The Modelling of Surface Roughness after the Ball Burnishing Process with a High ‐ Stiffness Tool by Using Regression Analysis, Artificial Neural Networks, and Support Vector Regression

: Surface roughness is an important indicator of the quality of the machined surface. One of the methods that can be applied to improve surface roughness is ball burnishing. Ball burnishing is a finishing process in which a ball is rolled over the workpiece surface. Defining adequate input variables of the ball burnishing process to ensure obtaining the required surface roughness is a typ ‐ ical problem in scientific research. This paper presents the results of experiments to investigate ball burnishing of AISI 4130 alloy steel with a high ‐ stiffness tool and a ceramic ball. The experiments were conducted following a randomized full factorial design for different levels of input variables. The input variables included the initial arithmetic mean roughness (the initial surface roughness), the depth of ball penetration, the burnishing feed, and the burnishing ball diameter, while the out ‐ put variable was the arithmetic mean roughness after ball burnishing (the final surface roughness). The surface roughness modeling was performed based on the experimental results, using regression analysis (RA), artificial neural network (ANN), and support vector regression (SVR). The regression model displayed large prediction errors at low surface roughness values (below 1 μ m), but it proved to be reliable for higher roughness values. The ANN and SVR models have excellently predicted roughness across a range of input variables. Mean percentage error (MPE) during the experimental training research was 29.727%, 0.995%, and 1.592%, and MPE in the confirmation experiments was 34.534%, 1.559%, and 2.164%, for RA, ANN, and SVR, respectively. Based on the obtained MPEs, it can be concluded that the application of ANN and SVR was adequate for modeling the ball bur ‐ nishing process and prediction of the roughness of the treated surface in terms of the possibility of practical application in real industrial conditions.


Introduction
Ball burnishing is a process in which the ball rolls on the surface and causes elastic, elastoplastic, and plastic deformations of the surface layer of the workpiece [1]. As a result, the workpiece changes shape, dimensions, roughness, hardness, stresses, etc. This process can provide excellent geometric specifications and high dimensional accuracy [2]. Prismatic and cylindrical workpieces are mainly processed, but complex-shaped surfaces can also be treated [3]. Ball burnishing can significantly increase wear resistance, fatigue resistance, corrosion resistance, etc. [4]. This process can manage various materials such as steel, copper, aluminum, brass, polymers, titanium, nickel-chromium-molybdenum, wood, etc. [5].
Previously, the ball burnishing process was studied in numerous ways, most often by the use of experiments. The literature cites a variety of experimental research that applied different techniques to study the relationship between input and output variables in ball burnishing. The investigations are mainly based on the development of new burnishing tool concepts, as well as analysis of the effect of input process variables (burnishing load, feed, number of passes, lubricant, etc.) on output process variables (residual stresses, surface roughness, hardness, etc.). The cost and time are merely some of the disadvantages of the conducted investigations. Furthermore, the obtained results can only be applied in conditions equal to those of the conducted experiments. The consequence of such investigations is the emergence of research on the prediction and modeling of the ball burnishing process.
Another research direction is the research of the ball burnishing process based on finite element analysis (FEA) [35][36][37][38][39][40][41][42]. Three-dimensional finite element models were developed to predict the deformation mechanics, plastic flow, hardness, and residual stress [41]. The disadvantage of these studies is time, especially if many FEA iterations with different input variables are to be conducted. This significantly prolongs the testing time, and thus the related costs. Furthermore, a considerable problem with FEA is the accurate determination of the coefficient of friction. The coefficient of friction can be variable not only between the ball and the surface to be treated but also between the ball and the supporting elements in the ball burnishing tool. This coefficient is an important input for accurate analysis in FEA and has a dominant impact on the results [42].
In earlier studies of the ball burnishing process, the Taguchi method [43][44][45][46][47][48][49][50] and response surface methodology (RSM) [51][52][53][54][55][56] were widely used. The Taguchi method reduces the number of experiments required for analysis but considers only the main effects on the process and does not allow consideration of the interaction between the input variables. RSM approach provides obtaining empirical, i.e., regression models, which quantify the effect of input variables on the output variables of the ball burnishing process. RSM is used to generate a polynomial model that includes ball burnishing process variables, as well as to diagnose their statistical significance. These traditional methods, i.e., the Taguchi method and RSM, are not characterized by great adaptability, especially if the experimentally obtained dependencies between the input and output variables of the ball burnishing are complex.
As computer technology developed, soft computing methods have also begun to be used to model ball burnishing processes. Soft computing differs from conventional computing in that it tolerates inaccuracy, uncertainty, and approximation. Soft computing techniques have attracted the attention of researchers because of their potential to study multidimensional, nonlinear, and complex problems. For example, Cagan et al. [49] investigated the effect of burnishing parameters such as the number of passes, burnishing force, burnishing speed, and feed rate on the surface roughness and hardness using different artificial neural network models. Basak and Goktas [57] used a fuzzy logic (FL) model to achieve the best parameters (number of revolutions, feed, number of passes, and pressure force), which affect surface roughness after the burnishing process. Al-Saeedi et al. [58] predicted surface roughness values by FL for dry and fluid burnishing in relation to bur-These three methods were compared based on the accuracy of the prediction of the output variable of the ball burnishing process. The comparison was made based on the obtained deviations, i.e., percentage errors between the current (experimental) and predicted values. The validation of the developed models was conducted on many confirmation experiments.
The remainder of the paper is organized as follows. Section 2 describes the framework of the research work. In this section, the methods and materials used in the research were also described. Section 3 presents detailed results of conducted experimental research, as well as results of surface roughness modeling using regression analysis, artificial neural network, and support vector regression. In order to objectively evaluate the efficiency of all three developed models, confirmation experiments were also presented in Section 3. Section 4 provides a discussion of the results and demonstrates the analysis of the results. Section 5 includes a summary in the form of main conclusions, limitations, and future research directions.

Materials and Methods
The ball burnishing process was conducted on a milling machine (HAAS-Toolroom Mill TM-1HE, Haas, Oxnard, CA, USA) ( Figure 1) in one pass at 1800 mm/min burnishing speed. During the ball burnishing process, the rotation of the main spindle was blocked. The ball burnishing process was performed only with translational movements of the workpiece that was located and clamped in a vise. By considering the increasingly stringent environmental requirements, ball burnishing was conducted in a dry environment. Four input variables were varied: the initial surface roughness, the depth of ball penetration, burnishing feed, and ball diameter. The output variable was the surface roughness after the ball burnishing process (the final surface roughness).
The ball burnishing process was performed with a high-stiffness tool [18][19][20], which ensures ball rolling without sliding. The rolling of the ball was ensured by the construction of the high-stiffness tool. The ball and three roller bearings are housed in a common support. The three-roller bearings arranged at an angle of 120° in relation to the direction of penetration of the ball into the material of the workpiece ensure that the ball rolls. The high-stiffness tool operates on the principle of the constant depth of ball penetration. The depth of penetration is given to the ball burnishing tool. The burnishing force necessary to perform the ball burnishing process is generated indirectly by setting the value of the depth of ball penetration into the workpiece. Ball burnishing with a high-stiffness tool, which is based on setting the depth of ball penetration, does not require additional equipment to achieve and monitor the required burnishing force. During ball burnishing, the surface layer of the workpiece is deformed. The peaks begin to deform and fill the valleys. This reduces the height parameters of surface roughness.
The research was conducted on prismatic workpieces made of steel AISI 4130. The chemical composition of workpieces is shown in Table 1, and the mechanical, physical, and thermal properties are presented in Table 2. This alloy 4130 steel is widely used, and some typical applications include commercial aircraft, aircraft engine mounts, military aircraft, automotive, machine tools, hydraulic tools, auto racing, aerospace, oil and gas industries, agricultural, defense industries, etc. The experiments were conducted with ceramic balls made from silicon nitride (Si3N4) with the following chemical composition: silicon nitride 97%, iron 0.5%, carbon 0.3%, silicon 0.3%. Si3N4 is characterized by its high strength over a wide temperature range, high fracture toughness, high hardness, excellent wear resistance, high thermal resistance and good chemical resistance, high shock, and thermal resistance. Balls were different only in diameter. Other balls' characteristics were identical: sphericity 99.9995%, density 3.27 g/cm 3 , elastic modulus 310 GPa, Poisson's ratio 0.24, hardness 78 HRC, thermal conductivity 29 W/mK, coefficient of thermal expansion 3.3 × 10 −6 °C, compressive strength 4055 MPa, maximum surface roughness 0.025 μm, diameter tolerance 0.5 μm, and sphericity tolerance 0.25 μm.
The surface roughness was measured before and after the experimental research. The measurement was performed on a Talysurf 6 measuring device with a diamond ball stylus, stylus radius 2 μm, stylus force 1 mN, and traverse speed 0.5 mm/s. The measurement was conducted with a cut-off length of 0.8 mm (Gaussian filter), a sampling length of 0.8 mm, and an evaluation length of 4 mm. Surface roughness values were calculated as mean values derived from five repeated measurements.
The workpiece surfaces treated with ball burnishing were pre-treated to the values of the arithmetic mean deviation of roughness profile Ra = 3 μm, 4 μm, 5 μm (the initial surface roughness). For these values of Ra, the following values of maximum profile peak height Rp = 12.05 μm, 16.31 μm, and 19.59 μm, maximum profile valley depth Rv = 11.75 μm, 15.96 μm, 20.06 μm, and the maximum height of roughness profile Rz = 23.8 μm, 32.27 μm, and 39.65 μm were obtained, respectively.
After the experiments, the ball burnishing process was modeled with the help of RA, ANN, and SVR. These three methods were used to predict the output variable-the final surface roughness (Ra). The theoretical foundations of these methods are presented below.

Regression Modelling
Regression modeling is a procedure by which statistical processing and analysis of experimental data give mathematical dependencies between the output and input variables, often using the method of least squares. This way, regression models are obtained, which, if there are two or more input variables, can be represented graphically using the response surface. It is best when the lowest order polynomial response surface can be obtained, i.e., one should not strive for an overly complex model of the higher-order. Models can be used to predict output variables for determining the values of input variables. They can also be a starting point or a basis for optimization, especially for optimizing the parameters of technological processes. The application of this method also indicates which of the input variables have a significant effect on the output variables.
For the derived regression model, it is necessary to carry out an analysis of variance and obtain significant statistics, e.g., mean, standard deviation, coefficient of variation, as well as ordinary, adjusted, and prediction coefficients of determination. Furthermore, the obtained model was analyzed graphically by the use of the following plots:  Normal probability plot of residuals;  Residuals versus input variables;  Residuals versus predicted values;  Residuals versus run.
The comparison of actual and predicted values calculated by the model is also significant.

Artificial Neural Networks
Artificial Neural Network (ANN) is a machine learning tool consisting of multi-processing elements called neurons. Neurons in ANNs are divided into an input layer, one or more hidden layers, and an output layer. Neurons are connected through interconnection weight coefficients, so the structure of ANN mimics the actual biological neural network. The number of hidden layers and the number of neurons in each hidden layer are problem-dependent.
ANNs are widely used in various applications, such as pattern recognition, classification, regression, time series prediction, etc. When used in regression problems, the architecture of ANN is usually a feed-forward multilayer perceptron, which means that signals are transmitted through the network only in one direction, from input to output layer [68]. The main task of such a network is to determine the relationship between input and output data, i.e., to substitute a classical mathematical regression function.
For the given training dataset x , ∈ R R, where xi is some m-dimensional input vector and yi is the corresponding output value, the weights in ANN need to be adjusted to obtain minimum output error compared to the training dataset. The process of weight adjustment is referred to as network training, and it is conducted using one of the optimization techniques (backpropagation, gradient descent, scaled conjugate gradient, Levenberg-Marquardt, Gaus-Newton, genetic algorithm, etc.) [69]. Once the network is trained, it can be used to obtain output values for any set of input parameter values.

Support Vector Regression
Support Vector Machines (SVM) is a very popular and powerful supervised machine learning technique used mostly to solve classification problems in a wide range of applications in various domains. However, SVMs can also be successfully applied to regression problems. In that case, they are commonly referred to as Support Vector Regression (SVR). The main advantages of SVR are their generalization ability, obtained by maximization of data margin, and the efficient learning of nonlinear functions by using kernel trick [70].
SVR is a technique of estimation of a linear function F that maps input data to a real number based on the provided training dataset x , ∈ R R. This function can be written as: where w is the vector of appropriate weights and b is the bias. The main task is to determine optimal values of these parameters to obtain an accurate regression model, i.e., to provide an accurate approximation of the training dataset by function F(x). Instead of trying to determine the regression function, which minimizes the error of the training dataset, as with most mathematical regression techniques, SVR tries to maximize the margin around the function F, which envelops the training data, considering the constraints that approximation error must be less than some specified value ε. In other words, SVR solves the following optimization problem: : where Lm is the actual margin width, and L is the alternative optimization criterion, which is more convenient for calculation. This optimization problem is usually solved by the method of Lagrange multipliers [69] in order to obtain optimal parameters w and b.
The above-explained SVR method is the simplest one, implying that all training data fall into " -tube". In general, some of the data will fall outside the boundary. These deviations need to be minimized, which is obtained using slack variables for any data value falling outside of . In this case, the optimization problem is modified in the following way: where represents the deviation of the predefined error limit , and C represents the adjustable penalty parameter. This extended SVR model is called soft margin SVR, which is explained in more detail in [69].
The graphical representation of the SVR problem with one-dimensional input vector x is depicted in Figure 2. If the regression function is nonlinear, which is a common case, SVR can still be used applying the kernel trick, meaning that some kernel function Φ(x) is used to transform the original m-dimensional feature space to some more-dimensional space in which the regression function is linear so that all previous considerations can be applied. The most common kernel functions are polynomial, Radial Basis Function (RBF), sigmoid, etc. [69].
After the RA, ANN, and SVR modeling processes, the actual (measured) Ra values were compared with the predicted values. The comparison between the measured and predicted Ra values were made based on the following equations: , , • 100%, , , where Raimv represents measured Ra value, Raipv is the predicted Ra value, PE is the percentage error, MPE is the mean percentage error.

Results
Experimental studies were conducted following a randomized full factorial experiment that allows the investigation of all combinations of input variable levels. Two experiments were conducted; the first one, named training experiment, was used for models obtaining, and the second one, named confirmation experiment, was used to examine the accuracy of models formed and trained on data from the training experiment. The structure of inputs and outputs was the same for both experiments.

Training Experiment
During the training experimental research, four processing parameters (input variables) were varied at the following levels: By considering the adopted levels of input variables, a total of 5 × 3 × 3 × 3 = 135 experiments were conducted within the training experiment. The measured values of the surface roughness are presented in Table 3.  Figure 3 provides a graphical representation of the measured data presented in Table  3. The value of output parameter Ra2 is depicted, related to pairs of input parameters, i.e., Ra1-ap, Ra1-fb, and Ra1-Db, respectively. The experimental data were approximated by a surface, obtained using cubic interpolation among all 135 data in Matlab's Curve Fitting Toolbox (MathWorks, Matlab version 2017b). The cubic interpolation method constructs the surface interpolating experimental data by bicubic interpolation function using the least-squares method. In comparison to the other two commonly used two-dimensional interpolation techniques, bilinear and nearest-neighbor methods, cubic interpolation obtains a smoother surface, which provides a more intuitive interpretation of experimental data [71]. Based on the statistical processing of the experimental data, using the Design Expert software (Stat-Ease, Inc., Design Expert version DX8, 8.0.7.1), a regression model was obtained. It was then transformed and reduced. The analysis of variance of the obtained regression model is presented in Table 4. Table 4 also shows other useful information. The ordinary coefficient of determination R 2 is close to 1, as well as its variations, R 2 adjusted and R 2 for prediction.  Figure 4 is a graphical representation of the significant interaction between the factors A-depth of ball penetration (ap) and B-the initial surface roughness (Ra1), for burnishing feed of 0.15 mm and burnishing ball diameter of 6 mm. These two factors, A and B are nonsignificant (Table 4), but their interaction is significant (Table 4), so they were added to the model after the backward elimination regression. It can be seen in Figure 4 that the different effects on the final surface roughness (Ra2) for the smallest (3 μm) and the largest (5 μm) initial surface roughness is in the range of the depth of ball penetration from 12 to 20 μm. For the smallest initial surface roughness, with the increasing depth of ball penetration from 12 μm to 20 μm, the final surface roughness also increases, while for the largest initial surface roughness, with the increasing depth of ball penetration from 12 μm to 20 μm, the final surface roughness reduces. For the depth of ball penetration lower than 12 μm, the final surface roughness reduces for both the smallest and the largest initial surface roughness, while for the depth of ball penetration greater than 20 μm, the final surface roughness increases for both the smallest and the largest initial surface roughness. Analysis of variance and the statistical parameters shown (Table 4), as well as Figures  5-7 above, indicate the adequacy of the regression model because the residuals are distributed normally (plot in Figure 5 resembles the straight line), and the internally studentized residuals are not higher than 3 or 4 standard deviations from zero and are quite structureless.   For modeling purposes of the burnishing process using ANN, input parameters were the depth of ball penetration, the initial surface roughness, burnishing feed, burnishing ball diameter, and the output was the predicted value of the final surface roughness. Two main characteristics must be defined for the ANN-its architecture and the training method used to optimize network weights. For the purposes of modeling, the most often used network type is a feed-forward multilayer perceptron network, which was also used in this study. The optimal architecture of the network was determined using the trial-and-error method, and it consists of three hidden layers with four, five, and four neurons, respectively (4-5-4 scheme), as shown in Figure 8. For neurons in the hidden layer, the sigmoid activation function is applied, while the neurons in the output layer are activated by the linear activation function, which is a common choice for feed-forward multilayer perceptron ANNs used in regression purposes [72]. Three popular training methods were applied, i.e., Levenberg-Marquardt, Scaled Conjugate Gradient, and Resilient Backpropagation. Among these methods, Levenberg-Marquardt was chosen because it provided the best results [73]. The optimization criterion, Mean Square Error (MSE), was used, which is common in similar applications. The ANN was implemented using Matlab's Deep Learning Toolbox. The available data set, consisting of 135 samples, was divided into three sets: the training set (95 randomly chosen data samples, which is approximately 70% of the whole data set), the validation set (20 samples, i.e., 15%), and the testing set (20 samples, i.e., 15%). The optimization of networks weights was conducted iteratively on the training data set, and results were verified using the validation set. When the error of the validation set reached the minimal value, the training was terminated to prevent overfitting, which is the standard procedure for ANN training. The test set was used to evaluate the quality of network prediction for the data that were not used in training.
In the modeling of the burnishing process using SVR, the same inputs and outputs as with RA and ANN modeling were used. Matlab's Regression Learner Application was used for software implementation. In order to avoid overfitting, k-fold validation was applied as a standard statistical method to estimate the SVR quality. In this procedure, the experimental data set was divided into k groups (k = 5 in this study), and a total of k models were fitted using the optimization procedure explained in Section 2.3. Each of these models was formed by holding out one group and using the remaining k − 1 group of data for the SVR training. The group of data that was held out is used to evaluate the SVR model since it represented the data "not seen" by the model. When all k models were evaluated, the average efficiency was used as the estimation of the SVR model quality.
Besides the linear SVR, three different kernel functions were also applied to construct nonlinear SVR, namely quadratic, cubic, and Gaussian. The best results were obtained using the quadratic kernel function. Table 5 shows all measured (actual), as well as regression (predicted) values and percentage errors (PE) calculated based on Equation (4).

Confirmation Experiment
In order to objectively evaluate the efficiency of all three models, additional 48 confirmation experiments were conducted, named confirmation experiment, as explained before. In these experiments, only the values of ball diameters remained unchanged, while the other three input parameters (the depth of ball penetration, the initial surface roughness, and burnishing feed) had different values from those used in the training experiment.
During the confirmation experimental research, four processing parameters (input variables) were varied at the following levels: After the confirmation experiments, the measurement and prediction of the final arithmetic mean roughness (Ra2) was performed for the three previously described models (RA, ANN, and SVR). Table 6 shows the obtained results of Ra2 measurements, Ra2 predictions, and the obtained percentage errors for the three models.

Discussion
Descriptive parameters of the measured values of surface roughness after ball burnishing during the training and confirmation experiments are shown in Table 7. The values of the surface roughness range from 0.13 μm to 4.39 μm. This points to two very important facts. First, a minimum surface roughness value of 0.13 μm indicates that the extremely high quality of the treated surface can be obtained by this procedure. Second, the ratio of maximum and minimum values (Ramax/Ramin) of 33.77 is extremely high. This indicates that with a suitable combination of process parameters, Ra can be significantly affected. In other words, a suitable choice of input parameters of the process can achieve the required roughness in a very wide range. This is important from the point of view of practical application, considering that it is not necessary to insist on the minimum but on the required surface roughness. The predicted values of Ra have the same trend as the measured values. The Ra values obtained by the training and confirmation experiments have close values for the RA, ANN, and SVR models. Differences between values are acceptable and occur as a consequence of the fact that the confirmation was performed on the input parameters on which no training was performed, i.e., on "unknown" input variables. Moreover, the MPE for the ANN and SVR models is lower for confirmation experiments than for training experiments, while for the RA model, it is negligibly higher. This confirmed the correct choice of training parameters and the correctly performed training process for all models. In most experiments (Tables 5 and 6), the surface roughness after the ball burnishing process is lower than the initial one (Ra2 < Ra1), which was the goal. However, in several experiments, it can be observed that the surface roughness obtained after the ball burnishing process was higher than the initial surface roughness, i.e., Ra2 > Ra1. This is a consequence of inadequate process parameters. These are also cases that represent the full expediency of the modeling process because the knowledge and experience of the operator are minimized if the process is well modeled, and inadequate process parameters that can lead to scrap are avoided.
Based on the results of experiments, the following can generally be stated:


The depths of ball penetration close to the maximum profile height before processing generate the lowest surface roughness. For depth values greater and smaller than this one, the surface roughness increases;  As the burnishing feed decreases, the surface roughness reduces;  As the ball diameter increases, the surface roughness reduces. Figure 3 shows an archlike dependence of the surface roughness obtained after ball burnishing (Ra2) in relation to the initial surface roughness (Ra1) and the depth of ball penetration (ap). For depths of ball penetration (ap) lower than the initial value of the profile height, when the ball reaches and "touches" the treated surface under the effect of high contact pressures, the peaks begin to flow and fill the valleys in the roughness profile and reduce surface roughness. This happens up to a certain depth of ball penetration (ap), approximately when ap ≈ Rp. For these values, the lowest surface roughness was achieved after processing. With a further increase in the depth of ball penetration (ap), the surface roughness (Ra2) worsens. Larger depths of ball penetration (ap) cause the emergence of larger valleys, which results in higher roughness (Ra2), i.e., a reduction in the quality of the treated surface. Larger depths of ball penetration (ap) also induce greater forces that can damage the surface and create surface cracks. At greater depths of ball penetration (ap), the depth of the valleys enlarges, which results in a large amount of plastic deformation that causes deeper traces of penetration. As a result, the surface roughness worsens (Ra2), and cracks can occur, which further damages the surface topography. The effect of the deterioration of surface roughness after a certain depth of ball penetration (ap) can be attributed to the greater hardening of the superficial workpiece layer due to compression. Furthermore, there may be a superficial peeling on the workpiece. With the reduction in burnishing feed (fb) and with the unchanged other process parameters, the roughness of the treated surface (Ra2) reduces. In all experimental studies, the surface roughness (Ra2) increased with increasing burnishing feed (fb), while the increased trend remains mild. The burnishing feed (fb) values were selected so as not to negatively affect the surface that was processed during the earlier tool movement. If two consecutive movements of the ball are close to each other, the material of the workpiece that has already filled the valleys could begin to overflow laterally (perpendicular to the direction of the movement) and thus worsen the roughness (Ra2) of the pre-treated surface on which the peaks have already filled valleys. This negative phenomenon, in the context of the movement, was avoided in this research. A larger ball diameter (Db), under the unchanged other processing conditions, reduces the surface roughness (Ra2). This can be explained by the very nature of the process, i.e., the contact that takes place between the ball and the treated surface. With larger ball diameters (Db), the contact area between the tool and the treated surface is larger, which reduces roughness (Ra2).
Descriptive parameters for PE obtained by modeling via RA, ANN, and SVR are shown in Tables 8 and 9. Figure 9 shows the PE for all conducted experiments. The ANN and SVR models predicted the obtained roughness for both experimental data and the "unknown" data very successfully during confirmation experiments, while prediction using the RA model indicated significantly lower accuracy, especially in cases when surface roughness values were very low. Based on the obtained data, identical trends can be observed for both the training experiment and the confirmation experiments. The lowest minimum and mean PE is provided by ANN, while the lowest maximum PE is suggested by SVR.
Out of a total of 183 experiments conducted, percentage errors PE greater than 5% occur only for eight predictions via ANN and six predictions via SVR. It should be noted that in these, from the point of view of prediction, the most unfavorable cases, the absolute error, i.e., the difference between the realized and the predicted value of Ra is in the range 0.01-0.03 μm, which is an acceptable deviation from a practical point of view. In all other cases, the errors obtained via ANN and SVR are significantly lower. Therefore, ANN and SVR models can be implemented in the actual industrial conditions, while the RA model can be successfully applied when it is not necessary to obtain extremely low surface roughness.

Conclusions
This paper discusses the application of the burnishing process on a workpiece made of AISI 4130 alloy steel with a high-stiffness tool and a ceramic ball. Different values of process input parameters were applied to assess their effect on the surface roughness of the workpiece after processing. Three methods were applied in the process modeling, RA, ANN, and SVR, respectively.
Based on experimental and modeling investigations, it is possible to draw several conclusions:  The obtained experimental results of surface roughness indicate the values of surface roughness range from 0.13 μm to 4.39 μm, which means that it is possible to obtain very low values of surface roughness, below 1 μm. The minimum value of surface roughness after ball burnishing was 0.13 μm, which relates to the N3 International Organization for Standardization (ISO) grade number. This is a very high quality of the treated surface;  The influence of input variables on surface roughness after ball burnishing is different. The depths of ball penetration close to the maximum profile height before processing generate the lowest surface roughness. For depth values greater and smaller than this one, the surface roughness increases. Moreover, the surface roughness decreases with decreasing burnishing feed and with increasing ball diameter;  The quality of RA, ANN, and SVM models was assessed based on the results of the training and confirmation experiments. The evaluation of the model was performed based on PEs. For a complete series of conducted experiments, the minimum and mean PEs are lower for the ANN model, but the maximum PE is lower for the SVR model. Bigger errors, but without a significant effect on applicability in practice, are obtained for values of Ra < 1 μm. For surface roughness values Ra ≥ 1 μm, these errors are negligible from a practical point of view since the majority of PE is lower than 1%;  The obtained PEs for ANN and SVR models are considered acceptable for practical application. ANN and SVR models have successfully predicted surface roughness, so they can be successfully implemented in the industry when planning the production process.
The main limitation of the applied methodology is that the process was modeled for certain experimental conditions in terms of defined tools, workpieces, and equipment. However, the applied approach is sufficiently general and universal so that it can be easily applied and implemented for other conditions of the ball burnishing process.
Future research will focus on investigating the ball burnishing process with other modeling methods. Furthermore, the aim is to cover a larger number of input and output processing parameters, as well as to perform multi-criteria optimization of the ball burnishing process.