Prediction of formation force during single-point incremental sheet metal forming using artificial intelligence techniques

Single-point incremental forming (SPIF) is a technology that allows incremental manufacturing of complex parts from a flat sheet using simple tools; further, this technology is flexible and economical. Measuring the forming force using this technology helps in preventing failures, determining the optimal processes, and implementing on-line control. In this paper, an experimental study using SPIF is described. This study focuses on the influence of four different process parameters, namely, step size, tool diameter, sheet thickness, and feed rate, on the maximum forming force. For an efficient force predictive model based on an adaptive neuro-fuzzy inference system (ANFIS), an artificial neural network (ANN) and a regressions model were applied. The predicted forces exhibited relatively good agreement with the experimental results. The results indicate that the performance of the ANFIS model realizes the full potential of the ANN model.


Introduction
Incremental sheet forming (ISF) is a flexible manufacturing technology that does not require special dies and uses a single tool to produce a variety of regular and multifaceted shapes. Further, it is economical when employed in the manufacturing of complex parts using simple tools as compared with other conventional sheet metal forming technologies (e.g., extrusion, hydroforming [1], and deep drawing [2]) because it does not require expensive dies or punches. In addition, it is used as a simple tool to incrementally develop the desired parts from sheets. However, it is time-consuming and thus not useful for mass production.
The ISF technique can be separated into two classes: two-point incremental forming (TPIF), which requires a partial die as a support for the sheet during the process [3], and single-point incremental forming (SPIF), which does not require any specific die [4]. Currently, focus is on SPIF, in which a small hemispherical tool is used to mold the sheet into the desired shapes; the tool is driven using a computer numerical control (CNC) machine along a predefined toolpath generated through computer-aided manufacturing software. The peripheral of the sheet is clamped using a fixture. Through this technique, complex parts can be manufactured in small batches and prototypes can be economically obtained. PLOS  using artificial intelligence methods as an alternative to traditional approaches [28]- [31]. This paper proposes an intelligent process model, founded on the concept of data mining, for predicting the forming forces in SPIF. Several researchers have addressed the limitations of this process, resulting in low-quality profile products. The predictive model for the forming forces described in this paper is based on an adaptive-neuro fuzzy inference system (ANFIS) and an artificial neural network (ANN), which have not been considered in previous studies to the best of our knowledge. An accurate model used to predict the forming forces in SPIF is essential in order to control the process quality. The rest of this paper is organized as follows: The experiments are presented in Section 2. The ANFIS, ANN, and regression models are presented in Sections 3 and 4. The results and discussions are detailed in Section 5. Finally, the conclusions are presented in Section 6.

Experiments
A vertical CNC milling machine, a specially designed fixture, forming tools, and a piezoelectric dynamometer were used to conduct the experiments. The sheet material selected for this study was a commercial aluminum alloy, AA1050-H14, which is a popular grade of aluminum for general sheet metal work owing to its excellent corrosion resistance, high ductility, and highly reflective finish. Further, the material composition, extracted using a SPECTRO machine, is presented in Table 1. Tensile tests were conducted on the specimens using a Zwick/Roell universal testing machine, the results of which are presented in Table 2. The sheet was clamped using the designed fixture in a working area of 200 mm × 200 mm. The tool used during this process was cylindrical with a hemispherical head. In this study, the tool motion was controlled numerically. Therefore, the required part was designed using SOLIDWORK software, and the design was then transferred to MASTERCAM software to generate the toolpath. The numerical control (NC) codes were obtained from the generated toolpath and transferred to the CNC machine. For the accurate formation of parts, it is important to select the best toolpath, which in this case is a spiral toolpath. A truncated conical geometry was built with a base diameter of 100 mm and a height of 50 mm. Important parameters considered for the incremental sheet metal forming are tool diameter, sheet thickness, feed rate, and step size, the values of which are listed in Table 3.
Measuring the forming force during this process is extremely important to prevent failure, determine the optimal process, and implement on-line control. Forming force tests were conducted using a KISTLER 2825A1 with eight freely selectable measuring signal-component force dynamometer controllers, which helped measure the force components in three directions (x, y, and z). In addition, the measuring system included charge amplifiers (a complementary KISTLER 5019B three-channel charge amplifier) and data acquisition cards to record the measured forces on a PC. The sampling rate of the force measurement was 50 Hz. Fig 1  shows the experimental system and procedure used to measure the performance of the forming forces. The workpiece fixture was mounted on top of a piezoelectric load cell. The experimental results for all responses that were used as training and testing data for both the ANN and ANFIS models are listed in Table 4.

Adaptive neuro-fuzzy inference system
ANFIS is an effective approach to building models of complex nonlinear systems. Here, a hybrid learning process is used to structure an input-output mapping based on human knowledge and training data pairs. The ANFIS is applied in the framework of adaptive networks. It consists of five network layers. Each layer is described by several node functions. The information is moved unidirectionally. A diagram of the ANFIS structure with three inputs and two membership functions for each input and one output is shown in Fig 2. The objective of the current work is to investigate the potential of ANFIS in SPIF. ANFIS consist of five layers to achieve the following fuzzy inference [32]: Layer 1: Fuzzy layer In this layer, the membership value is calculated using the following equation: where μ A i (x) is an appropriate parameterized membership function, and a i , b i , and c i form a parameter set that changes the forms of the functional movement screen () with a value between 1 and 0.
Layer 2: Multiplies the incoming signals and sends the product out. Table 4. Process parameters used with the corresponding experimental results of forming forces and predicted results. Each node output represents the firing strength of a rule.

Input Parameters Experiment forming force Predicted by ANFIS Predicted by Regression Predicted by ANN D f S t Fx (N) Fy (N) Fz (N) Fx (N) Fy (N) Fz (N) Fx (N) Fy (N) Fz (N) Fx (N) Fy (N) Fz (N)
Layer 3: Normalizes the firing strengths In this layer, the normalized firing strength is computed using the following equation: where w i denotes the output of layer i.

Layer 4: Defuzzification
In this layer, each node i is an adaptive node with a node function.
where p i , q i , r i , and s i make up the consequent parameter set of the node, which are identified during the training process.
Layer 5: Total output layer In this layer, all incoming signals are added (summation output). The circle node function is fixed whereas the indicated square function is adaptive. This can be calculated as follows:

Neural network model used for prediction
The ANN computational model involves three layers, output, hidden, and input layers. Each layer contains neurons and each neuron is related to all the neurons in the next layer. Fig 3  shows the layers in a model of the forming force (Fz). None of the processes are executed in the input layer, and the input for the neuron is obtained from the actual setting. The input vector is the weight of a neuron multiplied by the strength; the result obtained helps create the product. The output from the last neuron can be interconnected to the input of the next neurons or can be directly interconnected with the environment. The output comprises an activation function and a summation function. The activation function takes the weight of a neuron as an input and produces its activation as an output. The calculation of the net input from the processing neurons is the summation function. Using the ANN, the nonlinear relationships between the output and input owing to the contained activation function of the nonlinear and linear algebraic equations can be stored. After the weight is altered by the activation function, the neurons that have moved to other neurons make up the next layer. The output of the activation function accepts the results, and then presents them to either the external network or to the neurons in the next layer. The network output is compared with the target having the applied input, and the difference between them is then considered an error. Moreover, algorithms of different networks are applied to decrease the error [33].

Statistical analysis
An analysis of variance (ANOVA) was used to estimate the effects of all factors and their interaction on Fz. As a standard practice in ANOVA, terms with a p-value < α = 0.05 are  Table 5, indicate that the factors d, s, and t; the two-way interactions d � f, d � s, and d � t; and the three-way interaction between d, s, and t have a significant effect on Fz. The value of the adjusted R-squared value shows that the model can explain 91% of the variations in the data, and that 9% of the variations originate from unknown nuisance factors.

ANFIS results
The ANFIS model was developed as a function of SPIF for the forming force using training and testing data. The ANFIS tool that already exists in MATLAB was applied, which tests the relationship of the process parameters used to execute the perfect training and maximizes the prediction model accuracy for the selected responses (forming force). To obtain the results, the ANFIS algorithm was designed using the initial parameters. Table 6 lists the parameters used to help build the ANFIS model.
The training process was applied using 100 epochs for the forming force on three axes (Fx, Fy, and Fz). A training curve was obtained after the training process was complete, as shown in Fig 4. The figure shows the relationship between the number of epochs and the errors in the responses. An analysis of the curves shows that, after 35 epochs, the errors become steady, as shown in Fig 4(a). This occurs because the developed model was trained using limited experimental data. To obtain the initial predicted values of the outputs, such as the forming force, a set of fuzzy inference parameters (FIPs) were selected during the training process. The measured values were compared with the predicted value of the forming force obtained from the developed ANFIS model. The performance of this model was measured based on the difference between the measured and predicted values.
During the training process, FIPs were repeated multiple times until the errors were minimized. Different ANFIS parameters were used as the training parameters to validate the accuracy of the prediction model. Table 7 shows the different ANFIS architectures for a predictive   Table 7, the trimf function was chosen to train the ANFIS because it achieved the lowest testing error of 31.4218. In addition, Figs 5 and 6 show a comparison between the measured and predicted forming forces for the training and testing data.

Artificial neural network results
The results of the developed ANN model are used to predict the forming force based on the input process parameters in single-point incremental sheet metal forming. The numbers of training and testing samples are 28 and 6, respectively. Several training experiments were carried out to identify the optimal network structure and best training parameters of the neural networks, producing minimum errors during the training phase. Similarly, several training experiments with different numbers of hidden neurons, learning rates (0.60), and momentum values (0.80) were checked, as shown in Fig 7. The graph of the learning progress shows the maximum, average, and minimum training errors. The average validation error is 0.00138, which was obtained for a maximum of 38,650,000 learning cycles. The correlation coefficient (R value) can be used to gauge the performance of the established network. The R value is between the measured value and the predicted value for the testing (6) and training data (28). The measurement of the closeness of the dissimilarity in the output clarified by the target is known as the R value, which lies between 1 and 0. When the R value equals 1, the optimal correlation is observed between the output and target values for the forming force on the three axes. The R value obtained between the predicted values and the measured data is 0.981, which indicates a good correlation.

Regression model
A regression analysis helps in the development of a mathematical equation to characterize the relationship between two or more input variables and the response outputs. In this study, mathematical models are also developed using a regression analysis to fit the measured data for the three selected responses. Using Minitab software, regressions models for the forming force were developed, and a full quadratic model was initially selected for all responses. Later, the insignificant terms were removed based on their p-values and accuracy. The following equations can be used to predict the forming force as a function of significant factors:

Comparison of ANFIS with ANN and regressions
To assess the ability of the developed ANFIS model relative to that of a neural network and regression analysis, an ANN model and a regression algorithm were developed using the same input variables.  ANFIS, ANN, and regressions models for the training data. Fig 9 shows the same for the testing data. Fig 10 shows a comparison between the measured and predicted values obtained using ANFIS and the regression model for the training data. Fig 11 shows the same for the testing data.  The results obtained using the ANFIS prediction are very close to the measured values. Moreover, the absolute mean percentage errors were calculated for each of the developed models. Tables 8 and 9 present a comparison of the performance between the ANFIS, ANN, and regression models based on the mean absolute percentage errors (MAPE) for the training and testing data.
Based on the performances of the ANFIS and ANN models in terms of the average absolute percentage error for the training and testing data, it was observed that the ANFIS model outperforms the ANN and regression models, while retaining their full potential.

Conclusions
This paper proposed ANFIS and ANN models to predict the forming force in the context of sheet metal forming, particularly SPIF. In addition, the influences of the tool diameter, feed rate, sheet thickness, and step size on the main forming force were investigated. Considering the ANOVA for the forming force (Fz), it was concluded that the significant factors are the tool diameter, step size, and sheet thickness. The results of the ANFIS and ANN models were compared with both the experimental data and those predicted using a regression model. The comparison showed that the ANFIS model can accurately predict the forming force for both training and testing data; in addition, the ANFIS model exhibited a better prediction performance for the selected responses. Moreover, the results showed that the ANFIS model can predict the forming force along the three axes for the training data with a MAPE of 7.25%, 6.42%, and 8.98%, respectively, and for the testing data with a MAPE of 5.85%, 9.61%, and 15.44%, respectively. It can therefore be concluded that the developed model using the ANFIS approach can be effectively used to measure the forming force during ISF and provide more reliable results than the ANN and regression models.