Fuzzy Clustering Applied on Mobile Agent Behaviour Selection

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1 Fuzzy Clusterng Appled on Moble Agent Behavour Selecton Andrej Turčan, Eva Ocelíková Dept. of Cybernetcs and Artfcal Intellgence Faculty of Electrcal Engneerng and Informatcs, TU of Košce Abstract: The goal od ths paper s to provde a concept of possble herarchcal structructure of mechanzm for moble agent behavour selecton. Herarchy of the controler s composed of fuzzy classfer and several smple fuzzy systems, where each can deal wth one smple stuaton. Fuzzy systems are able to treat and cope wth uncertan and mprecse nformaton, they have capablty of expressng the knowledge n the form of lngustc rules, whch sometmes are not easy to desgn. Fuzzy classfers on the other hand do not need requre antecendent knowledge of the space where they operate. Combnaton of those two techques can leed nto smple and effectve system, where fuzzy classfer s employed to select the approprate behavor, wth wch one of the fuzzy controlers can deal wth. Keywords: classfcaton, fuzzy clusterng, fuzzy classfer, moble agent 1 Introducton Fuzzy systems have the attrbute of expressng knowledge n the form of lngustc rules. They offer a possblty to mplement expert human knowledge and experence n form of those lngustc rules. Even though they are relatvely easy to desgn, no unversal systematc procedure exst to desgn them. 2 Fuzzy Inference Systems and Fuzzy Clusterng The frst and most common applcaton of fuzzy logc technques n the doman of moble agents s the use of fuzzy control to mplement ndvdual behavour unts. Fuzzy logc controllers ncorporate heurstc control knowledge n the form of IF- THEN rules, and are convenent choce when a precse lnear model of the system, whch s to be controlled, cannot be easly desgned. They have also shown a good degree of robustnes n face of large varablty and uncertantty n parameters.

2 These characterstcs make fuzzy control partcularly suted to the needs of autonomous moble agents. On the other hand, fuzzy clusterng, and clusterng n general, have the possblty to dvde a set of objects nto reasonable groups (called clusters). The dvson of objects s based on measurng of smlarty or dssmlarty between the par of objects. Thus, result of clusterng s a set of clusters, where object wthn one cluster are more smlar to each other, than to object n another cluster. Why not treat those objects as typcal or basc behavor types of the moble agent? 2.1 Fuzzy Clusterng Fuzzy clusterng algorthms allows each datum to belong nto one or more clusters wth certan membershp degree. Larger membershp values ndcate hgher confdence n the assgnment of the datum to the cluster General Fuzzy C-means Algorthm Fuzzy c-means algorthm has been descrbed by [4] conssts of teraton over the followng steps untl the termnaton crteron s not met: (1) Select an ntal fuzzy partton of the N objects nto K clusters by selectng the N K membershp matrx U. An element uj of ths matrx represents the grade of membershp of object x n cluster cj. Typcally, uj [0,1]. (2) Usng U, fnd the value of a fuzzy crteron functon, e.g., a weghted squared error crteron functon, assocated wth the correspondng partton. One possble fuzzy crteron functon s J ( Z, U, V) C N = = 1 k = 1 m k k 2 µ z v (1) Reassgn patterns to clusters to reduce ths crteron functon value and recompute U. (3) Repeat step (2) untl entres n U do not change sgnfcantly. Fuzzy c-means (FCM) algorthm s probably the most popular fuzzy clusterng algorthm. Even though t s better than the hard k-means algorthm at avodng local mnma, FCM can stll converge to local mnma of the squared error crteron. The desgn of membershp functons s the most mportant problem n fuzzy clusterng; dfferent choces nclude those based on smlarty decomposton and centrods of clusters. A

3 2.1.2 Gustafson-Kessel Algorthm The Gustafson-Kessel (GK) algorthm s an extenson of the FCM algorthm. The GK algorthm employs an adaptve dstance norm n order to detect the clusters wth the dfferent shapes. Contrary to the FCM algorthm the GK algorthm gves the clusters wth the dfferent szes n the dfferent szes n the dfferent dmensons (clusters are ellpsods). In ths case every cluster has ts own norm matrx A. The matrces A are an optmzaton varable n followng objectve functon: J (, U, V, A) µ C N = = 1 k= 1 m k ( z v ) A ( z v ) T Z (2) k But the objectve functon cannot be mnmzed drectly because J depends on A lnearly. It means that J can be made as small as the consdered postve defnte matrces A. Therefore A s constraned by: det A = ρ (3) The followng result s obtaned: 1/ n 1 [ det( F )] F A ρ (4) = where n s the dmenson of the data (patterns) and F s the fuzzy covarance matrx of the -th cluster and the fuzzy covarance matrx s defned as: F N m µ k k k= 1 = N ( z v )( z v ) k= 1 µ m k k T Such these cluster shape provdes an easer and more accurate coverage of the pattern space and n the followng secton s shown that ths method s better for a generatng of the membershp functons and the rules. The GK algorthm s also data scale ndependent,.e. f the data n any dmenson are multpled by a constant than the relatve coordnates of the cluster centers and the matrx of membershp degrees are dentcal. Moreover the convergence of ths algorthm s ensured, but locally only. k (5)

4 a) b) Fgure 1 a) set of data used for clusterng; b) resultng clusters after the applcaton of the Gustafson- Kessel clusterng algorthm (only the maxmal values of degree of membershp are vsualzed) 2.2 Fuzzy Clusterng vs. Fuzzy Inference System Fuzzy nference s the process of formulatng the mappng from a gven nput to an output usng the fuzzy logc. For our purposes a Takag-Sugeno method of nference s used, where output membershp functons are only lnear or constant. A typcal fuzzy rule n ths model has the form If x s A and y s B then z=k (6) where A and B are fuzzy sets n the antecendent, whle k s a crsply defned constant n the consequent part of the rule. Output surface of a such system wth two nput varables and one output can be seen on Fg. 2. As sad before, fuzzy clusterng algorthms assgn each datum degree of membershp for each cluster. After havng look on Fg. 3; and comparng t wth Fg. 2 we can fnd certan smlarty n both surfaces. Resultng surface of the fuzzy clusterng could be used for controll as well as the fuzzy controler.

5 Fgure 2 Surface representng fuzzy nference model Fgure 3 Surface representng membershp functon for one of the clusters from Fg. 1 3 Buldng a Herarchy In behavor-based control, a complex control problem s dvded nto a set of smpler controllers, known as behavors, that collectvely should solve the orgnal problem. To do ths, t s thus necessary to coordnate the actvtes of the behavors to satsfy the ntal complex system s control objectves. Ths s known as the behavor coordnaton problem (or acton selecton). At frst we are gong to create buldng blocks for the ndvdual behavor unts. 3.1 Smple Behavors Concept of the smple behavors s taken from the dvde and conquer approach to solve problems. It s easer to decompose one complex behavor nto a group of smple behavours, whch nteracton and composton can lead to the desred complex behavor. A good example s the moble agent. The hgh-level moton control system s decomposed nto a seto of specal-puropose behavors that acheve dstnct tasks when subject to partcular stmul. For ths example we employ herarchy wth two levels. The lowest level of behavor represents the smplest (or prmtve) behavors. These are smple, self-contaned behavors that serve a sngle purpose by operatng n a reactve or reflexve fashon. Each of them would be nsuffcent

6 for the autonomous task. They are combned to produce behavor sutable for accomplshng goal-drected operaton. 3.2 Behavor Coordnaton Mechansms Behavor coordnaton mechansms can be dvded nto two man clases: arbtraton and command fuson Arbtraton Arbtraton mechansms select one behavor, from a group of competng ones and gve t ultmate control of the system untl the next selecton cycle. Ths approach s sutable for arbtratng among the set of actve behavors n accord wth the system s changng objectves and requrements under varyng condtons. It can focus the use of scarce system resources on task that are consdered to be relevant. Fnal behavor can be chosen ether usng prorty-based arbtraton mechansm, where behavors wth hgher prortes are allowed to supress the output of behavors wth lower prortes, or wnner-take-all mechansms, where acton selecton results from the nteracton of the behavor that compete untl one wns and takes the control of the system Command Fuson Command fuson mechansms combne recommendatons from multple behavors to form a control acton that represents ther consensus. Thus, ths appproach provdes for a coordnaton scheme, that allows all behavors to smultaneously contrbute to the control of the system n a cooperatve rather than a compettve manner. The technques mostly used are dealng wth superposton of the smple behavor outputs. In the proposed archtecture a fuzzy classfer s employed to compute the weghts for each behavor. The fnal acton s carred out as the supperposton of all smple behavors. 4 Moble Agent For the experments wth the herarchcal system composed of fuzzy clusterng system for the behavor selecton and Takag-Sugeno fuzzy nference to control the smple behavor was chosen. Ths system was used to acomplsh sngle (and qute smple) goal. Ths goal was to navgate the agent from pont A to pont B. Ths smple task has been chosen, because the focus has been put on the desgn of the behavor selecton and the fuzzy clusterng.

7 As can be seen n Fg. 2; the surface of the functon s dependant on the placement of the clustered data. So for our purposes, we are gong to do some reverse work. We wll construct a set of typcal stuatons, whch we wll descrbe by feature vector. After applyng the fuzzy clusterng algorthm (Gustafson-Kessel s used for ts capabltes to group data nto the clusters of elptc shapes) on data set we have created, we get a functon, whch calculates for each pont of ts unverse wage for the fnal output of the fuzzy nference systems representng smple behavors. Fgure 5 Structure of the system On the Fg. 5; the structure of the system s depcted. Only two smple behavors have been chosen goto-xy and avod_obstacle. Over these two fuzzy nference system a fuzzy classfer s lnked n order to set the wages for the computaton of the fnal acton. a) b) Fgure 6 Wages computaton for a) goto-xy b) avod_obstacle

8 Fgure 7 Path of the moble agent; navgaton to xy posston wth avodance of the obstacle Concluson The herarchy of fuzzy-behavors provdes an effcent approach to controll moble robots. Such a herarchy allows us to decompose overall behavor nto subbehavors that are actvated only when applcable (when condtons for actvaton of a sngle or more are stsfed), and there s no need to process all the rules that do not apply. Proposed desgn focused on use of the fuzzy classfer to ether combne smple behavors or to choose the most approprate behavor. Superposton of smple behavors allows realzaton of a contnuum of behavoral responses governed by respectve degrees of applcablty. Acknowledgement Ths work s supported by the VEGA Project No. 1/2185/05 References [1] Pudl, P., Novovčová, J., Ktler, J.: Floatng Search Methods n Feature Selecton. Pattern Recognton Letters, Vol. 15, No. 11, 1994, pp [2] Hual, L., Hrosh, M.: Extracton, Constructon and Selecton. Kluwer Academc Publshers, Boston, 1998, 440 pp [3] Klmešová D., Sac S.: Feature Selecton Algorthm and Correlaton Cobweb, Pattern Recognton Letters 19(1998), , 1998

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