modeling error (the definition of average model error Eave and RMSE will be given later) and is thus chosen in our study. It is also noticed that the current adjustment (dCurrent: ΔI (k) = I (k) –I(k–1)) instead of absolute current is used to model the human behavior. This is because when the human welder makes adjustments on the current by using the current regulator, he/she doesn’t know the absolute current value but the increase/decrease amount. Figure 6 plots the inputs (filtered weld pool width, length, and convexity) and output (current adjustment dCurrent) of the human intelligence model. Figure 7 shows the distribution of the input parameters in these experiments. It can be observed that the weld pool parameters have filled the certain range of the input space and are nearly uniformly distributed. This distribution implies that the resultant model can be used during prediction of the human welder’s response if the weld pool parameters are in this range. Once a model is identified, its quality/ performance can be evaluated using the average model error, root mean square error (RMSE), and maximum error defined by (2) (3) (4) where N is the number of samples, ΔI(k) is the model estimated human welder’s response at instant k. In general, the human intelligent model can be written as ΔI(k) = f(Wf (k–3), Lf (k–3), Cf (k–3), ΔI(k–1) (5) where the used human response delay of 1.5 s (three sampling periods) was discussed in the Experimental Data section. The simplest form of Equation 5 can be expressed by the following linear model: ΔI(k) = α1Wf (k–3) + α2Lf(k–3) + α3Cf (k–3) + α4ΔI (k–1) (6) Using the standard least squares method, the linear model can be fitted from the raw data. The identified linear models for skilled welder are shown in Equation 6A. ΔI(k) = –0.16 Wf (k–3) –0.082 Lf (k–3) + 1.81 Cf (k–3) + 0.26 ΔI (k–1) (6A) The linear modeling result is depicted in Fig. 8. The average model error, maximum model error, and RMSE for the identified linear model can be seen in Table 5 where all comparative models will be listed. In the next section, nonlinear neuro-fuzzy modeling is constructed to improve the model performance, and the modeling results are analyzed. Neuro-Fuzzy Modeling Results and Analysis As a human inference mechanism, the human welder’s response to the 3D weld pool surface is inevitably fuzzy and nonlinear. However, the abstract thoughts or concepts in human reasoning are difficult to extract from the domain knowledge. In this study, ANFIS algorithm developed by Jang (Ref. 20) will be used to model a skilled human welder response. Neuro-Fuzzy Modeling The term neuro-fuzzy modeling is used to refer to the application of algorithms developed through neural network training to identify parameters for a fuzzy model. In neuro-fuzzy modeling, the abstract thoughts or concepts in human reasoning are incorporated with numerical data so that the development of fuzzy models becomes more systematic and less time consuming. Most neuro-fuzzy systems have been developed based on the Sugeno-type fuzzy model. A typical fuzzy rule in a Sugeno-type model has the form (Ref. 20) If x is A and y is B, Then z = f(x,y) (7) where A and B are fuzzy sets, and z = f(x,y) is a linear function. ANFIS can construct an input–output mapping in the form of Sugeno-type If-Then rules by using a hybrid learning procedure (Ref. 20). The membership functions that correspond to the fuzzy antecedents as well as functions that form the consequence parts are parameterized. The hybrid learning proposed is composed of a forward pass and a backward pass. In the forward pass, by keeping the antecedent parameters fixed, consequence parameters are optimized by a least square estimation. A fuzzy logic control/ decision network is constructed automatically by learning from the training data. The ANFIS architecture thus can enable a change in rule structure during the evaluation of the fuzzy inference system. The ANFIS optimizes itself given the number of iterations by providing a change in rules, by discarding unnecessary rules, and by changing shapes of membership functions, which is called modifications. It is common practice to use the domain knowledge about the addressed problem for determining the fuzzy model structure, i.e., selecting the relevant inputs, partitioning the fuzzy sets, etc. In this study, we selected filtered topside welding parameters, the width, length, and convexity of the weld pool as three relevant E N 1 ˆ = Σ Δ ( ) − Δ ( ) N I k I k ave k 1 = N = ( ( )− ( )) = Σ 2 RMSE I k I k N k Δˆ Δ / 1 max ˆ , () () =Δ−Δ = EI k I k kN max () 1,..., WELDING JOURNAL 49-s WELDING RESEARCH Fig. 6 — Filtered weld pool parameters (width, length, convexity) and current adjustments made by skilled welder. Fig. 7 — Distribution of the inputs for skilled welder modeling. Table 2 — Partition of Fuzzy Input Variables Fuzzy Number of Partition Variables Fuzzy Sets Width 2 wide, narrow Length 2 long, short Convexity 2 large, small dCurrentp 2 large, small ˆ
Welding Journal | February 2014
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