Optimization of Type-2 Fuzzy Logic Controllers for Mobile Robots Using Evolutionary Methods Ricardo Martinez and Antonio Rodriguez UABC, Tijuana, México mc.ricardo.martinez@hotmail.com, ardiaz@uabc.mx Oscar Castillo, Patricia Melin Instituto Tecnológico de Tijuana Tijuana, México {ocastillo, epmelin}@hafsamx.org Luis T. Aguilar CITEDI Tijuana, México luis.aguilar@ieee.org Abstract— We describe in this paper the application of evolutionary methods for the optimization of type-2 fuzzy logic controllers. These optimal type-2 fuzzy logic controllers are used for the trajectory tracking control of autonomous mobile robots. The evolutionary method to consider is the genetic algorithm, presenting simulations results in Simulink © of Matlab. Keywords— Autonomous Mobile Robot, Evolutionary Methods, Genetic Algorithms, Type-2 Fuzzy Logic. I. INTRODUCTION Evolutionary Computing is the collective name for a range of problem-solving techniques based on the principles of biological evolution, such as natural selection and genetic inheritance. These techniques are being increasingly widely applied to a variety of problems, ranging from practical applications in industry and commerce to leading-edge scientific research [1]. There are many techniques for evolutionary computing and for this study we used genetic algorithms (GA). This technique is used to optimize a type-2 fuzzy system. Once the optimized type-2 fuzzy logic controllers (FLC) are obtained, we apply them to an autonomous mobile robot for the trajectory tracking control. This paper is organized as follows: Section II presents the theoretical basis and problem statement; Section III presents the Type-2 Fuzzy Logic Controller design. Section IV provides simulation results of the evolutionary computing techniques using the controller described in Section III. Finally, Section V presents the conclusions. II. THEORETICAL BASIS AND PROBLEM STATEMENT A. Type-2 Fuzzy Logic Systems Figure 1. a) Type-1 membership function and b) Blurred type-1 membership function. If we have a type-1 membership function, as in Figure 1 (a), and we are blurring it to the left and to the right as illustrated in Figure 1 (b), then, for a specific value ' x , the membership function ( ' u ), takes on different values, which are not all weighted the same, so we can assign an amplitude distribution to all of those points. Doing this for all X x ∈ , we create a three-dimensional membership function –a type-2 membership function– that characterizes a type-2 fuzzy set [1, 11]. A type- 2 fuzzy set A ~ , is characterized by the membership function: ( ) { } ] 1 , 0 [ , | ) , ( ), , ( ~ ~ ⊆ ∈ ∀ ∈ ∀ = x A J u X x u x u x A μ (1) in which 1 ) , ( 0 ~ ≤ ≤ u x A μ . Another expression for A ~ is, ) , /( ) , ( ~ ~ u x u x A X x J u A x ∈ ∈ = μ ] 1 , 0 [ ⊆ x J (2) where denotes union over all admissible input variables x and u. For discrete universes of discourse is replaced by [11]. In fact ] 1 , 0 [ ⊆ x J represents the primary membership of x, and ) , ( ~ u x A μ is a type-1 fuzzy set known as the secondary set. Hence, a type-2 membership grade can be any subset in [0,1], the primary membership, and corresponding to each primary membership, there is a secondary membership (which can also be in [0,1]) that defines the possibilities for the primary membership [10]. This uncertainty is represented by a region called footprint of uncertainty (FOU). When ] 1 , 0 [ , 1 ) , ( ~ ⊆ ∈ ∀ = x A J u u x μ we have an interval type-2 membership function, as shown in Figure 2. Figure 2. Interval type-2 membership function. The uniform shading for the FOU represents the entire interval type-2 fuzzy set and it can be described in terms of an upper membership function ) ( ~ x A μ and a lower membership function ) ( ~ x A μ . A Fuzzy Logic System (FLS) described using at least one type-2 fuzzy set is called a type-2 FLS. Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 978-1-4244-2794-9/09/$25.00 ©2009 IEEE 4909