Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a significant amount of time to run and hence are not very practical if solutions are of immediate need. For this reason, Machine Learning (ML) is an attractive option since the ML models,
Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a significant amount of time to run and hence are not very practical if solutions are of immediate need. For this reason, Machine Learning (ML) is an attractive option since the ML models, * Corresponding author. Tel.: +1-704-687-8321 . E-mail address: harish.cherukuri@uncc.edu (Harish Cherukuri). once built, can provide real-time solutions. Many researchers have successfully applied ML techniques in studying various manufacturing problems.
Nomenclature α rake angle (deg) f uncut chip thickness (mm) V c cutting speed (m/min) insensitive loss parameter in SVR objective function C regularization parameter in SVR γ RBF kernel parameter in SVR l length scale hyper-parameter in GPR σ f amplitude hyper-parameter in GPR Cho et al. [5] used Support Vector Machines (SVMs) to provide a detection system which is able to define process abnormalities during the milling process based on cutting forces and power consumption. Their investigation shows that the SVR model performs with high accuracy compared with a multilinear regression model. They conclude that the SVR model can potentially lead to a reduction of machine downtime and production costs. Jurkovic et al. [8] compared three different ma-
Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a significant amount of time to run and hence are not very practical if solutions are of immediate need. For this reason, Machine Learning (ML) is an attractive option since the ML models, * Corresponding author. [5] used Support Vector Machines (SVMs) to provide a detection system which is able to define process abnormalities during the milling process based on cutting forces and power consumption. Their investigation shows that the SVR model performs with high accuracy compared with a multilinear regression model. They conclude that the SVR model can potentially lead to a reduction of machine downtime and production costs. Jurkovic et al. [8] compared three different ma-
Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a significant amount of time to run and hence are not very practical if solutions are of immediate need. For this reason, Machine Learning (ML) is an attractive option since the ML models, * Corresponding author. Tel.: +1-704-687-8321 . E-mail address: harish.cherukuri@uncc.edu (Harish Cherukuri). once built, can provide real-time solutions. Many researchers have successfully applied ML techniques in studying various manufacturing problems.
Nomenclature α rake angle (deg) f uncut chip thickness (mm) V c cutting speed (m/min) insensitive loss parameter in SVR objective function C regularization parameter in SVR γ RBF kernel parameter in SVR l length scale hyper-parameter in GPR σ f amplitude hyper-parameter in GPR Cho et al. [5] used Support Vector Machines (SVMs) to provide a detection system which is able to define process abnormalities during the milling process based on cutting forces and power consumption. Their investigation shows that the SVR model performs with high accuracy compared with a multilinear regression model. They conclude that the SVR model can potentially lead to a reduction of machine downtime and production costs. Jurkovic et al. [8] 
In this paper, machine learning (ML) models, namely Support Vector Regression (SVR) and Gaussian Process Regression (GPR), are presented for the prediction of specific cutting forces and maximum tool temperatures during orthogonal machining processes. The training/test data for building the ML models is generated from finite element (FE) simulations. The simulations are performed using the commercial FE package Abaqus/Explicit and validated using experimental results. The FE generated data consists of cutting speed, uncut chip thickness, and rake angle as the input parameters. The response variables are the cutting force and maximum tool temperature. The data is split into training and test sets using 80-20 split. The optimal SVR and GPR models are selected using grid search on the training data. The predictions on the test data sets show that both the models perform well with high accuracy in predicting cutting force and maximum tool temperature. Between the two models, the mean square errors (MSE) for SVR are less than those for GPR.
Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a signif- Nomenclature α rake angle (deg) f uncut chip thickness (mm) V c cutting speed (m/min) insensitive loss parameter in SVR objective function C regularization parameter in SVR γ RBF kernel parameter in SVR l length scale hyper-parameter in GPR σ f amplitude hyper-parameter in GPR Cho et al. [5] used Support Vector Machines (SVMs) to provide a detection system which is able to define process abnormalities during the milling process based on cutting forces and power consumption. Their investigation shows that the SVR model performs with high accuracy compared with a multilinear regression model. They conclude that the SVR model can potentially lead to a reduction of machine downtime and production costs. Jurkovic et al. [8] compared three different ma-
Specific cutting force is defined as the cutting force required to remove unit area of work material. Prediction of specific cutting forces prior to the physical machining process is important for estimating power and torque requirements. Similarly, knowledge of maximum tool temperatures is essential to improving the machining process and tool life. For a given workpiece material and cutting tool material, several parameters including the cutting speed, the rake angle, uncut chip thickness affect these two quantities. A model that incorporates these parameters to predict specific cutting force and maximum tool temperatures in real-time is of practical importance. One approach is to use the finite element method. However, a drawback of this approach is that the simulations can take a signif- insensitive loss parameter in SVR objective function C regularization parameter in SVR γ RBF kernel parameter in SVR l length scale hyper-parameter in GPR σ f amplitude hyper-parameter in GPR Cho et al. [5] used Support Vector Machines (SVMs) to provide a detection system which is able to define process abnormalities during the milling process based on cutting forces and power consumption. Their investigation shows that the SVR model performs with high accuracy compared with a multilinear regression model. They conclude that the SVR model can potentially lead to a reduction of machine downtime and production costs. Jurkovic et al. [8] chine learning models to predict cutting parameters in a highspeed turning process. They studied Support Vector Regression (SVR), polynomial (quadratic) regression, and Artificial Neural Networks (ANNs) based on input parameters namely surface roughness, cutting force, and tool life. Their study demonstrates that the SVR and polynomial regression models not only have the same performance for predicting cutting force and surface roughness, but also their accuracy is higher than ANN predictions. However, the neural network model generated the predicted tool life more accurately. The effect of cutting parameters such as the cutting speed, feed, and depth of cut on surface roughness in the face milling processes was studied by Lela et al. [11] . Results of three different machine learning models namely regression analysis, SVM, and Bayesian neural network were compared to each other in their studies. Their results show that although the Bayesian neural network is the best model, the accuracies of the other methodologies were close. A new technique for diagnosis of tool breakage in face milling using SVM was introduced by Hsueh and Yang [6] . Krizek et al. [10] applied response surface methodology, evolutionary algorithms (GA & GP), SVR and ANNs on their data sets. Their objective was to find the best approach to determine the cutting force based on input variables namely spindle speed, feed rate, and cutting depth. Kong et al. [9] introduced a new technique using Gaussian Process Regression (GPR) to predict tool wear. Their studies show that Gaussian process approach results in a more accurate model than ANNs and SVMs. A Gaussian process model for predicting surface roughness under different cutting conditions in end face milling was introduced by Zhang et al. [24] . They reached an accuracy about 84.3%. This model prevents expensive trial-and-error procedure for selecting suitable cutting parameters. Liu [12] investigated sequential design for quantification of stability in the machining process. He obtained a model using GPR which predicts stability boundary with accuracy greater than 98%.
In this paper, we propose two machine learning models for predicting cutting forces and maximum tool temperatures during orthogonal machining under stable cutting conditions. The primary objective of the present study is to demonstrate the feasibility of using machine learning models for prediction purposes. The two models considered are based on SVR and GPR. Both SVR and GPR are known to work well when the sample sizes are small. As described below, the data needed for training these models are generated computationally using the finite element simulations of orthogonal machining.
The rest of the paper is organized as follows: Section 2 reviews the finite element models used to generate the data for developing the SVR and GPR models. The theory behind SVR and GPR models is introduced in Section 3. Section 4 explains the procedures used to train our models, an assessment of the error between model predictions and data, and a discussion of these results. Section 5 contains our conclusions and future work on this topic.
A summary of the finite element model used to generate the data for the machine learning models is present here. For the complete details of the finite element model, the reader is referred to [16] .
The general purpose commercial finite element software ABAQUS/EXPLICIT was used to model orthogonal machining of the Aluminum alloy A2024-T351. The cutting tool is made of Tungsten Carbide. A schematic of the computational model is shown in Fig. 1 . For model verification purposes, the geometry and the material properties are taken to be the same as those used by Mabrouki et al. [13] . The workpiece and cutting tool are meshed using plane-strain, quadrilateral elements (CPE4RT) and triangular elements (CPE3RT) with reduced integration with the total number of elements being 22447. The nodes on the bottom and left boundaries of the workpiece are fully constrained whereas the tool is given only horizontal velocity V c in the negative x direction. The clearance angle and the tool nose radius are fixed at 7 • and 20 µm respectively. The constitutive response of the workpiece is taken to be governed by the Johnson-Cook (JC) [7] constitutive model:
The first term in the equation accounts for strain-hardening, whereas the second and third terms account for strain-rate hardening and thermal softening respectively. The material parameters A, B, n, C and m for the JC model are given in Table 2 and are the same as those used by [13] and [19] . The quantityT in 2 The interaction between the tool and the chip is defined by the penalty stiffness contact formulation where the tool is considered as master surface and the chip is considered as slave surface. In addition, self contact of the chip is defined using penalty contact formulation. Guided by the work of Patel et. al [16] , the contact region between the workpiece and the cutting tool is broken up into two distinct regions, the stick region and the slip region (Fig. 2) . The stick region is defined according to the Zorev friction model [25] whereas the slip region is defined according to the model proposed by Yang and Liu [21] .
The Johnson-Cook damage model [7] is used for chip separation, serration, and breakage. According to this model, the overall damage in material occurs in two steps [1] : damage initiation followed by damage evolution. Damage initiates in a material when the damage parameter, ω, defined as:
exceeds or equals one. The numerator, ∆¯ , is the increment in equivalent plastic strain, whereas the denominator,¯ d , is equivalent plastic strain at the onset of damage initiation and is given Fig. 2 . Normal stress σ n (x) and shear stress τ distribution at the tool rake face as per Zorev's friction model [16] . Note that the direction of motion of the tool is vertical. The tool-chip interface is along the x axis.
The term p σ in the equation 3 is a dimensionless pressuredeviatoric stress ratio (where p is the pressure stress andσ is the Mises equivalent stress) and the term˙¯ ˙ 0 represents a non dimensional plastic strain rate ( where˙ 0 is the reference strain rate and˙ is the equivalent plastic strain rate). The parameters D 1 to D 5 are shown in Table 3 . The damage evolution in ABAQUS can be modeled either as linear evolution or exponential evolution. Exponential evolution is defined across the area of uncut chip thickness ( f ) whereas linear evolution is defined for the remaining area (see Fig. 3 ). This approach and the values of G f (Critical energy release rate) are adopted from Patel & Cherukuri in [15] . 
To validate the model, the specific cutting force obtained from the FEA simulations was compared with the experimental results available in the literature [2] for similar cutting parameters, geometry, and material properties. Figure 4 shows the comparison and the corresponding differences.
The difference (%) obtained for uncut chip thickness of 0.4 mm is between 16% to 19% and 15% to 17% for uncut chip thickness of 0.3 mm. Moreover, the specific cutting forces remain constant with the increase in cutting speed for the experimental data. A similar trend is observed for the FE results. In addition, although not shown here, the FE predictions of chip shape are in agreement with experimental observations.
The data for machine learning models are generated using the FE model described in section 2 by varying the cutting parameters: the rake angle, uncup chip thickness and cutting speeds. Seven rake angles, four uncut chip thickness values and seven cutting speeds ( Table 4) were considered. These different parameters result in a total of 196 (7 × 4 × 7) simulations. The tool nose radius and clearance angle were kept constant for all the simulations. The results were sampled with a sampling time of 1µs. The total run time for all the simulations is 2352 hrs with each simulation taking about 12 hrs on average. The required output parameters (specific cutting force and maximum tool temperature) are obtained for all 196 simulations.
The data generated from these simulations are used for training the machine learning models discussed next. 1 1 The data is available in [14] . 
The machine learning models considered in the present work are based on SVR and GPR. In the following, the theory behind the two models is summarized.
The objective of SVR is to find a hyperplane that has at most deviation from the actual target value for all training data in order to balance the trade-off between complexity and prediction error. Adopting a convex −insensitive loss function (shown in Figure 5 ), a symmetric tube is defined around the estimated function of hyper-plane and the absolute errors less than a pre-specified threshold are tolerated and ignored below or above the estimated function [3] . One-dimensional form of SVR can be described with a geometrical perspective as in Figure 6 .
The estimated function can be written as
with f (x), b ∈ R and x, w ∈ R M . Here, M is the order of the polynomial considered as hyper-surface. The problem in SVR is to tune between minimizing the amount of error and finding widest tube centered around the hyper-plane. Hence, the objective function to be minimized is
where y is the target value. Linear −insensitive loss function minimizes the predicted value of the function for an input and the actual output. Note that SVR can adopts different loss functions such as linear, quadratic, Huber, etc. [4] . In some cases a part of error is tolerated but deviation larger than cannot be avoided. Therefore, nonnegative slack variables ξ, ξ * can be introduced to measure objective errors [22] :
Most data set are not even close to being linearly separated by a hyperplane. The most common technique to tackle these kinds of nonlinear problems is to use a kernel trick. Kernel functions map the data from low dimensional feature space to a higher space where a linear separation can be performed. Radial basis function (RBF) is the most common kernel function in SVR:
where γ, the RBF kernel's hyperparameter, controls the influence of a single data point on the other points far from it.
GPR is a supervised machine learning algorithm that interpolates the observations by proposing many probabilistic prediction functions and validating them over the observations. Unlike many supervised approaches, GPR can work with small data sets. More importantly, it will calculate a distribution for the predicted values that can be translated into empirical confidence intervals for the predictions. GPR works by extending the initial idea of defining the probability distribution of numbers to the more advanced idea of having the probability distribution of functions. In the initial idea, one may assume that data points in the space are sampled from a probability distribution determined by a mean vector and covariance matrix. In the more advanced idea, we can look after a probability distribution of functions determined by a mean function and a covariance function. Given a certain sample of values and a family of candidate functions, we can fit a probability distribution of functions to them by choosing the distribution parameters that closely match the given function values. A Gaussian process is defined by its mean function and covariance function. For GPR, the covariance function is determined by a chosen kernel function that describes how much influence one point has on another. This effectively controls the smoothness of the functions in the distribution [18, 20] .
Suppose we have n data points in our training data X, and m points in the test data X * which are zero-mean normalized:
Also, the training and test labels are defined as: {y 1 , y 2 , . . . , y n } and Y * = {y * 1 , y * 2 , . . . , y * m }.
In GPR, the main task is to find the best set of latent functions F = { f 1 , f 2 , . . . , f n } whose joint distribution match the data. Figure 7 represents a Gaussian process model graphically. As shown in the figure, all the latent functions are connected to each other by covariance function which is defined based 5 on their impact on each other. Each latent function f i is corresponding to an x i . Also having f i and x i suffices to predict y i . Nevertheless, each f i is related to all other latent functions and therefore to all other values [17] .
Gaussian process is introduced as a way to describe and then derive the unknown latent functions. A prior distribution is assigned over the functions before observing any data points. The prior distribution updates by observing data. The posterior distribution is calculated by making use of the Bayes theorem that assumes the combination of the prior distribution and data distribution gives the posterior. Considering the whole fitted distribution of functions, we may predict y * i for each test data point x * i as well as a confident interval for the prediction. This gives us not only our prediction function, but also the probabilistic bands on the predictions.
The distribution over a set of latent functions F = { f 1 , f 2 , . . . , f n } can be defined as:
Most commonly, the mean function for the prior is considered to be zero. The covariance matrix of the multivariate normal distribution K defines the relations among inputs. This sym-metric matrix can be calculated based on different assumptions. Notably, the shape of the distribution and also the characteristics of predicting functions are determined by the covariance matrix. Here, a popular approach is to use Radial Basis Function (RBF) kernel defined by [18] 
where the length scale l and the amplitude σ f are the hyperparameters. Increasing the length parameter increases the impacts of data points on each other where they are further away. According to the kernel function, data points which are close to each other are considered as highly covariant, and points with further distance are modeled as low covariant.
To link the expected test labels Y * to the train labels Y,we may assume a joint distribution for them as Y Y * ∼ N(0,K) (13) in which
where K(X, X) is the covariance matrix of training data points, K(X * , X * ) is the covariance matrix of test data points, and K(X, X * ) is the covariance matrix representing the correlation between training and test data points. Note that previously, the kernel function was defined for a pair of data points. Here we may define the kernel function K(A, B) of any two sets of data points A = {a 1 , a 2 , . . . } and B = {b 1 , b 2 , . . . } as K(A, B) i, j = K(a i , b j ). Now we can predict the test labels Y * conditioned on the training labels with the probability distribution of Y * |Y ∼ N(µ, Σ) where µ and Σ are calculated with the kernel functions:
and Σ = K(X * , X * ) − K(X * X)K(X, X) −1 K(X, X * ).
Note that to tune the hyper-parameters σ f and l in the kernel function K, we need to maximize log marginal likelihood log P(Y|X).
Two different SVR models are utilized to predict maximum tool temperate and speed cutting force. The input variables are rake angle (α), uncut chip thickness ( f ), and cutting speed (V c ) which are normalized to 0-1 range prior to hyperparameters tuning. The Scikit-learn library in Python is used to develop the SVR model. The data set consisting of 196 records is split into 80% training and 20% testing. Linear -insensitive loss function is used for this model. , the regularization parameter C, and the RBF kernel parameter γ are tuned by grid search in this work. In our experiment, we run the cross-validation with 10 folds and measured the mean square error (MSE) and the coefficient of determination (R 2 ) defined by
and
respectively. Here, y i , y p i , andȳ represent the actual output, predicted output, and mean of actual output, respectively. The objective in SVR is to find the model with the least value of MSE and the largest value of R 2 . For predicting maximum tool temperature, hyper-parameters γ = 0.4, = 0.001, and C = 20 were found to achieve the best model. The MSE and R 2 values for this model are shown in Table 5 . This model results in values 0.0015 and 0.0012 of MSE for the training and testing data, respectively. The small values of MSE confirm that the SVR model estimation is considerably accurate. Comparing the MSE values on both training and testing data reflects the idea that the model performs well since the value of MSE on testing is less than the MSE value on training data. This also shows that the model does not overfit on the training data points.
Similarly, an SVR model for predicting specific cutting force is developed by applying a grid search on the training data. Optimized hyper-parameters are obtained as γ = 0.4, = 0.001, and C = 15. Table 6 presents the MSE and R 2 results for SVR model. It can be seen that the MSE magnitude for both training and testing data is not only considerably small but also this value on the testing is less than training data. Thus, it can be concluded that the model with the chosen hyper-parameters performs with great accuracy. Also, the values of R 2 for both models are very close to 1 for both training and testing data which indicates that these models have been learned very well. Figure 8 shows the relation between the actual and the predicted values for both maximum tool temperature and specific cutting force using the SVR model. It can be seen that the test data are so close to the bisector line which means the predicted values are in close agreement with the actual outputs. On a system with Intel-i7 CPU (2.0 GHz clock rate) and 16 Gb of RAM, training each SVR model (predicting the specific cutting force and the tool temperature) took around 10 ms, while predictions on the test data set took less than 1 ms in each case. In comparison, each of the simulations took over ten hours on a Linux workstation with Intel-i7 CPU (3.6 GHz clock rate) and 32 Gb of RAM.
In this section, models based on GPR are developed to predict maximum tool temperature and specific cutting force. The goal is to minimize the MSE while maximizing R 2 by using cross-validation with 10 folds. To achieve this goal, hyperparameters σ f and l need to be tuned. The data are normalized prior to tuning hyper-parameters. Like the SVR models, the data is again split into 80% training and 20% testing. Hyperparameters of the prior distribution are initialized to find the model to predict the maximum tool temperature. Maximizing log marginal likelihood on training data, the prior distribution is updated to a posterior with the optimized values of σ f = 0.324 and l = 0.21. Table 7 shows the results of applying this model on training and testing data points. It can be seen from Table  7 that MSE for training data is equal to zero since the data is noise-free. Thus, the predicting distribution function goes exactly through data points, and the predicted value for each data point is exactly equal to the actual value. The MSE value for testing data is 0.0027 which represents the excellent performance of the GPR model on the data set.
Applying the same procedure to find a suitable model to predict specific cutting force results in optimal hyper-parameters σ f = 0.287 and l = 0.196. Table 8 represents MSE and R 2 for specific cutting force prediction by GPR model. The MSE values zero and 0.0021 are reported for training and testing data 7 points, respectively. Since the values of R 2 are so close to 1 for testing data for both GPR models, it can be concluded that these models, just as the SVR models, predict accurate temperatures and cutting forces. This is further confirmed by Fig. 9 where the predictions of temperature and specific cutting force values by the GPR models are plotted against the true values.
The running time to train GPR was around 6.5 s for the specific cutting force model and 1.9 s for the tool temperature model. Predictions on the test data took less than 1 ms for each case. Table 7 . GPR Results for maximum tool temperature prediction.
Training Data 0.0000 1.0000 Testing Data 0.0027 0.9533 Table 8 . GPR Results for specific cutting force prediction.
Training Data 0.0000 1.0000 Testing Data 0.0021 0.9423
In summary, machine learning approaches to predicting specific cutting force and maximum tool temperature during cutting are presented in this paper. Two methods, namely the Support Vector Regression and the Gaussian Process Regression, are presented. The data used for learning models are generated using the finite element method. It is observed that both SVR and GPR models predict the maximum tool temperature and specific cutting force accurately although the former is marginally better than the latter.
Future work includes generating more data to make the models more robust. In addition, additional machining parameters 8 such as the tool tip radius will also be introduced as an input parameter. Work is under way to also use experimental data in addition or in place of the numerically generated data to further strengthen the predictive models.
The Center for Self-Aware Manufacturing and Metrology, and the research described in this paper, are supported under a multi-year grant from the University of North Carolina's Research Opportunities Initiative.