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Fuzzy relation method: In this case, a fuzzy relation is constructed to make pair wise comparisons between the fuzzy quantities involved. Let M be an ordering method on E. The statement two elements A1 and A2 in E satisfy that A1 has a higher ranking than A2 when M is applied will be written as A1  A1 by M. A1  A1 and A1  A1 are similarly interpreted. The following reasonable properties for the ordering approaches are introduced by Wang and Kerre (Wang & Kerre 2001). 54 1. 2. 3. 4. 5. 6. Recurrent Neural Networks and Soft Computing For an arbitrary finite subset  of E and A1   , A1  A1 .

An alternative choice is to consider the coordinates (x, y) of each point on the path; as inputs; and the corresponding actions (speeds); as output. The third choice is to input the path as a whole as a single input vector and the corresponding sequence of actions (speeds) as a single output vector. All these choices share two fundamental disadvantages. First, it is impossible to interpret the trained weights as fuzzy rules. Furthermore, the ANN does not learn the "concept" of path tracking in general.

Therefore, sigmoid membership functions have been chosen, for the purpose of comparison with our approach. In this case, the membership function of the jth rule takes the form:   Aij = sig aij xi  c ij  Our approach can be viewed as a modified ANFIS system with the 'Product' operator replaced by the 'ior' operator and with pijk = 0 . As indicated by the results (Fig. 15), these modifications enhance the performance considerably. ANFIS training involves the estimation of the parameters pijk , rjk for each rule contributing to the output as well as the membership functions parameters aij ,cij of each rule.

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