Keywords

1 Introduction

The modern machining industries aim to produce high quality product which has desired dimensional accuracy, surface roughness in a cost and time effective manner. CNC machining plays a vital role in modern industry for making products on scales from small individual parts like screw, mold to large, heavy-duty operations such as military and automobile products [4]. CNC milling is one of the major machining operations which makes complex shapes and finishing of the machined products. Surface roughness and the milling accuracy are two major output parameters regarded as the index for evaluating performance of milled product. ML models are capable of learning the non linear relationships between input and output variables. Hence, ML Models can be used in the CNC milling process to model the relationship between input and output parameters [1].

Several studies have been done in the literature which investigate the effect of milling parameters on performance index values. The effect of milling parameters on the surface roughness is modelled using Gaussian Process Regression (GPR) and the surface roughness value is predicted using the milling parameters [11]. A statistical multiple regression technique has been proposed for the prediction of surface roughness using aluminium 6061 alloy for end milling process [7]. A statistical model to analyze the effect of machining parameters and geometrical parameters on the vibration amplitude is proposed in [10]. A second order mathematical model have been proposed and the results were verified using ANOVA. An ANN based prediction model for predicting surface roughness of the CNC face milled product is proposed in [1], where authors have used Taguchi method for designing the experiments and have considered various factors such as depth of cut, cutting speed, engagement and wear of cutting tool, etc. The surface roughness of the face milled product is predicted using feed forward ANN. Although ANNs are capable of learning non-linear relationships, the process of training and learning happens in a black-box manner. Hence, for improved interpretability and understandability of the model, in this paper we have used regression trees for prediction of the performance index values.

Most of the papers studied about the effect of input parameters on the surface roughness. To the best of our knowledge, no studies have been done which investigate the effect of input parameters on geometric tolerances such as parallelism and perpendicularity.

2 Methodology

2.1 Regression Trees

Regression trees [2, 6] are one of the machine learning approaches which helps in construction of prediction models using data. The prediction models are developed by partitioning the data and then fitting a small model on that partitioned data. The procedure of partitioning is recursively followed thereby giving various set of small models. The procedure is graphically represented as tree hence called decision tree. The decision tree consists of nodes as features, branch as rules and leaf as the output. Decision Tree have clear representation of the information which makes it one of the most useful machine learning methods for linear problems. Classification and Regression Tree [2] is one of the algorithm for decision tree which uses Gini Index to determine the point of partition of the data [3]. The linear regression model can be represented as:

$$\begin{aligned} y=w_{0}+\sum _{j=1}^{k}w_{j}x_{j} \end{aligned}$$

where y is a output, \(x_1,x_2, \ldots , x_k\) are independent variables, \(w_1,w_2, \ldots , w_k\) are regression coefficients. The squared error is used as the cost function for the regression tree.

$$\begin{aligned} Error= \sum _{j=1}^{k} {(\hat{y} -y)}^2 \end{aligned}$$

where \(\hat{y}\) and y represents the predicted output and actual output respectively. CART algorithm derives rules from the information and it does not require any pre-specified values. For the prediction of values, it takes “if and then” rules predicates. Each branch represents a conditional statement, each leaf node is derived by joining “if” statements from root node to leaf node.

In this paper, we have used regression tree for the prediction of performance indexes, as the tree representation created using CART algorithm provides more understandability about the system. Also, the interpretability of the model is high with respect to the tree with which rules can be derived. Thus the manufacturer can decide values of the input parameters accordingly.

2.2 Milling Process

Milling is a type of CNC machining in which the rotating cutter move towards the workpiece and removes the material. The milling is accomplished by varying the parameters such as spindle speed, direction, pressure etc. Milling has various types which includes pocket milling, slot milling, drilling etc. Slot milling is considered for the experiments for building the predicition model. The input parameters considered for the model are spindle speed, feed rate, depth of cut. In the milling process, slots of dimensions 33 mm \(\times \) 6 mm \(\times \) 3 mm were made with the combinations of the input parameters. The combination of values is chosen by considering four levels for each input parameter. With the levels for each input value, 96 combinations are chosen. The milling is accomplished in aluminium alloy (Al 7075) by varying the three input parameters such as feed rate (FR) (mm/rev), spindle speed (SS) (rpm), depth of cut (DC) (mm) in MAX MILLPLUS milling machine. The steps followed in the process are listed below.

Fig. 1.
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Graphical representation of geometric tolerances

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  1. 1.

    The aluminium alloy workpiece is face milled for the smoother and flat surface such that the measurement errors and surface errors can be minimized.

  2. 2.

    The dimension of the slots and the parameters are given in G-code files to the CNC. It should be noted that each slot has different values of parameters but the same dimension.

  3. 3.

    The surface roughness value of each slot is measured using surface roughness tester.

  4. 4.

    Geometric tolerances of the milled product is measured using CMM as follows: The parallelism is measured considering the surface of the object as datum plane and the milled surface as feature plane. The value measured is referred as ‘\(parallel_a\)’. The parallelism of the two sides of the milled surfaces considering one of the surface as datum plane and other as feature plane, the value is measured, which is referred as ‘\(parallel_b\)’. The graphical representation of the two parameters \(parallelism_a\) and \(parallelism_b\) is as shown in the Fig. 1.

  5. 5.

    The perpendicularity is measured considering the surface of the object being datum plane and two sides of the milled slot being feature planes, two values are measured which are referred a ‘\(perpendicular_a\)’ and ‘\(perpendicular_b\)’ respectively. The graphical representation of the two parameters \(perpendicular_a\) and \(perpendicular_b\) is as shown in the Fig. 1.

3 Experimental Results

In this section, the specifications of the machine used for the experiments, analysis of data and the results obtained are discussed. The experimental data has been collected by performing experiments using 3 - axis vertical axis machine MTAB MAXMILLPLUS as shown in the Fig. 2(a). SINUMERIK 808 controller is used in the machine which is designed by SIEMENS [9]. After milling process, the average surface roughness (\(R_a\)) data is collected using surface roughness tester belongs to Mitutuyo SJ-210 series as shown in Fig. 2(c). The surface roughness tester can measure upto 17.5 mm with three different measuring speeds such as 0.25, 0.5, 0.75 mm/s which are widely used in the literature [5].

Fig. 2.
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Machines used for the experiments

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The geometric tolerances such as parallelism and perpendicularity of the slots are collected using SPECTRA CMM designed by ACCURATE which is shown in the Fig. 2(b). The CMM can be operated in both motorized or CNC mode [8]. The major milling parameters considered for the milling process are feed rate, spindle speed and depth of cut. Aluminium alloy (Al 7075) is used as the object for milling. The minimum values considered for FR, SS and DC are 0.08 mm/rev, 2000 rpm and 0.1mm respectively. The maximum values considered for FR, SS and DC are 0.2 mm/rev, 6000 rpm and 0.8 mm respectively. The values considered are found to be the optimized range for the milling process in aluminium alloy [1, 4, 7, 10].

Fig. 3.
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Regression trees for all the output variables

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The regression tree model built is validated using 10-fold cross validation method. The mean square error and the root mean square error of the model is calculated. Figure 3(a), (b), (c), (d), (e) shows the regression trees created for the prediction of the average surface roughness, \(parallel_a\), \(parallel_b\), \(perpendicular_a\), \(perpendicular_b\) respectively. From the Fig. 3(a), it can be seen that the first split of the data, the information gain of DC is higher compared to other parameters. The impact of depth of cut is higher in surface roughness value compared to other parameters. Also, the surface roughness value is higher when depth of cut is more than 0.22 mm. For the geometric tolerance - parallelism, feed rate has higher impact other than any parameters which can be observed from the Fig. 3(b), (c).

Fig. 4.
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Actual vs estimated value - surface roughness

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Fig. 5.
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Actual vs estimated value - \(parallel_a\)

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Fig. 6.
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Actual vs estimated value - \(parallel_b\)

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Fig. 7.
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Actual vs estimated value - \(perpendicular_a\)

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Fig. 8.
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Actual vs estimated value - \(perpendicular_b\)

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Figures 4, 5, 6, 7 and 8 represents the actual value and predicted value of the all the output parameters average surface roughness, \(parallel_a\), \(parallel_b\), \(perpendicular_a\), \(perpendicular_b\) respectively. It can be observed from the figures that the decision tree model predicted the output parameter values with lesser error. The actual value line almost overlaps estimate line which symbolizes decision tree model is trained properly to learn the relationship present in the data. The validation root mean square error obtained for all performance indexes are 0.5107 mm, 0.3659 mm, 0.0551 mm, 0.6334 mm, 1.6157 mm respectively.

4 Conclusion

In this paper, we proposed a prediction model using regression tree to predict performance index values such as average surface roughness and geometric tolerances of slot milled product. The experiments were conducted in 3 - axis vertical milling machine. The performance index values are collected using surface roughness tester and co-ordinate measuring machine. The dataset collected is analysed and the regression tree model is built for prediction. Results shows that model achieves lesser root mean square error. Also, to the best of our knowledge no studies have built a model which predicts the geometric tolerances of the milled product. Our proposed model will be useful in manufacturing the parts with desired performance index values, where manufacturer can choose the parameters using the model instead of trial and error which is not an efficient way.