Fig. 10 — Fuzzy input membership functions before training (left) and after training (right). Fig. 11 — ANFIS modeling result. Fig. 8 — Linear modeling result. fuzzy variables for representing the welder’s impression about the geometrical characteristics of the weld pool geometry. Human welder’s previous response dCurrentp is also considered and added to the fuzzy variables. These four variables are denoted as pj (j = 1, 2, 3, and 4) where p1 = W, p2 = L, p3 = C, and p4 = dCurrentp. Based on the knowledge of the human welder, we have assumed that each variable has no more than three fuzzy sets. By modeling trials of three partitions, it is found that no noticeable improvement has been observed regarding the overall fitting impression. Moreover, the fuzzy rule sets as well as model parameters for three partitions are much larger than those with two fuzzy sets, and the obtained fuzzy models using three fuzzy sets are more difficult to interpret than those with two fuzzy sets. Thus, the partition shown in Table 2 is obtained. Figure 9 shows the proposed four inputs one output ANFIS scheme. Because of the smoothness and concise notation, Gaussian membership function (MF) and generalized bell MF are becoming increasingly popular methods for specifying fuzzy sets. Generalized bell MF is adopted in this study, which is specified by three parameters (Ref. 20) (8) ( ; , , ) ( ) ji j ji ji ji 1 j ji ji b 1 / 2 ji where pj is the fuzzy variables and aji , bji, and cji are the input fuzzy membership function parameters. For a given set of input variables (for example, p1, p2, p3, and p4), the following rule is implemented (Ref. 20): (9) ( 1 , 2 , 3 , 4 ) , 1 1 2 3 4 1 ( ) i i i p 2 3 4 3 + ( ) + ( ,i 4 ) where dj s' are the so-called consequent parameters. The final output of the fuzzy model is (10) 2 2 2 2 Σ Σ Σ Σ i 1 1 i 2 1 i 3 1 i 4 1 ( ) ( ) where w(i1, i2, i3, i4) is the weight representing the truth degree for the premise: p1 is A1i1 , p2 is A2i2 , p3 is A3i3, and p4 is A4i4, and is expressed by the following equation: (11) ( 1 2 3 4 ) = Π ji ( ) kj The output Equation 10, together with the weighting Equation 11, membership function Equation 8, and the fuzzy rule Equation 9 form an ANFIS model. Its model parameters aji, bji, cjib , and dj can be identified using the Matlab ANFIS toolbox. Modeling Results In order to further improve the modeling accuracy and better model the inherent fuzzy inference mechanism of the human welder, human intelligence model Equation 5 is realized using the proposed ANFIS nonlinear model. The fuzzy input variables are partitioned by 2. The input fuzzy membership functions before and after training are depicted in Fig. 10. Table 3 lists the trained parameters for these input membership functions. The estimation result is plotted in Fig. 11. The resultant model RMSE and maximum model error are listed in Table 5. All criteria, including the average model error, RMSE, and maximum model error are improved by the proposed ANFIS model. Table 4 lists the output membership parameters for the proposed ANFIS model. A p a b c p c a = + − Rule i i i i : If is is p A i p A i 1 1 1 2 2 2 3 3 3 4 4 4 1 2 3 4 , , , , , , p A p A z i i i i is i is i Then ( ) = ( ) + ( ) + d i i i i p d i i i i p d i 2 1 2 3 4 2 3 1 , , , , , , , , , , , , , , d i i i i p d i i i 4 1 2 3 4 4 0 1 2 3 z w i ,i ,i ,i z i ,i ,i ,i 1 2 3 4 1 2 3 4 = = = = = w i ,i ,i ,i A p j 1 4 = FEBRUARY 2014 VOL. 93 50-s WELDING RESEARCH Fig. 9 — Four inputs one output ANFIS scheme. Table 3 — Fuzzy Input Parameter for Skilled Welder MF1 Parameters MF2 Parameters Skilled Width 2.565 1.993 1.103 2.558 1.996 6.189 Welder Length 2.673 2 1.598 2.677 1.996 6.937 Convexity 0.0382 2.004 0.0561 0.0249 2.004 0.2277 dCurrentp 4.008 1.995 -3996 4.004 1.993 4
Welding Journal | February 2014
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