Chico CSCI 397 - Introduction to Fuzzy Logic Control With Application to Mobile Robotics

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Introduction to Fuzzy Logic Control With Application to Mobile RoboticsEdward Tunstel, Tanya Lippincott and Mo JamshidiNASA Center for Autonomous Control EngineeringDepartment of Electrical and Computer EngineeringUniversity of New MexicoAlbuquerque, NM 87131ABSTRACT: A brief introduction to fuzzy set theory and itsapplication to control systems is provided. Fuzzysets do not have sharp boundaries and are thereforeable to represent linguistic terms which may beconsidered "gray" or vague. Aspects of fuzzy settheory and fuzzy logic are highlighted in order toillustrate distinct advantages, as contrasted toclassical sets and logic, for use in control systems.Using a mobile robot navigation problem as anexample, the synthesis of a fuzzy control system isexamined..Keywords: mobile robots, fuzzy logic control, fuzzysets, rover, autonomy1. INTRODUCTION"The world is not black and white but only shades ofgray." In 1965, Zadeh [1] wrote a seminal paper inwhich he introduced fuzzy sets, sets with unsharpboundaries. These sets are considered gray areasrather than black and white in contrast to classicalsets which form the basis of binary or Boolean logic.Fuzzy set theory and fuzzy logic are convenient toolsfor handling uncertain, imprecise, or unmodeled datain intelligent decision-making systems. It has alsofound many applications in the areas of informationsciences and control systems.In this paper, fundamental concepts of fuzzy setsand logic are briefly presented. Its utility forsynthesis of control systems is discussed in thecontext of an application to mobile robot motioncontrol. In mobile robotics, a fuzzy logic basedcontrol system has the advantage that it allows theintuitive nature of collision-free navigation to beeasily modeled using linguistic terminology. Due tothe relative computational simplicity of fuzzy rule-based systems, intelligent decisions can be made inreal-time, thus allowing for uninterrupted robotmotion. Moreover, accurate (expensive) sensors anddetailed models of the environment are not absolutelynecessary for autonomous navigation [2]. . B.S. degree in electrical engineering, University ofNew Mexico. Currently pursuing M.S. degree, electricalengineering, with an expected graduation date of May,1997; Networks and Control Systems concentrationwith emphasis on fuzzy control of mobile robots.2. FUZZY SET THEORYIn classical set theory a set, C, is comprised ofelements, x∈U, whose membership in C is describedby the characteristic, or membership functionµCx U( ): { , }→ 0 1 (1)where U is the universe of discourse, a collection ofelements that can be continuous or discrete. Themembership function µC(x) implies that the element xeither belongs to the set (µC(x) = 1) or it does not(µC(x) = 0). In fuzzy set theory a fuzzy set, ˜F, isdescribed by the membership functionµ˜( ): [ , ]Fx U → 0 1 (2)where elements, x∈U, have degrees of membership in˜F with any value between 0 and 1 inclusive. Notethat a fuzzy membership function is a so-calledpossibility function and not a probability function. Amembership value of zero corresponds to the casewhere the element is definitely not a member of thefuzzy set. A membership value of one corresponds toelements with full membership in the fuzzy set.Membership values in the open interval (0, 1)correspond to partial membership and indicate ameasure of uncertainty or imprecision associated withthe element.A comparative example of a crisp set and a fuzzyset can be illustrated by using the linguistic termÔfarÕ in reference to relative distance between objects.The term ÔfarÕ can take on different meanings todifferent individuals, and in different contexts. Forillustrative purposes, let ÔfarÕ be 2 meters(approximately 2 meters in the fuzzy set case). Agraphical representation of a crisp set and a fuzzy setfor ÔfarÕ is shown in Figure 1.Membership functions can be defined as functionswhich take on a variety of possible shapes determinedat the discretion of the fuzzy system designer.Commonly used function shapes (fuzzy logicterminology given in parentheses) include triangular(Λ), trapezoidal (Π), delta (singleton), positivelysloped ramp (Γ), and negatively sloped ramp (L).These are shown in Figure 2. The ramp functions aresometimes referred to as right shoulders (Γ) and leftshoulders (L)..00.250.50.751Degree of membership0 1 2 3 4Distance to object00.250.50.751Degree of membership0 1 2 3 4Distance to objectb. Fuzzy seta. Crisp set0.50.750.5mmFigure 1 Graphical representations of ‘far’.SingletonΠLΓΛFigure 2 Common fuzzy membership functions.Fuzzy sets, like classical crisp sets, are subject toset operations such as union, intersection, andcomplement [1] which are used to express logicstatements or propositions. The union of two fuzzysets ˜A and ˜B with membership functions µ˜( )Ax andµ˜( )Bx is a fuzzy set ˜C=˜˜A B∪, whose membershipfunction is related to those of ˜A and ˜B as follows:µ µ µ µ˜ ˜˜˜ ˜( ) ( ) max[ ( ), ( )]C A B ABx x x x= =∪ (3)The operator in this equation is referred to as themax-operator and is represented by the logical termOR. The intersection of ˜A and ˜B is a fuzzy set˜D=˜˜A B∩ whose membership function is given by:µ µ µ µ˜ ˜˜˜ ˜( ) ( ) min[ ( ), ( )]DA B ABx x x x= =∩ (4)The operator in this equation is referred to as themin-operator represented by the logical term AND.For details on complements and other fuzzy logicaloperations see [1] or [3].Consider the Cartesian product of two universes Uand V defined byU V u v u U v V× = ∈ ∈{( , ) | ; }which combines elements of U and V in a set ofordered pairs. A fuzzy relation R is a mapping:R: [ , ]U V× → 0 1where µ µ µ µR( , ) ( , ) min[ ( ), ( )]˜˜˜ ˜u v u v u vA B AB= =× (5)The composition of two relations, R(u,v) and S(v,w),is denoted by T R S= o. Its membership value can bedetermined by the following expressionµ µ µT RS( , ) max[ ( , ) ( , )]u w u v v w= • (6)which is called the max-product


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Chico CSCI 397 - Introduction to Fuzzy Logic Control With Application to Mobile Robotics

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