed random values. From the array a partcular vector s selected at random and presented to the lattce as the nput. The wnner s selected and the wnner and the neghborng neurons are updated as descrbed n Appendx A. The network s traned for several teratons. The number of teratons should be of two orders more than the total number of neurons n the lattce 30 for the SOM to capture the essental topology of the nput space, n ths case to dentfy clearly varous knds of landmark. Once the SOM has been traned for varous landmarks the robot s agan made to navgate across the same landmark wth reduced szes for obtanng the vector of lower bounds for that landmark. The smallest of such landmarks that the SOM can recall s dentfed. The number of occurrences of each class n the sequence for the smallest landmark s stored as a lattce neuron whose weghts represent the lower bounds. The neuron s stored n the poston correspondng to the lattce poston of the wnnng neuron n the SOM. Thus for each neuron n the SOM representng a partcular landmark by the vector c1 c2 c3 T there exsts a correspondng neuron whch stores the vector of lower bounds t1 t 2 t3 T where t1 represents the mnmum number of tmes c1 must occur n a sequence Ž Fg 3.. Otherwse, for example, the sequence 3, 1, 0 wthout consderng the lower bounds can possbly be somethng other than a narrow dead-ended corrdor. Fgure 6 shows how the SOM has been able to map smlar landmarks to neghborng postons n the lattce upon tranng. Tranng the SOM for a Fgure 5. Sensor sequences obtaned for SOM learnng by navgatng the robot across a rght bend. Fgure 6. Some of the landmarks coded by the Kohonen neurons. The weght vector s shown along wth landmarks. partcular landmark nvolves consderable vsualzaton by the user wth respect to the partcular shape of a landmark and varous szes of that shape. It becomes an nvolved affar to tran the SOM for the exhaustve combnaton of 648 sequences and ther correspondng lower bounds. Wthout the vector of lower bounds a map of 648 neurons can dentfy any sequence n real-tme but the obvous advantage of SOM s lost. The same can be done wth much less memory by the fuzzy ART archtecture and wthout offlne learnng. The classfcaton by fuzzy ART archtecture s a relatvely smpler affar. As the robot navgates through the envronment the sensory nputs are obtaned and classfed by the frst layer of the classfer network. The temporal order of classes s extracted. A sequence of three such classes wth no two consecutve classes beng dentcal forms the nput vector for the fuzzy ART. The fuzzy ART maps the sequence wth an earler one or adds a new spato-temporal pattern to the exstng set of patterns. The decson for addng a new pattern s decded by the vglance parameter, whch s set to 0.9 n our algorthm. A revew of the fuzzy ART algorthm s gven n Appendx B. 4. SIMULATION RESULTS AND COMPARISON To test the effcacy of the algorthm a graphcal smulator has been developed on a Pentum machne. The robot has been modeled as a crcle of sze 5 pxels and magnary sensors, seven n number, are placed n the form of an arc along the crcumference of the robot subtendng an angle of 105 degrees at the center. The robot rotates about ts center whch s a reasonable assumpton consderng the fact that real robots such as the LABMATE and ROVER do rotate about ther centers. Each sensor sends rays wthn a cone of 15 degrees. The mnmal dstance obtaned wthn the cone of each sensor s

8 556 Journal of Robotc Systems 2000 consdered as the dstance of the obstacle from that sensor and s avalable as nput to the algorthm. Accordng to the nput vector the fuzzy algorthm makes ts decson. In our smulaton the poston of all the obstacles n the workspace s unknown to the robot. The robot s aware of only ts start and fnal postons. The purely fuzzy algorthm performs well n a hghly cluttered envronment as shown n Fgure 7. Ths s because the goal attractng and the obstacle repulsng modules work n tandem and steer the robot to the target Immedate Memory Due to SOM The mmedate memory Ž IM. can be termed as the property by whch the robot understands and remembers ts most recent envronment. It can reduce path lengths as seen from Fgures 8 and 9. Fgure 8 s a landmark n the form of a rght angled corner such as meetng of the two walls. The target les on the other sde of the bend. The locatons marked a and b Ž Fg. 8. represent the two ends of a stretch of the wall, whle b also represents the corner. The robot s unable to get attracted Fgure 7. Navgaton n a densely cluttered envronment. to the target due to the presence of a stretch of the wall ab. Along the sde ab of the wall the obstacle avodance behavor offsets the target attractng behavor. Ths prevents the robot from bangng nto the wall but follow t. Untl the robot arrves at the corner b the sensors on the left sde keep detectng the wall whle the center sensor and those on the rght do not. The presence of the wall on the left forces the obstacle repulsng module to turn the robot to ts rght. But the presence of the target on the robot s left makes the target attractng module turn the robot to ts left. Thus the acton spaces of the target attractng and obstacle repulsng module offset each other and the robot moves straght, followng the wall. When t reaches the corner b all the sensors begn to detect the obstacles. But because of the corner on the left, the left sensor readngs become slghtly larger than those on the rght;.e., at the bend, the corner on the left s slghtly further than the wall Ž bc. whch s n front and extendng to the rght of the robot. These larger readngs can also arse due to multple reflectons n real world mplementatons. When ths occurs both the obstacle avodance and target reachng behavors force the robot to turn to ts left only to meet the wall agan. Ths results n a navgaton path as shown n Fgure 8. Thus the robot seems to have forgotten that t had been seeng the wall all ths whle on ts left, beng captvated by a slght ncrease n the sensor readng on ts left sde and turnng to ts left. Ths scenaro s avoded by the IM provded by the SOM. The SOM automatcally dentfes the meetng of the two walls at rght angles on the robot s left as the temporal sequence T. Ths sequence essentally can be nterpreted as wall on the left of the robot that bends at near about rght angles to ts rght. The classfcaton of the seven dmensonal sensory vector u Ž t. s nto a sngle group or class flters the mnor varatons n the sensor readngs. Near the corner the sensory vectors are classfed as 1 or 0. Ths feature of the classfer network s partcularly helpful as t gves a generalzed classfcaton of the envronment the robot has recently wtnessed. When the lattce neuron wth Fgure 8. Path traversed by the robot wthout ncorporatng IMAS. S and T represent start and target postons. Fgure 9. Robot traversal of landmark n Fg. 8 after ncorporatng IMAS.

9 Krshna and Kalra: Local Mnma Problem for a Moble Robot 557 the weght vector T s trggered the mmedate memory acton space Ž IMAS. of the neuron that won fres the rules whch turns the robot sharply to ts rght Ž n Fg. 9. and results n a shorter path The Infnte Loop Problem When the obstacles are long wth many bends and knks the target attractng and goal repulsng behavor conflct and the robot gets tself nto an nfnte loop Fg. 10Ž. a. Along the sde ab of the wall the obstacle avodance behavor offsets the target attractng behavor. When at b the IMAS of the SOM turns the robot rght and the robot contnues to see the wall on ts left after ts rght turn at b. Henceforth both the target attractng and obstacle repuls- Fgure 10. Ž. a The robot gettng trapped between the two corners a and b though IMAS acts makng the robot turn approprately at b. Ž b. Robot guded out by DMASc. Ž. c Fuzzy ART detects local mnma Ž 7, 8, 5. earler than SOM, nvokes DMAS. A shorter path results. ng behavors make turn the robot to ts rght. After ths the robot sees no obstacle and only the target attractng module nfluences the overall behavor of the robot. Thus the robot begns to rush toward the target and encounters the same wall agan. Ths contnues ad nfntum and s termed the local mnma problem. A possble means by whch the robot can come out of ths loop s to recognze ts repeated traversal n the same envronment and execute a sequence of steps that pulls t out of the trap, dscussed n the followng secton The Dstant Memory ( DM) The dstant memory feature s common to both the ART and SOM networks. It s called so because of the ablty of the robot to correlate an mmedate envronment that t sees to a smlar envronment experenced earler n ts traversal. To become aware that t s passng through the same envronment agan the robot keeps a record of ts spatal poston every tme the neuron codng a landmark wns. For every neuron n the output lattce there s a queue whch s dynamcally allocated whenever a neuron wns. An element of the queue s the nstantaneous spatal locaton of the robot when the neuron wns. The queue stores a maxmum of sx such spatal locatons of the robot, a spatal locaton characterzed by the Cartesan coordnates of the center of the robot. If a neuron wns more than sx tmes the least recent locaton n the queue s dropped from the end of the queue and the most recent spatal locaton s stored at the begnnng of the queue. Whenever a neuron wns more than once the robot understands t s seeng another landmark of the same category, lke the robot may be seeng another half open door n ts traversal. But to make sure that t s seeng the same half open door and not an another one t must compare ts prevous spatal locatons n the queue wth ts present one. If any one of the prevous locatons stored n the queue matches the current one the robot understands ts encounter of the same landmark. The dstant memory acton space Ž DMAS. of the wnnng neuron fres a sequence of steps that gudes the robot out of the trap. The sequence of steps can be understood as follows through Fgure 10Ž b.. At the nstance DMAS s actvated a record s made on the drecton of the target wth respect to the robot s left or rght flanks. Smlarly the sensor whch obtans the nearest range readng determnes the drecton of the closest obstacle. In Fgure 10Ž b. both the obstacle and the target are on the robot s

10 558 Journal of Robotc Systems 2000 left at pont b, the nstance of DMAS actvaton. There arse four cases and accordngly the robot executes a sequence of steps. Case () Target and closest obstacle are on the left. The followng steps are executed untl the robot reaches the target. 1. Fuzzy rule base gudes the robot untl the target s on the left. 2. The robot swtches to obstacle followng when the target comes on the rght whle the obstacle contnues to appear on the left. 3. Durng the course of obstacle followng f a break n the obstacle s detected ether because the obstacle ends or has turned n a drecton away from the robot locaton c n Fg. 10Ž b., the robot turns around the break to contnue followng the obstacle. 4. If durng such a break whle turnng around the obstacle the target appears on the left the fuzzy rule base s actvated agan or else obstacle followng s contnued through step 2. If the robot comes agan under the control of fuzzy rules and both the target and obstacle are on the rght steps 1 4 are executed accordng to Case Ž. below;.e., the occurrences of left n the steps 1 4 are replaced wth rght and vce versa. Ths occurs at poston c n Fgures 11Ž d. and 12Ž b.. Case () Target and obstacle are on the rght. The algorthm executes steps 1 4 of the prevous case wth the occurrences of left replaced wth rght and vce versa. Case ( ) Target s on the left and obstacle s on the rght. The robot turns n such a manner that the target comes on the rght and obstacle on the left. Steps 1 4 of the frst case are repeated untl the target s reached. Case ( v) Target s on the rght and obstacle s on the left. The robot turns n such a manner that the target comes on ts left and obstacle on ts rght. Steps 1 4 Fgure 11. Ž. a Another nfnte loop. Robot oscllatng between a and b. Ž b. Robot guded out of oscllatons. Ž. c Double walled obstacle. Ž d. Gudng out of the double layered wall. of the frst case are executed wth a swap n left and rght occurrences untl robot reaches the target. Ths occurs, for example, at poston c n Fgure 10Ž. c. The above sequence of steps s competent to carry the robot out of complcated meshes and loops, as Fgures llustrate. The sequences of steps are the same rrespectve of whether the DMAS s nvoked by the SOM or by the ART. In all these fgures the nfnte loop s shown n part Ž. a where only the fuzzy algorthm s mplemented, and the gudance out of the loop when the local mnma s detected by the SOM s shown n Ž b.. The temporal sequence of classes that leads to the dentfcaton of the landmark s also shown n the fgures. Fgure 13Ž. a shows a smulaton where the robot s unable to come through an openng n the maze at locaton

11 Krshna and Kalra: Local Mnma Problem for a Moble Robot 559 c due to conflctng behavors of the two modules. Fgure 13Ž b. llustrates crcumventon of the trapped stuaton wthout the DMAS nvoked. The IM capacty of the SOM understands through the weght vectors the unduly long traversal through the maze and executon of IMAS pulls the robot through the hole n the maze Fg. 13Ž b.. Whle DMAS s exe- Fgure 12. Fgure 13. Ž. a Pont c robot unable to pull tself out because of conflctng target attractng and obstacle avodance behavors. Ž b. IMAS for 3, 4, 8 pulls the robot out. Ž. c Wthout SOM, Fuzzy ART pulls the robot out of the maze. Absence of IMAS s seen as robot completes a full traversal between the 2nd and 3rd obstacle loops from the center.

12 560 Journal of Robotc Systems 2000 cuted the obstacle followng behavor s also modeled through a separate set of fuzzy rules whch has not been dscussed snce t s not the theme of the paper The Role of Fuzzy ART Fgure 14Ž. a shows a smulaton envronment where the SOM s unable to detect the local mnma stuaton. Ths s because the SOM s unable to map the real-tme patterns to a partcular landmark learned durng offlne. To overcome ths the fuzzy ART network has been ncorporated to classfy new patterns onlne. Repeated occurrences of such a pattern trgger the DMAS when the spatal locatons of the robot match. Here the ART network nvokes the DMAS n Fgure 14Ž b. and pulls the robot out of the local mnma. Another advantage of the ART s shown n Fgures 10Ž. c and 12Ž. c. Here the ART s able to detect a repeated occurrence of a pattern earler than SOM. Ths results n a reduced path. The SOM has to wat for a pattern that matches a landmark stored by ts lattce neurons through the weght vectors. Ths can result n a longer traversal. Ths, however, cannot be generalzed as t depends on the knd of envronment present once the robot gets out of the local mnma through the two routes. But t can be ntutvely seen that onlne learnng by fuzzy ART helps n detecton of the local mnma stuaton earler than the SOM, whch can probably Fgure 14. Ž. a An envronment where the SOM s unable to detect the local mnma. Ž b. Fuzzy ART pulls the robot out of the local mnma. lead to shorter traversals. Fgure 13Ž. c shows an example when only the ART network s present n the second layer. Hence the IMAS advantage of SOM s absent. Stll the ART network alone s suffcent to detect patterns onlne and pull the robot out of the maze. The absence of IMAS results n the robot makng a complete traversal of the free space between the second and thrd obstacle loops from the center of the mazelke structure wth four obstacle loops. To summarze the ART network n the second layer alone s capable of detectng the local mnma and nvokes the DMAS. The SOM provdes for modelng and understandng the local envronment based on landmarks learned offlne. When the SOM s also present n the second layer of the classfer network the ART plays the role of a backup, detectng the local mnma when the SOM s unable to do. It also helps n detectng the local mnma earler than the SOM Comparng wth an Earler Approach In a prevous approach 20 the determnaton of the local mnma stuaton s done n an emprc way by comparng the dfference n orentaton of the robot between successve nstants. If ths orentaton dfference has a value greater than 160 degrees the robot s consdered trapped or boxed. Ths can result n a stuaton where the robot begns to track the obstacle contour though t s not trapped. We take an example from the same paper 20 for comparson. Fgure 15Ž. a portrays the path obtaned by the authors n ther paper whle Fgure 15Ž b. s the path obtaned by the current method whch s shorter as the robot does not experence a smlar scenaro twce durng ts traversal. Another ssue, whch merts consderaton, s the nstant when the robot swtches from obstacle followng mode to fuzzy rule base mode or from the track mode to Heurstc mode n the termnology of the prevous paper. The prevous approach says the robot swtches from the T mode to H mode when a, b, c are collnear and b s between a and c. Here a s the pont where robot swtches from H mode to T mode, b s the pont where the robot fnshes trackng the obstacle, and c s the target locaton. In the present approach the robot fnshes trackng the obstacle accordng to step 4 dscussed n secton 4.3. Accordng to the present approach the robot swtches from obstacle followng mode to fuzzy rule base mode when the sensors detect the end of the obstacle, and durng the course of turnng around the obstacle s end, the target

13 Krshna and Kalra: Local Mnma Problem for a Moble Robot 561 robot reaches the end of the obstacle and fuzzy rules are able to pull the robot toward the target, resultng n a hghly optmal path compared to Fgure 16Ž. a. Also Fgures 16Ž. c and Ž. d llustrate the same pont when the robot begns trackng n the counterclockwse drecton. In Fgure 16Ž d. the earler algorthm has been modfed wth the crtera for breakng from the T mode set accordng to the present algorthm. Thus t s observed that the present crtera gves shorter traversals as the robot stops trackng the obstacle much earler than the prevous approach, and n the absence of further obstacles the fuzzy target reachng module steers the robot to ts destnaton Lmtatons and Practcalty Issues Fgure 15. Ž. a An example from a prevous paper. Heurstc determnaton results n msdentfcaton of a stuaton to a local mnma. Ž b. The robot traverses a shorter path by the current method. poston relatve to the current drecton of moton gets swapped. A swtch also occurs f the target poston relatve the robot gets swapped durng obstacle followng mode and s mantaned untl the robot reaches the end of the obstacle. It has been observed that f there exsts no further obstaclež. s beyond the concave obstacle nto whch the robot gets trapped and the path tracked by the robot from the nstant of obstacle followng s consdered n ether methods, the present method gves a shorter path. Ths s verfed from Fgures 16Ž. a and 16Ž. b whch s once agan an example from the prevous paper. In Fgure 16Ž b. the robot turns around the end of an obstacle between the par of ponts a, b and c, d. At b after turnng around the end there s no change n target poston and hence the obstacle followng mode s contnued. At d a change n poston occurs and the fuzzy rule base s executed but s unable to pull the robot to the target Ždue to the wall on the left.. A change n target poston once agan occurs at e whle the robot follows the wall on ts left and s mantaned untl f. At f the Durng navgaton the robot can wtness smlar scenes along dfferent paths. To avod gettng confused between two smlar scenes and a same scene seen twce the robot compares ts spatal locaton at the nstant of regsterng a scene wth ts spatal poston when t saw a smlar scene earler. If the spatal postons concde the robot understands ts encounter of the same landmark agan. In real-tme mplementaton errors creep up due to dead reckonng and other sources and the estmates of the robot s poston n a global frame can vary from ts actual poston. Ths can lmt the performance of the algorthm especally f the robot encounters smlar scenes at rapd ntervals such that the robots global poston does not vary much between these ntervals. The algorthm, however, can tackle a problem commonly encountered durng mplementatons, that of errors n range readngs due to multple reflectons at the corners. Ths has been dscussed n secton 4.1. In our experence wth real world sensory data, a system of seven sensors has been found to be adequate for dentfcaton of typcal landmarks. The algorthm does not demand storage of a long sequence of sensory data as only changes n sequence are to be processed and consderable memory s saved due to the dmensonalty reducton by the spatal classfer. The net space requred s a lattce of 64 neurons that dentfy the landmarks along wth 49 neurons of lower bounds, each neuron characterzed by a three dmensonal vector of ntegers. Spatal postons of the robot are stored only at nstances of regsterng a landmark and not more

14 562 Journal of Robotc Systems 2000 Fgure 16. Ž. a Accordng to the earler method the robot swtches from T to H mode at pont b when ponts a, b and c are almost collnear. Ž b. A more reduced path results by the current method. Ž. c The robot tracks the obstacle n counter clockwse drecton. Ž d. Path obtaned when the earler algorthm modfed wth the new crtera for swtchng from T mode to H mode. than 20 spatal postons need to be stored on an average at any nstant durng navgaton. 5. CONCLUSION Real-tme navgaton nvolves decson makng accordng to the percepton of the local envronment. The fuzzy nferencng method has been shown to be successful n real-tme navgaton wth cluttered envronments. But when the envronment s flled wth obstacles n the form of loops, mazes, and other complcated structures the robot tends to lose track of drecton and gets trapped. The paper proposes a new approach for dentfyng the robot s trapped state that seems more consstent wth how a human would normally understand hs trapped condton n an envronment by recallng the landmarks he had seen earler n hs traversal. A spato-temporal classfer network s employed for learnng and classfyng temporal sequences of spatal sensory data that enables the robot to comprehend ts mmedate envronment n terms of landmarks and remember prevous experences of a smlar envronment. In ths way the algorthm dffers from other methods that surmount the local mnma problem by recollectng prevous experences to understand ts trapped condton. The algorthm has been tested on cluttered, concave, and mazelke envronments and ts effcacy has been establshed. A possble extenson of ths work could be to recognze dynamc objects n the vcnty of the robot based on smlar classfcaton of range patterns. APPENDIX A: SOM NETWORKS AND KOHONEN LEARNING Kohonen developed the SOM to transform an nput sgnal of arbtrary dmenson nto a lower Žone or two. dmensonal dscrete representaton preservng topologcal neghborhoods. Let : U A denote the

15 Krshna and Kalra: Local Mnma Problem for a Moble Robot 563 SOM mappng from an nput space U and the dscrete output space A. The SOM defnes n Kohonen s words an elastc net of ponts A that are ftted to the nput sgnal space U to approxmate ts densty functon n an ordered way. In order to acheve ths goal the dscrete grd A of neurons ndexed by A s descrbed by reference vectors w whch take ther values n the nput space, U. The response of a SOM to an nput u U s determned by the reference vector ww of the two d- mensonal lattce whch produces the best match to the nput. w arg mn dstž w u., 1,..., N, where w refers to the wnnng neuron of the two dmensonal lattce and dstž. s the Eucldean metrc. The tranng of the SOM can be accomplshed generally wth a compettve learnng rule as Ž. Ž. Ž.Ž.Ž Ž. Ž.. w n 1 w n h, w n u n w n where h Ž, j. s an unmodal functon that decreases monotoncally for ncreasng j wth a characterstc decay constant. The usual choce for such a functon s a Gaussan gven by h Ž, j. e Ž r s. 2 2, where r and s represents respectve lattce poston vectors correspondng to th and jth neural unts, respectvely. h s also known as the neghborhood functon that adjusts apprecably the neurons close to the wnner and those far away wth lttle change. Both the learnng parameter and the neghborhood parameter are annealed wth tme. APPENDIX B: FUZZY ART ALGORITHM The fuzzy ART algorthm s capable of unsupervsed classfcaton of both bnary and analog nputs n real-tme. The man advantage of ths network s that t s very stable to prevously stored nputs and plastc to new nputs. Ths mples that new classes can be allocated dynamcally and the number of classes that can be formed s only lmted by the total memory avalable. The fuzzy ART network conssts of two processng layers. The frst layer corresponds to the nput layer whle the second layer conssts of neurons where each neuron represents a self-organzed category of the nput. The two layers are connected by adaptve weghts Z. Intally the components of the weght vector Z are set at 1.0. Then the nput vector c and ts complement y are stored as Y Ž y, y c. n the frst layer where y c 1 y. Ths process of storng both the nput and ts complement s called complement codng and helps n preventng a category prolferaton problem. 25 The nput actvates the node n the second or category layer as Ý j Ž. mn Y, Z Y Z j Tj. Z z j Here the conjuncton operator represents the fuzzy AND operator and the norm used s an L1 norm. The network then makes a hypothess by selectng the node J that has the maxmum Tj to be the category that stores the presented nput. In case of a te, the node wth the least ndex s chosen. Then ths hypothess s tested usng a smlarty test called the vglance crteron as Y Z Y j j, where s called the vglance parameter. The rato on the left of the above smlarty test represents the degree of match between Y and the category J n the second layer. If the node J satsfes the vglance crteron then the weghts are updated as Z j Zj Y Ž 1. Z, where Ž 1.0. j s called the for- gettng factor. In our smulatons we have fxed to be If node J does not satsfy the vglance crteron, t s shut down. Then, a search process s ntated n the category layer to see f any other node satsfes the vglance crteron. If there exsts one such node, then ts weghts are updated as above. If there exsts no such node then a new node s recruted to store the spato-temporal nput. Thus the vector Y s classfed nto a category node n the second layer. REFERENCES 1. L.L. Wang and W.H. Tsa, Car safety drvng aded by 3-D mage analyss technques, Proc. Mcro Elect. and Info. Sc. and Tech. workshop, Hsnchu, Tawan, R.O.C., 1986, pp J. Borensten and Y. Koren, Real-tme obstacle avodance for fast moble robots, IEEE Trans SMC 19 Ž 1989., H.G Tlleme Ž Ed.., An overvew of the moble autonomous robot twente project MART 93, Unv. Twente, WA-315, 1993, pp T. Lozano-Perez and M.A. Wesley, An algorthm for plannng collson-free paths among the polyhedral Ž. obstacles, Commn ACM , Ý j

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