Surface Roughness Prediction of Bearing Ring Precision Grinding Based on Feature Extraction

Abstract

Grinding, as the most crucial finishing process for bearing rings, influences the surface integrity of bearings through the roughness of the ground surface. In order to improve the surface roughness of bearing ring grinding under multiple working conditions, a prediction model of bearing ring surface roughness based on feature extraction was proposed. Firstly, the signal was decomposed using the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) algorithm, and the sensitive components were selected based on the correlation coefficient between Intrinsic Mode Functions (IMFs) and the original signal. The time-domain, frequency-domain, and entropy-domain features of the selected IMF components were extracted. Then, Principal Component Analysis (PCA) was employed for signal feature fusion, and a feature set was constructed in combination with grinding parameters. A prediction model based on Support Vector Regression (SVR) was established to achieve regression prediction of the grinding surface roughness. The proposed method for predicting the surface roughness of precision cylindrical grinding of bearings demonstrated that the determination coefficient (R²), mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) were 0.9953, 0.0020, 0.0050, and 0.0187, respectively. The results indicate that the incorporation of entropy features and grinding parameters in the model provide more information pertinent to grinding surface roughness, thereby effectively enhancing the predictive accuracy.

1. Introduction

Precision bearings serve as critical load-bearing interfaces in mechanical systems, where the outer ring’s machining quality dictates performance metrics including dynamic load capacity, operational stability, and service lifespan. Grinding plays a vital role in bearing production and is the most important process in manufacturing high-precision, high-quality bearings. Therefore, the bearing outer ring grinding process directly influences the performance of the bearing ring, making it highly important to enhance precision grade. The topographical features of ground surfaces govern multiple functional aspects, from interfacial contact mechanics to long-term reliability indicators such as assembly precision, fatigue strength, corrosion resistance, and contact stiffness. They serve as a crucial indicator for evaluating the grinding process. Consequently, to improve the quality of bearing grinding, it is very important to predict the surface roughness of the bearing outer ring during the grinding process.

Conventional post-process surface characterization techniques present notable limitations in terms of throughput and operational economics, often relying on roughness meters for measurement after grinding. This approach may lead to a significant number of workpieces being deemed unqualified, thereby increasing costs. However, roughness issues may arise in the early stages of the grinding process. Early detection and handling can effectively reduce scrap rates, improve processing efficiency, and also play a crucial role in intelligent processing. Therefore, the importance of real-time prediction of bearing ring surface roughness is increasingly prominent, as it not only enhances production efficiency and reduces production costs but also improves product quality levels. As an important cutting process, grinding has attracted extensive research from many scholars focusing on monitoring and prediction during the grinding process. The monitoring parameters are also quite diverse, ranging from workpiece roundness [1,2,3] and grinding burns [4,5,6] to abrasive belt wear [7,8,9], as well as thermal damage in cylindrical rough grinding [10]. Developing models that can accurately predict the surface roughness of workpieces has always been a challenge in the field of grinding. State-of-the-art approaches for surface topography prediction in grinding operations typically adopt one of three methodological frameworks: prediction models based on machining theory, empirical models based on experimental design and analysis, and prediction models based on signal data.

Prediction modeling strategies derive surface generation predictions from fundamental abrasive–workpiece interaction mechanisms. Gu et al. [11], based on a normal distribution model of the grinding wheel surface and grit size, and considering the interaction between the grinding wheel and the workpiece, established a three-dimensional theoretical model for the surface topography of inner-diameter ground bearing rings. The theoretical model for the grinding surface topography was validated by examining the surface morphology of bearing rings under various machining parameters. Jin et al. [12] defined the surface characteristics of the grinding wheel using grinding wheel parameters. Based on the relationship between bearing raceway grinding and surface grinding, as well as the movement trajectory of the abrasive grains on the grinding wheel surface during processing, a prediction model for the surface roughness of bearing raceway grinding was established. Based on the experimental and simulation results of grinding surface roughness, the effects of grinding wheel speed, workpiece speed, and grinding depth on surface roughness were investigated, and the surface roughness at high grinding wheel line speeds was predicted. Chen et al. [13] considered the ultrasonic vibration of the workpiece and the shape of abrasive grains, and they established a surface equation for abrasive grain trajectories as a function of time. Subsequently, by subdividing the workpiece into a grid, and based on the minimum grain size at each grid point, they proposed a novel simulation model to describe the surface topography during the grinding process. This model utilizes simulation techniques to predict the surface roughness in ultrasonic vibration-assisted grinding and has been experimentally validated. Furthermore, an in-depth discussion was conducted on the influence of ultrasonic amplitude on surface roughness.

Empirical modeling paradigms establish process–property relationships through systematic experimental investigations. Trung et al. [14] studied multi-objective optimization of the external circular grinding process, selected cutting speed, feed speed, and cutting depth as input parameters of the experimental process, designed the experimental matrix using the Taguchi method, and applied the data envelopment-based sorting (DEAR) method to determine the values of the input parameters. At the same time, both minimum surface roughness and maximum material removal rate were ensured. Yan et al. [15] conducted orthogonal experiments to investigate the impact of grinding parameters (wheel speed, workpiece speed, and grinding depth) on the surface quality of ceramic internal cylindrical grinding. As a result of these experiments, two mathematical models for predicting surface roughness were established. The surface roughness prediction model derived from empirical equations exhibited better predictive performance compared to the theoretical prediction model. The empirical model based on experimental design and analysis requires large cost and time investment in experimental design, data acquisition, and processing, especially in the complex grinding process, which also requires a large number of tests and data processing. At the same time, the experimental data are affected by experimental conditions, environment, equipment, and other factors, and they may have certain limitations.

The proliferation of industrial IoT systems has enabled comprehensive process digitization through multi-modal sensor networks. Commonly used monitoring signals include force signal [16,17], vibration signal [18,19], acoustic emission signal [20,21], current signal [22,23], etc. These signals contain dynamic multi-physical field properties in the cutting process. Li et al. [24] proposed a time–spatial spectrum analysis method based on the monitored grinding force signal and the grinding surface texture curve, established a Chi-square distribution model, and predicted the amplitude–frequency response of the surface texture through the monitoring of the force signal. They found that the surface roughness (Ra) could be calculated from the surface texture curve. Wang et al. [25] established a surface roughness prediction model that introduced cutting power as a decision variable and considered the dynamic changes of grinding parameters and grinding wheel state. Compared with common models, the prediction accuracy was improved, especially in the case of grinding wheel wear. Guo et al. [26] successfully improved the accuracy of online prediction of nodular cast iron surface roughness by extracting 13 parameters that reflect the characteristics of grinding acoustic emission signals and combining a genetic algorithm and BP neural network optimized using particle swarm optimization. Pan et al. [27] proposed a multi-sensor signal fusion method based on principal component analysis to extract fusion features from the force and vibration signals in the grinding process, which retains the physical significance of the original features and achieves stable and high-precision surface roughness prediction. Pan et al. [28] studied the influence of vibration characteristic frequency on surface roughness by extracting the vibration characteristics of fluorophlogopite in speed point grinding, and they analyzed the influence of five process parameters such as grinding speed, table feed speed, grinding depth, deflection angle, and inclination angle on the deviation rate of characteristic frequency and surface roughness. On this basis, A modified model containing vibration characteristics was proposed, which successfully realizes the goal of real-time vibration signal monitoring and surface roughness control. Guo et al. [29] analyzed the correlation between the characteristic values in different frequency bands of grinding acoustic emission signals and the grinding surface roughness values. They selected the best sensitive frequency band and the characteristic matrix of grinding acoustic emission signals as the input parameters of a CNN-BiLSTM neural network. A method for predicting the surface roughness of PSZ ceramic grinding based on correlation analysis and a convolutional bidirectional long short-term memory neural network (CNN-BiLSTM) was proposed. Lin et al. [30] established three surface roughness prediction models based on vibration signals, namely a fast Fourier transform–deep neural network, fast Fourier transform–long short-term memory network, and one-dimensional convolutional neural network (1D-CNN). Through comparative analysis, it was shown that the 1D-CNN had strong feature extraction ability. Siamak et al. [31] designed a suitable artificial neural network to predict grinding surface roughness and grinding force by installing a single integrated acoustic emission sensor on the machine tool. They trained and tested two models, one using only grinding parameters and the other using acoustic emission signals and grinding parameters as input data. A feedforward neural network was selected to model Bayesian backpropagation, and the model was verified using experiments with different grinding parameters and neural network parameters. The experimental results show that the acoustic emission signal, as an additional input parameter to the grinding parameters, can significantly improve the efficiency of neural network prediction of grinding force and surface roughness. The surface roughness prediction method based on signal data avoids strict mathematical derivation of complex mechanisms in the grinding process. The signal data in the grinding process can be collected in real time, which can reflect changes in the machining process in real time so the model can be quickly adjusted and optimized to adapt to dynamic changes in the grinding process.

In summary, the monitoring and prediction of grinding processes have been widely studied, but the following issues remain:

(1)

Most surface roughness prediction models are established using a single sensor, and the influence of process parameters and other factors on surface roughness is rarely considered at the same time, which greatly limits the stability and universality of prediction models.

(2)

In actual processing, the number of signal samples is small, and the deep learning model does not provide clear advantages in the case of small samples and strong noise due to the excessive number of parameters, the need for a large amount of data for training, complex parameter adjustment processes, and other reasons.

To address the aforementioned issues, this study proposes an innovative feature engineering method that integrates CEEMDAN decomposition, multi-domain feature extraction, and grinding process parameters. It adaptively decomposes signals to screen sensitive components and effectively captures the nonlinear dynamics of the grinding process through entropy-based features, demonstrating significant advantages over traditional single-sensor or single-domain feature analysis methods. This enables high-precision monitoring of grinding surface roughness for bearing outer rings under conditions of small sample sizes and multiple operations. The research findings provide theoretical and technical guidance for real-time monitoring of grinding surface roughness in the outer ring grinding process during bearing production.

2. CEEMDAN Denoising Technology

2.1. CEEMDAN Principles

The CEEMDAN decomposition method was proposed based on the Ensemble Empirical Mode Decomposition (EEMD), primarily targeting the decomposition of nonlinear and non-stationary signals. Performing Empirical Mode Decomposition (EMD) multiple times provides better modal separation with relatively lower computational costs.

The main process and calculation formula of signal decomposition are as follows:

Step 1: Add Gaussian white noise δ_i(t) to signal X(t) to be decomposed and construct sequences X_i(t), i = 1, 2, 3, …, n.

$$X_i(t)=X(t)+\epsilon {\delta }_i(t)$$

(1)

where ε denotes the weighting coefficient of the Gaussian white noise.

Step 2: Perform EMD decomposition on the sequence to be decomposed, and take the average of the first modal component from the EMD decomposition to obtain the first IMF (Intrinsic Mode Function) of the CEEMDAN decomposition.

$$IM{F}_1(t)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}IM{F}_1^i(t)}$$

(2)

$$r_1(t)=x(t)-IM{F}_i(t)$$

(3)

where IMF₁(t) denotes the first-order IMF (Intrinsic Mode Function) obtained from the CEEMDAN decomposition, and r₁(t) denotes the first-order residual signal.

Step 3: After adding noise to the decomposed k−1th order margin signal, calculate the kth-order modal components.

$$IM{F}_k(t)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{E}_1(r_{k-1}(t)+{\epsilon }_{k-1}{E}_{k-1}({\delta }_i(t)))}$$

(4)

$$r_k(t)=r_{k-1}(t)-IM{F}_k(t)$$

(5)

where IMF_k(t) denotes the k−1th order IMF (Intrinsic Mode Function) obtained from the CEEMDAN decomposition, and E_k−1(t) denotes the k-1th order modal components.

Step 4: Repeat the above steps until the residual signals cannot be further decomposed and the final residual signal is monotonic. If the total number of final modal components is N, then the formula for the final residual signal is as follows:

$$R(t)=X(t)-{\displaystyle \sum _{n=1}^{N}IM{F}_n(t)}$$

(6)

2.2. Calculation of Correlation Coefficient

Although the CEEMDAN decomposition method can address the issue of modal aliasing, it still poses the problem of generating pseudo-modes. Furthermore, noise typically represents a non-dominant component within signals. Therefore, in order to eliminate unnecessary components arising during the decomposition process, retain the primary components pertinent to the grinding of bearing rings within the signal, select modal components sensitive to the grinding process, and reduce data dimensionality, it can effectively eliminate noise interference in the signal and enhance its quality by calculating the correlation between each modal component and the original signal, retaining modal components with high correlation levels.

To represent the degree of correlation between each modal component and the original signal, this section introduces the Pearson Correlation Coefficient (PCC) as a measurement standard. Its value ranges from −1 to 1, and the higher the correlation between the two, the larger the correlation coefficient.

The Pearson correlation coefficient between the nth-order modal component IMF_n(t) and the original signal X(t), calculated using Equation (7):

$$R_n={\displaystyle \frac{{\displaystyle \sum _{t=1}^T(IM{F}_n(t)-\overline{IM{F}_n(t)})}(X(t)-\overline{X(t)})}{\sqrt{{\displaystyle \sum _{t=1}^T{(IM{F}_n(t)-\overline{IM{F}_n(t)})}^2}{(X(t)-\overline{X(t)})}^2}}}$$

(7)

The decomposed modal components are ranked according to the calculated Pearson Correlation Coefficient (PCC) values. The top eight modal components are retained, while those with lower correlation to the original signal are removed. This approach helps to preserve the main characteristic information related to the grinding process of bearing rings, facilitating subsequent extraction of signal features.

3. Grinding Surface Roughness Feature Extraction and Fusion

3.1. Time-Domain and Frequency-Domain Characteristics

Time-domain features refer to the extraction of statistical characteristics of a signal in the time-domain. By observing the amplitude, time series, waveform, and other characteristics of the signal, information such as its time-varying nature, stability, and peak values can be reflected.

Table 1. Time-domain statistical features.

Time-Domain Features	Calculation Formulas	Time-Domain Features	Calculation Formulas
Mean Value	$x_1={\displaystyle \frac{{{\displaystyle \sum }}_{n-1}^{N}x\left(n\right)}{N}}$	Kurtosis	$x_7={\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{\left(x\left(n\right)-x_1\right)}^4}{\left(N-1\right)x_2^4}}$
Standard deviation	$x_2=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{\left(x\left(n\right)-x_1\right)}^2}{N-1}}}$	Peak value	$x_8={\displaystyle \frac{x_5}{x_4}}$
Absolute mean value	$x_3={\left({\displaystyle \frac{{{\displaystyle \sum }}_{n-1}^N\sqrt{\left|x\left(n\right)\right|}}{N}}\right)}^2$	Kurtosis	$x_9={\displaystyle \frac{x_5}{x_3}}$
Root Mean Square, RMS	$x_4=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{\left(x\left(n\right)\right)}^2}{N}}}$	Pulse indicator	$x_{10}={\displaystyle \frac{x_4}{\frac{1}{N}{{\displaystyle \sum }}_{n=1}^N\left|x\left(n\right)\right|}}$
Peak value	$x_5=max\left|x\left(n\right)\right|$	Waveform factor	$x_{11}={\displaystyle \frac{x_5}{\frac{1}{N}{{\displaystyle \sum }}_{n=1}^N\left|x\left(n\right)\right|}}$
Skewness	$x_6={\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{\left(x\left(n\right)-P_1\right)}^3}{\left(N-1\right)P_2^3}}$

Note: (n) denotes the signal sequence, n = 1, 2, 3, …, N; N denotes the number of data points.

presents the 11 time-domain statistical features extracted from vibration and acoustic emission signals along with their specific calculation formulas.

The mean value, absolute mean value, root mean square value, and the amplitude and energy characteristics of the peak signal in the time-domain are given. The kurtosis index, margin index, skewness index, waveform index, and pulse index describe the distribution and characteristics of the signal in the time-domain, which can reveal the sharp impact and other characteristics existing in the signal.

Frequency-domain features refer to the extraction of information about the frequency components of a signal by transforming the signal in the frequency-domain. During the grinding process of bearing rings, vibrations or energy releases at different frequencies may exist, and extracting frequency-domain features can help capture information about these frequencies. This section presents 13 statistical frequency-domain features extracted from vibration and acoustic emission signals, along with specific calculation formulas, as shown in

Table 2. Frequency-domain features.

Frequency-Domain Features	Calculation Formulas	Frequency-Domain Features	Calculation Formulas
Sample mean of spectral amplitude	$x_{12}={\displaystyle \frac{{{\displaystyle \sum }}_{k-1}^{K}s\left(k\right)}{K}}$	Change in the position of the main frequency band	$x_{19}=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{k-1}^K{f}_k^{4}s\left(k\right)}{{{\displaystyle \sum }}_{k-1}^K{f}_k^{2}s\left(k\right)}}}$
Sample variance of spectral amplitude	$x_{13}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(s\left(k\right)-x_{12}\right)}^2}{\left(K-1\right)}}$	Concentration of frequency energy	$x_{20}={\displaystyle \frac{{{\displaystyle \sum }}_{k-1}^K{f}_k^{2}s\left(k\right)}{\sqrt{{{\displaystyle \sum }}_{k-1}^{K}s\left(k\right){{\displaystyle \sum }}_{k-1}^K{f}_k^{4}s\left(k\right)}}}$
Skewness of spectral amplitude	$x_{14}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(s\left(k\right)-x_{12}\right)}^3}{K{\left(\sqrt{x_{13}}\right)}^3}}$	Coefficient of variation	$x_{21}={\displaystyle \frac{x_{17}}{x_{16}}}$
Kurtosis of spectral amplitude	$x_{15}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(s\left(k\right)-x_{12}\right)}^4}{K{x}_{13}^2}}$	Skewness of frequency	$x_{22}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(f_k-x_{16}\right)}^{3}s\left(k\right)}{K{x}_{17}^3}}$
Mean frequency	$x_{16}={\displaystyle \frac{{{\displaystyle \sum }}_{k-1}^K{f}_{k}s\left(k\right)}{{{\displaystyle \sum }}_{k-1}^{K}s\left(k\right)}}$	Kurtosis of frequency	$x_{23}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(f_k-x_{16}\right)}^{4}s\left(k\right)}{K{x}_{17}^4}}$
Variance of frequency	$x_{17}=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(f_k-x_{16}\right)}^{2}s\left(k\right)}{K}}}$	Standardized mean of spectral amplitude	$x_{24}={\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^K{\left(f_k-x_{16}\right)}^{\frac{1}{2}}s\left(k\right)}{K\sqrt{x_{17}}}}$
Root Mean Square (RMS) frequency	$x_{18}=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{k-1}^K{f}_k^{2}s\left(k\right)}{{{\displaystyle \sum }}_{k-1}^{K}s\left(k\right)}}}$

Note: s(k) denotes the spectrum of signal sequence x(n), k = 1, 2, 3, …, K; K denotes the number of spectral lines.

. Among them, the mean of spectral amplitude samples reflects the amplitude and energy characteristics of the signal in the frequency-domain. Parameters such as frequency skewness, spectral amplitude skewness, frequency variance, spectral amplitude kurtosis, frequency kurtosis, spectral amplitude sample variance, coefficient of variation, and standardized spectral mean provide more specific descriptions of the signal’s distribution characteristics in the frequency-domain. The mean frequency, root mean square frequency, main frequency band position variation, and frequency energy concentration reflect the main frequency peaks of the signal, representing changes in the position of the main frequency in the spectrum.

3.2. Entropy Characteristics

In the grinding process of bearing rings, there may be a complex nonlinear relationship between various factors. The entropy feature has a high sensitivity to this nonlinear relationship and can reflect signal volatility and nonlinear characteristics more accurately. In addition, the entropy also changes with the interaction between the particle and the workpiece during grinding. Therefore, singular spectrum entropy, power spectrum entropy, permutation entropy, and sample entropy were selected, as shown in

Table 3. Features of entropy-domain.

Entropy Feature	Calculation Formulas
Power spectrum entropy	Obtained from Equation (10)
Singular spectrum entropy	Obtained from Equation (13)
Permutation entropy	Obtained from Equation (17)
Sample entropy	Obtained from Equation (21)

.

3.2.1. Information Entropy

Information entropy is a concept that describes the amount of information, and its definition involves the concept of self-information. Directly calculating the information entropy of the time-domain signal only reflects the uncertainty in time, ignoring the information in space. Therefore, the signal is usually preprocessed using methods such as power spectrum analysis and singular value decomposition to generate power spectrum entropy and singular spectrum entropy.

(1) Power spectrum entropy

Power spectrum entropy is an index used to measure the uncertainty of a signal via power spectrum division. Therefore, power spectrum entropy provides a quantitative description of the complexity of the energy distribution of a signal in the frequency-domain [32].

To solve the signal power spectrum and power spectrum density, the following equation is used:

$$s(k)={\displaystyle \frac{1}{N}}{\left|y(k)\right|}^2$$

(8)

where k denotes frequency point, k = 1, 2, 3, …, N − 1; y(k) denotes the application of a discrete Fourier transform to the signal.

$$p_i={\displaystyle \frac{s(k)}{{\displaystyle \sum _{k=0}^{N-1}s(k)}}}$$

(9)

Here, p_i denotes the power spectral density corresponding to the frequency of i.

Standardized power spectral density is used to calculate power spectrum entropy:

$$H_{pse}={\displaystyle \sum _{i=1}^N{p}_i{\log }_2{p}_i}$$

(10)

where N denotes the total frequency count.

(2) Singular spectrum entropy

The singular spectrum entropy method is an advanced method for signal analysis. It combines singular spectrum analysis and information entropy. By calculating the singular spectrum of the signal and further obtaining its information entropy, the complex state features of the time series can be quantitatively described [33].

By selecting the appropriate embedding dimension m, the time sequence signal X is reconstructed into the trajectory matrix A, and the singular spectrum of the time sequence signal X is obtained via singular value decomposition:

$$A_{k\times m}=U_{k\times k}{S}_{k\times m}{V}_{m\times m}^T$$

(11)

where k × m denotes the dimension of trajectory matrix A, k = N − m + 1.

The singular value spectral density of the signal is calculated as follows:

$$p_i={\displaystyle \frac{{\lambda }_i}{{\displaystyle \sum _{i=1}^r{\lambda }_i}}}$$

(12)

According to the definition of information entropy, the singular spectrum entropy of the signal is calculated as follows:

$$H_{sse}={\displaystyle \sum _{i=1}^r{p}_i\log }{p}_i$$

(13)

3.2.2. Permutation Entropy

Permutation entropy is a method designed to measure the complexity and irregularity of timing signals. The calculation process is based on converting the timing signal into permutations and further analyzing the frequency distribution of these permutations.

The time sequence signal x(i) of N data points is reconstructed in phase space, and each component after reconstruction is arranged in increasing order to obtain the position index symbol sequence of elements in the component:

$$X(N-(m-1)\tau )=\{x(N-(m-1)\tau ),x(N-(m-2)\tau ),\dots ,x(N)\}$$

(14)

where τ denotes the delay coefficient, and X(i) contains m elements.

$$\left\{x\left(i+\left(j_1-1\right)\tau \right)\le x\left(i+\left(j_2-1\right)\tau \right)\le \dots \le x\left(i+\left(j_m-1\right)\tau \right)\right\}$$

(15)

Here, j₁, j₂, j₃, …, j_m denotes the index of the column in which each X(i) element resides.

Thus, each reconstructed component X(i) can be mapped to a set of symbolic sequences S(l):

S(l) = (j₁,j₂,j₃,…,j_m)

(16)

where l = 1, 2, …, k, k ≤ m.

The sequence [j₁, j₂, j₃, …, j_m] can form $m!$ (m factorial) different permutations. Therefore, each S(l) is one of these $m!$ symbol sequences. To calculate the probability of each symbol sequence, the ratio of the occurrence count of each symbol sequence S(l) to the total number of all distinct symbol sequences is taken as probability p₁, p₂, p₃, …, p_k. Then, ${\displaystyle {\sum }_{l=1}^k{p}_l}=1$, and we calculate the permutation entropy of the signal.

$$0\le H_{pe}={\displaystyle \frac{-{\displaystyle \sum _{l=1}^k}{p}_l\ln \left(p_l\right)}{\ln (m!)}}\le 1$$

(17)

Based on the above calculation steps and formulas, it can be seen that the embedding dimension m has a great influence on the permutation entropy, so it is necessary to study its selection. The results show that when m = 5, 6, 7, permutation entropy can better reveal the dynamic characteristics of time series. Therefore, the embedding dimension m is set to 6 in this research.

3.2.3. Sample Entropy

Sample entropy is a new complexity parameter of time series. It has a series of advantages, such as low data length requirement, strong anti-noise and anti-interference ability, and good consistency in parameter selection.

The timing signal x(i) is constructed as N-d d-dimensional vectors, where d is the embedding dimension.

$$X(N-d+1)=\{x(N-d+1),x(N-d+2),\dots ,x(N)\}$$

(18)

The maximum difference of the corresponding elements between different vectors X(i) and X(j) is computed, defined as the distance between the two vectors, where i is not equal to j.

$$L[X(i),X(j)]=\underset{k\in \left(0,d-1\right)}{\max }(|x(i+k)-x(j+k)|)$$

(19)

Given the tolerance r, we count the number of times satisfying L[X(i), X(j)] ≤ r, calculate its ratio to the total number of vectors N-d, and average the results obtained for all vectors.

$$B^d(r)={\displaystyle \frac{1}{N-d+1}}\sum _{i=1}^{N-d+1}{\displaystyle \frac{1}{N-d}}num\{L[X(i),X(j)]<r\}$$

(20)

After the vector dimension is changed to d + 1, the above steps are re-computed to obtain B^d+1(r).

The sample entropy of the timing signal is calculated as follows:

$$E_{sam}=-\ln \left({\displaystyle \frac{B^{d+1}(r)}{B^d(r)}}\right)$$

(21)

For a given value of d and r, the lower the value of the sample entropy, the greater the similarity of the given timing signals. In this research, the value of parameter r is set to 0.2 times the data standard deviation, and the embedding dimension d is set to 2 [34].

3.3. Multi-Transformation Domain Feature Fusion

Through the above analysis and calculation, a multi-transform domain combined feature set can be obtained, which can reflect the grinding processing state from many aspects. However, using these features directly as feature vectors may lead to a large amount of data, which makes it difficult to establish a mapping relationship between feature vectors and the grinding surface roughness of bearing rings. At the same time, these features may have redundant information, which will not only increase the number of parameters in model training but also affect the accuracy and training efficiency of the model. In order to solve this issue, this research employs principal component analysis (PCA) to synthesize the information of different signal sources, and it integrates the multi-transform domain features of vibration signal and acoustic emission signal extraction to explore potential data structures.

The core idea of principal component analysis is dimensionality reduction, which reduces the dimensionality of the original feature space by transforming the original feature into a few comprehensive indicators (i.e., principal components), effectively eliminating the correlation between features and improving the learning efficiency and accuracy of the model. Principal component analysis can find the main direction of change in the dataset and eliminate redundant information in the data by keeping the main components and discarding the minor components. The feature set after dimensionality reduction is more representative and can better explain changes in the original data. Dimensionality reduction through principal component analysis can reduce the dimensionality of the dataset, thereby simplifying the model, improving training speed and generalization ability while reducing the risk of overfitting. The specific calculation steps for data fusion in principal component analysis are as follows:

Step 1: Perform dimensionless processing on the signal feature set.

Different features may have different dimensions and scales; for example, one feature may have a value range of tens to hundreds, while another feature may have a value of between 0 and 1. If no dimensionless processing is carried out, these scale inconsistencies will lead to unreasonable weight allocation in PCA, which will affect the extraction results of the PCA.

Common non-dimensional processing methods include min–max normalization, z-score normalization, and normalization regularization. Among them, the z-score standardization method aims to standardize the original dataset into a dataset with a mean of 0 and a variance of 1 that is close to the standard normal distribution, thus eliminating the dimensional influence between the data. This makes the importance of each feature more equal to the impact of the model, converting the values of different features into values with the same scale so they can be compared and analyzed on the same scale. In order to ensure that the data of each dimension can play the same role in the distance calculation and avoid the significant impact of data of different dimensions on distance calculation, this study adopts the z-score standardization method to carry out dimensionless processing on the combined feature set.

The main formula for z-score standardization is as follows:

$$z_{ij}={\displaystyle \frac{x_{ij}-\overline{x_j}}{s_j}}$$

(22)

where i = 1, 2, 3, …, n; j = 1, 2, 3, …, p; $\overline{x_j}$ denotes the mean of feature j, and S_j denotes the standard deviation of the feature number, where:

$$\overline{x_j}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_{ij}}$$

(23)

$$s_j=\sqrt{{\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{i=1}^n{(x_{ij}-\overline{x_j})}^2}}$$

(24)

Step 2: Solve the principal component and its variance contribution rate and cumulative contribution rate.

The raw data matrix can represent X_n×p, where n represents the number of samples contained and p represents the feature number. The covariance matrix is calculated, and the covariance matrix is obtained. Through eigenvalue decomposition, the eigenvalues of the covariance matrix and corresponding eigenvectors are obtained, and the eigenvalues are sorted from small to large. By obtaining λ₁, λ₂, λ₃, …, λ_p and the corresponding eigenvector T₁, T₂, T₃, …, T_p, the decomposed principal component Y_i can be expressed as follows:

$$Y_i=X·T_i,1\le i\le p$$

(25)

Then, the variance contribution rate and the cumulative contribution rate of the eigenvalues are calculated.

The variance contribution rate of principal component analysis refers to the proportion of total variance explained by each principal component, which reflects the contribution degree of each principal component to the degree of data variation. The specific formula is as follows:

$${\phi }_k={\displaystyle \frac{{\lambda }_k}{{\displaystyle \sum _{i=1}^p{\lambda }_i}}}$$

(26)

where λ_k denotes the eigenvalue corresponding to the kth principal component.

Under normal circumstances, the cumulative contribution rate of the first m principal components of the sample can be calculated using the following formula:

$${\mathsf{\Psi }}_m={\displaystyle \frac{{\displaystyle \sum _{k=1}^m{\lambda }_k}}{{\displaystyle \sum _{k=1}^p{\lambda }_k}}}={\displaystyle \sum _{k=1}^m{\phi }_k}$$

(27)

If the variance contribution rate of the principal components is larger, this means that more information of the original data will be saved. Therefore, this study takes the calculated variance contribution rate and cumulative contribution rate as the basis for selecting the number of retained principal components to achieve the purpose of dimensionality reduction.

Step 3: Solve the score coefficient.

The score coefficient of the principal component in a principal component analysis is not only used to find the principal component but also to deeply understand the meaning of each factor for problem analysis. It is used to calculate the coefficient of the projected value of the original data on the principal component and to quantify the projection of each sample in the direction of the principal component. The score coefficient reflects the importance and relative position of the samples in the principal component direction, which is helpful to understand the distribution and mutual relationship of the samples in the principal component space. By analyzing the score coefficients, it is possible to obtain a clearer picture of how each sample performs in the principal component direction, thus better interpreting the meaning of the principal component and the structure of the data.

The original data matrix is expressed as X_n×p, where n represents the number of samples and p represents the feature number. When the first m principal components are selected for dimensionality reduction, the dimension of the projection matrix V is p × k. Each column of the projection matrix V is an eigenvector of a principal component, and the eigenvector contains the weight of each original feature on the principal component.

Then, the calculation formula of the score coefficient matrix T is as follows:

T = XV

(28)

where the dimension of T is n × k, and each row corresponds to the projected value of a sample on the selected principal component.

Grinding parameters are the key control factors in the grinding process, which directly affect the removal of materials, the distribution of grinding heat, and the size of grinding force. Incorporating these parameters into the prediction model can more accurately capture the dynamic changes in the grinding process, thus improving the prediction accuracy of the model. At the same time, the addition of grinding parameters can expand the information dimension considered by the model, making it more comprehensively consider various influencing factors in the grinding process, where the influence of grinding parameters on the final surface roughness is particularly important. Therefore, the grinding parameters are also added to the fusion feature set, and the total feature set reflecting the grinding surface roughness is formed.

4. Surface Roughness Prediction Model of Bearing Cylindrical Precision Grinding Based on Feature Extraction

4.1. Surface Roughness Prediction Model of Bearing Cylindrical Precision Grinding

Support vector regression (SVR) is a machine learning method for regression analysis derived from the theory of support vector machines (SVMs). Similar to the SVM classification model, SVR fits the data by finding the optimal hyperplane. This non-probabilistic algorithm uses a kernel function to map the data into a high-dimensional space and find the optimal hyperplane in that space to maximize the interval between the training data and the hyperplane, thereby building a regression model. SVR is based on the structural risk minimization principle in statistical learning theory and aims to obtain the optimal solution with limited sample data, taking into account the empirical risk and confidence range. Compared with traditional regression methods, SVR pays more attention to the effective use of data and performs well with small sample sizes. This method uses a kernel technique to map raw data to high-dimensional space, which is helpful to deal with nonlinear relations. By using support vectors to define decision boundaries, SVR usually selects only a small number of sample points as support vectors, thereby obtaining sparse solutions, reducing the complexity of the model and effectively avoiding overfitting problems. Therefore, SVR is used to make regression predictions for the grinding feature set of the bearing ring.

Suppose the current training dataset is {(x₁, y₁), (x₂, y₂,), …, (x_n, y_n)}, where x_i is the eigenvector of the i-th sample and y_i is the corresponding target value.

SVR maps the eigenvector {x₁, x₂, …, x_n} nonlinearly in a high-dimensional space, followed by a linear fit in a high-dimensional space. Its basic model can be expressed as follows:

$$y=f(x)={\omega }^{\mathrm{T}}\mathsf{\Phi }(x)+b$$

(29)

where ω denotes the corresponding weight vector after the eigenvector is mapped to the higher-dimensional space, Φ(x) denotes the feature mapping function, and b denotes a bias term.

In the calculation of SVR, the following formula is usually used as the loss function:

$$E_{\epsilon }(f(x)-y)=\left\{\begin{array}{cc}0,& |f(x)-y|<\epsilon \\ |f(x)-y|-\epsilon ,& |f(x)-y|\ge \epsilon \end{array}\right.$$

(30)

where $\epsilon$ denotes a tolerance parameter.

If the fitted mathematical model is regarded as a curve in a high-dimensional space, a pipeline that envelopes this curve and the sample points is formed through the tolerance parameter; that is, the difference between the predicted value f(x) and the true value y is considered to lose 0 within a certain range. Finding the optimal hyperplane is equivalent to finding the maximum interval so that all sample points are within a pipe formed by two boundary lines.

At this time, two relaxation variables ξ_i ≥ 0 and ξ* ≥ 0 are introduced, where the former corresponds to the data point of y_i ≥ f(x_i) + ξ and the latter corresponds to the data point of y_i ≥ f(x_i) − ξ. The optimal regression function can be obtained by finding the minimum extreme value of Formula (31):

$${\displaystyle \frac{1}{2}}\parallel \omega {\parallel }^2+C(\sum ^{\underset{n}{i=1}}{\xi }_i+\sum ^{\underset{n}{i=1}}{\xi }_i^{*})$$

(31)

where C denotes the value of the penalty factor, representing the degree of punishment for exceeding the error sample.

Therefore, the issue of finding a regression function at this time translates into the following quadratic programming issue:

$$\underset{\omega ,b,{\xi }_i,{\xi }_i^{*}}{min}C\sum ^{\underset{n}{i=1}}({\xi }_i+{\xi }_i^{*})+{\displaystyle \frac{1}{2}}\parallel \omega {\parallel }^2$$

(32)

$$Constraints: \begin{array}{c}\left[\right(\omega \cdot x_i)+b]-y_i\le \epsilon +{\xi }_i\\ y_i-\left[\right(\omega \cdot x_i)+b]\le \epsilon +{\xi }_i^{*}\\ {\xi }_i\ge 0,\hspace{1em}{\xi }_i^{*}\ge 0,\hspace{1em}i=\mathrm{1,2},...,n\end{array}$$

(33)

As long as the optimization solution meets the KKT condition, the Lagrange dual problem solution is usually adopted. The dual form of the optimization problem is as follows:

$$\underset{{\alpha }_i,{\alpha }_i^{\ast }}{\max }\left\{-{\displaystyle \frac{1}{2}}\sum _{i,j=1}^n({\alpha }_i-{\alpha }_i^{\ast })({\alpha }_j-{\alpha }_j^{\ast })〈x_i,x_j〉+\sum _{i=1}^n({\alpha }_i-{\alpha }_i^{\ast })y_i-\sum _{i=1}^n({\alpha }_i+{\alpha }_i^{\ast })\epsilon \right\}$$

(34)

$$Constraints: \begin{array}{c}{\displaystyle \sum ^{\underset{n}{i=1}}}({\alpha }_i-{\alpha }_i^{*})=\mathrm{0,0}\le {\alpha }_i,{\alpha }_i^{*}\le C,\hspace{1em}i=\mathrm{1,2},...,n.\end{array}$$

(35)

Lagrange multipliers a_i and a_i* can be solved, and the weight vector ω can be obtained as follows:

$$\omega =\sum _{i=1}^n({\alpha }_i-{\alpha }_i^{\ast })x_i$$

(36)

The value of b can be obtained according to the KKT condition, so the nonlinear function f(x) can be obtained:

$$f(x)={\displaystyle \sum _{i=1}^n({\alpha }_i-{\alpha }_i^{\ast })K(x_i,x_j)}+b$$

(37)

where K(x_i, x_j) denotes a kernel function.

Selecting the appropriate kernel function can not only improve the accuracy of the prediction model but also reduce the influence of random noise on the prediction model and reduce the calculation cost. Therefore, radial basis function (RBF) is chosen as the kernel function of the model. The specific formula is as follows:

$$k(x,y)=\exp \left(-{\displaystyle \frac{x-y^2}{2{\sigma }^2}}\right)$$

(38)

RBF maps the input sample space to the high-dimensional space through the exponential relationship to achieve high-precision nonlinear fitting, which has the characteristics of low sample requirements, wide application, and high flexibility.

4.2. Algorithm Flow

The process for predicting the surface roughness of bearing outer ring precision grinding, based on the feature extraction method proposed in this section, is shown in Figure 1. The specific steps are as follows:

Step 1: Through a grinding test of thin-walled bearing rings, the vibration and acoustic emission signals in the grinding process are collected. The CEEMDAN algorithm is used to decompose each vibration and acoustic emission signal into a set of modal component IMFs, and the Pearson correlation coefficient between each IMF and the original signal is calculated. According to the size of the Pearson correlation coefficient, the first eight IMFs are retained in order from largest to smallest, removing noise and fake IMFs.

Step 2: The time-domain, frequency-domain, and entropy features were extracted from the first eight IMFs retained, and principal component analysis (PCA) was used for fusion dimensionality reduction of the extracted features. Then, the grinding parameters and the signal features after dimensionality reduction were combined into a total feature set, which was then divided into a training sample set and test sample set.

Step 3: The SVR parameters were set, and the SVR was trained using grid search and 5x cross-validation based on the training sample set and test sample set.

Step 4: The trained SVR fitted the surface roughness of the thin-walled bearing ring grinding and obtained the prediction results of the surface roughness.

5. Experimental Results and Discussion

The hardware setup consisted of a KELLENBERGER CNC internal and external cylindrical grinding machine (AG UR225/1500, Kellenberger, St. Gallen, Switzerland, with roundness accuracy for fly grinding of 0.2 μm, max. grinding diameter of 432 mm, grinding length of 1500 mm, and center height of 225 mm). The grinding wheel was made of CBN (cubic boron nitride) with a diameter of 400 mm. The workpiece as a thin-walled bearing ring (model 719–182B), manufactured from GCr15 bearing steel, and featured an outer ring diameter of 214 mm [35]. This configuration ensured high-precision grinding for the bearing ring.

Grinding surface roughness is affected by many factors, such as grinding wheel type, workpiece parameters, grinding parameters, and so on. Compared with other processing, grinding processing usually takes a long time, and the feed speed is slow. If all the influencing factors are considered in the experimental scheme, the experimental workload will become huge. In order to explore the grinding surface roughness law of the outer ring of the bearing ring and comprehensively consider the experimental cost, the orthogonal experiment design included 16 test conditions with 3 factors and 4 levels. The varying parameters for grinding wheel linear speed (m/s), workpiece speed (r/min), and grinding depth (μm) are shown in

Table 4. Three-factor four-level L16 orthogonal experiment table.

Serial Number	Grinding Wheel Linear Speed/m·s−1	Workpiece Speed/r·min−1	Grinding Depth/μm
1	25	20	20
2	25	40	30
3	25	60	40
4	25	80	50
5	30	40	40
6	30	60	50
7	30	80	20
8	30	20	30
9	35	60	50
10	35	80	20
11	35	20	30
12	35	40	40
13	40	80	30
14	40	20	40
15	40	40	50
16	40	60	20

. While the parameter ranges in Table 4 reflect current production standards, existing data typically provide local optima within fixed operational constraints. Our model aims to generalize process behavior beyond historical parameter combinations (e.g., predicting untested scenarios), as well as to quantify interactions between parameters (e.g., nonlinear effects of wheel speed and feed rate on Ra) that may not be evident in routine production records.

The method of reverse grinding was used in the experiment. At the same time, considering that electrical components such as sensors and signal data lines are easy to damage, each processing time is short, and the bearing surface temperature rise is low, the experiment adopts a dry grinding method. The grinding repeats 5 times and holds for 10 s under the same parameters.

According to Nyquist’s sampling law, in order to avoid aliasing, the sampling frequency should be greater than twice the highest frequency in the signal. Therefore, the sampling frequency set by the vibration sensor acquisition system is 51.2 KHz, and the sampling frequency set by the acoustic emission sensor data acquisition card is 4 MHz.

The probe sensor of the roughness meter was placed at the highest point of the bearing outer ring, and five points were randomly selected along the vertical direction of the grinding direction for measurement. The sampling length was 0.8 mm, the evaluation length was 0.8 × 5 mm, and the measurement parameter was the arithmetic average deviation Ra value of the evaluation profile. After measurement, the average value of 5 points is taken as the surface roughness value.

The test equipment layout and test site are shown in Figure 2 and Figure 3. An acoustic emission sensor and vibration sensor were placed on the specially designed bearing ring fixture, and the acoustic emission signal and vibration signal were collected at the same time.

In this research, grinding experiments were conducted to obtain 80 sets of vibration and acoustic emission data and the corresponding surface roughness values. The experimental results are shown in

Table 5. Partial grinding experimental results.

Serial Number	Grinding Wheel Line Speed/m·s−1	Workpiece Speed/r·min−1	Grinding Depth/μm	Ra/μm
1	25	20	20	0.1126
2	25	20	20	0.1128
3	25	20	20	0.1056
4	25	20	20	0.1102
⋮	⋮	⋮	⋮	⋮
38	30	80	20	0.2534
39	30	80	20	0.2526
40	30	80	20	0.2438
41	30	20	30	0.0960
42	30	20	30	0.0998
⋮	⋮	⋮	⋮	⋮
76	40	60	20	0.0936
77	40	60	20	0.0916
78	40	60	20	0.0916
79	40	60	20	0.0946
80	40	60	20	0.0916

.

In order to reduce the CEEMDAN decomposition and feature extraction time of the signal, reduce the influence of random noise of the signal, and improve the signal-to-noise ratio of the signal, the average value of the vibration signal was extracted every 5 data points, and the average value of the acoustic emission signal was extracted every 400 data points to form a new vibration and acoustic emission signal. Considering the feed and retreat time of the grinding wheel, 5 s of data from the middle of each data file—that is, 50,000 data points of the new vibration and sound emission signals—are intercepted, as shown in Figure 4.

In the small sample dataset, 80% of the data is divided as the training set, i.e., 64 sets of data are used as the training set and 16 sets of data are used as the test set.

Each set of vibration and acoustic emission signal samples is decomposed into a set of IMF components by the CEEMDAN algorithm, and the Pearson correlation coefficients between the decomposed IMF components and the original signals are calculated. The first eight IMF components are retained according to the magnitude of their values, as shown in Figure 5.

Multi-transform domain feature extraction for the first eight retained IMF components results in a multi-transform domain feature set for a single vibration or acoustic emission signal, with all feature metrics shown below:

The time-domain eigenvector T consists of 88 time-domain metrics, as follows:

$$T=\left[x_1^1 x_2^1 \cdots x_{11}^1 x_1^2 x_2^2 \cdots x_{11}^2 \cdots x_1^7 x_2^7 \cdots x_{11}^7 x_1^8 x_2^8 \cdots x_{11}^8\right]$$

(39)

The frequency-domain feature vector F consists of 104 frequency-domain indicators, as follows:

$$F=\left[x_{12}^1 x_{13}^1 \cdots x_{24}^1 x_{12}^2 x_{13}^2 \cdots x_{24}^2 \cdots x_{12}^7 x_{13}^7 \cdots x_{24}^7 x_{12}^8 x_{13}^8 \cdots x_{24}^8\right]$$

(40)

The entropy-domain eigenvector E consists of 32 entropy-domain indicators, as follows:

$$E=\left[x_{25}^1 x_{26}^1 x_{27}^1 x_{28}^1 x_{25}^2 x_{26}^2 x_{27}^2 x_{28}^2\cdots x_{25}^7 x_{26}^7 x_{27}^7 x_{28}^7 x_{25}^8 x_{26}^8 x_{27}^8 x_{28}^8\right]$$

(41)

where the superscript i of $x_n^i$ denotes the i-th IMF component.

T, F, and E form the multi-transform domain feature vector I of a single signal, where I is a 224-dimensional vector, as follows:

I = [T F E]

(42)

For each grinding acquisition of three channels of vibration signals and two channels of acoustic emission signals—a total of five channels of signals—the multi-transform domain feature vector of the five channels of signals will be combined together to obtain the initial signal feature vector, a 1120-dimensional vector, as follows:

Feature = [I_AE1 I_AE2 I_V-X I_V-Y I_V-Z]

(43)

where I_AE1 denotes the narrow-frequency acoustic emission signal; I_AE2 denotes the wide-range acoustic emission signal; I_V-X denotes the X-axis vibration signal; I_V-Y denotes the Y-axis vibration signal; and I_V-Z denotes the Z-axis vibration signal.

In order to reduce the correlated features and the dimensionality of the dataset, the initial signal feature vectors are fused using principal component analysis (PAC), and a total of 54 principal components are obtained. These have a cumulative contribution value of 98.065%, which is able to retain most of the useful information in the original signal feature vectors. The results are shown in

Table 6. Partial PCA solution results.

Principal Component	Eigenvalue	Variance Contribution Rate/%	Cumulative Contribution Rate/%
1	330.850	29.171	29.171
2	177.041	15.610	44.781
3	88.175	7.774	52.555
4	57.692	5.087	57.642
$⋮$	$⋮$	$⋮$	$⋮$
51	1.796	0.158	97.622
52	1.709	0.151	97.773
53	1.684	0.148	97.921
54	1.625	0.143	98.065

.

The score coefficients of the raw features indicate the importance or weight of the feature in each principal component. These score coefficients represent the magnitude of the role that each raw feature plays in composing the principal components, with larger score coefficients indicating a higher influence in the corresponding principal component and smaller coefficients indicating a lower influence. By looking at the score coefficients, it is possible to understand which raw features play an important role in composing the principal components to better understand the structure of the data and the relationship between the features. The score coefficients for some of the raw features are shown in

Table 7. Component score coefficient matrix.

	Principal Component	1	2	$\cdots$	53	54
Primary Feature
$I_{AE1}(x_1^1)$		−0.035	0.014	$\cdots$	0.007	0.008
$I_{AE1}(x_2^1)$		−0.020	0.020	$\cdots$	0.003	0.023
$I_{AE1}(x_3^1)$		−0.046	0.012	$\cdots$	0.011	−0.004
$I_{AE1}(x_4^1)$		−0.020	0.020	$\cdots$	0.003	0.023
$⋮$		$⋮$	$⋮$	$\cdots$	$⋮$	$⋮$
$I_{V-Z}(x_{26}^8)$		0.028	−0.035	$\cdots$	−0.008	0.017
$I_{V-Z}(x_{27}^8)$		0.022	−0.046	$\cdots$	−0.007	−0.002
$I_{V-Z}(x_{28}^8)$		0.025	−0.044	$\cdots$	−0.016	0.006

.

After the initial signal feature vectors Feature are fused by the principal component analysis, 54-dimensional signal feature vectors Feature_signal are obtained, which are connected with the grinding parameters in Table 5 to form the final set of feature vectors Feature_total as shown below:

Feature_total = [Feature_signal Feature_argument]

(44)

The obtained 57-dimensional total feature vector Feature_total was used as the input of the support vector regression (SVR) model to obtain the final predicted grinding surface roughness.

In order to validate the effectiveness and performance of the established feature extraction-based surface roughness prediction method for precision grinding of bearing outer rings, different feature sets and different regression models were compared in two dimensions. In order to quantitatively assess performance of the proposed prediction method, R², MAE, RMSE, and MAPE are selected as the criteria for evaluation.

(1) Comparison of feature sets

In order to verify the effectiveness of the signal multi-transform domain feature fusion scheme and the combination of signal features and grinding parameters to predict the surface roughness of the bearing collar outer ring grinding, three different feature combinations, T&F (Figure 6a), T&F&E (Figure 6b), and T&F&E&P (Figure 6c), are compared, respectively, with T&F indicating the combination of the signal time-domain and frequency-domain features, T&F&E indicating the combination of the signal-time, frequency, and entropy-domain multi-transform domain features, and T&F&E&P indicating the combination of the signal multi-transform domain features and grinding parameters proposed in this research. The signal features of the three different feature combinations are fused via PCA, then the final feature vectors are input into the SVR for surface roughness fitting. The SVR model parameters are set up via grid search and 5-fold cross-validation, and the fitting results are shown in

Table 8. Quantitative evaluation of different parameter combinations.

Feature Combination	R2	MAE/μm	RMSE/μm	MAPE
T&F	0.9706	0.0041	0.0125	0.0361
T&F&E	0.9838	0.0034	0.0093	0.0311
T&F&E&P	0.9953	0.0020	0.0050	0.0187

and Figure 6.

The vibration and acoustic emission signals collected during the grinding experiments captured subtle changes in the grinding process, which contained information about the changes in the surface roughness of the workpiece during the grinding process of the bearing collar. After using the entropy feature as an input to the model, the R² is improved from 0.9706 to 0.9838, which represents a significant improvement in the fitting degree of the model, and the model is more capable of interpreting the surface roughness and can more accurately characterize the changes in the surface roughness. the MAE, RMSE, and MAPE are reduced from 0.0041, 0.0125, and 0.0361, respectively, to 0.0034, 0.0093, and 0.0311 respectively, which demonstrates that the average error, variance, and average percentage error of the model prediction have been reduced. The absolute error of the model can also be seen in the figure, which indicates that the model’s prediction results are more accurate, and the prediction accuracy is more balanced and reliable. After again using the processing parameters as one of the model inputs, the R² is further improved to 0.9953, and the MAE, RMSE, and MAPE are further decreased to 0.0020, 0.0050, and 0.0187. The absolute error of the model is also further reduced, which represents a further improvement in the degree of fit, prediction accuracy, precision, and reliability of the model.

In summary, using the signal entropy feature as one of the inputs to the model can better capture the complexity and dynamic changes in the data during the grinding process, while the processing parameters have an important influence on predicting the surface roughness, which can provide additional information that can help the model to more accurately capture the changes in the working conditions during the grinding process, thus improving the accuracy and reliability of the prediction.

(2) Model comparison

In order to verify the advantages of the SVR model (Figure 7b) in processing small sample data, the BP neural network model (Figure 7a) is used as a comparison to analyze the prediction effect. The T&F&E&P dataset is taken as the input sample, and the same partition ratio and random seed are given when dividing the training set and test set samples to ensure that the training set and test set are the same each time, to eliminate random errors, and to debug and optimize the parameters of the two algorithms, respectively. The regression results of the bearing grinding surface roughness obtained are shown in

Table 9. Quantitative evaluation of different models.

Model	R2	MAE/μm	RMSE/μm	MAPE
BP	0.9473	0.0088	0.0167	0.0612
SVR	0.9953	0.0020	0.0050	0.0187

and Figure 7.

The R² value of the BP model is 0.9473, which is smaller than the R² value of the SVR model, indicating that the BP model has a lower degree of fit in explaining surface roughness relative to the SVR model. Meanwhile, the MAE, RMSE, and MAPE values of the BP model are 0.0088, 0.0167, and 0.0612, respectively, which are larger than the MAE, RMSE, and MAPE values of the SVR model with the corresponding values, which indicates that the mean error, variance, and mean percentage error between the prediction results and the actual values of the BP model are higher than those of the SVR model, and the prediction error of the BP model is relatively large. It can be seen that the BP model has poor regression of bearing grinding surface roughness under small sample conditions, and the degree of regression fitting and accuracy of the SVR model are significantly improved compared with the BP model under the same sample data conditions, which indicates that the SVR model has a significant advantage in dealing with small sample sizes. Therefore, the selection of the model for bearing grinding surface roughness monitoring is effective and can meet the demand.

6. Conclusions

This research proposes a feature extraction-based surface roughness prediction method for precision grinding of bearing outer rings, which integrally considers the grinding processing parameters and a variety of sensor signals and extracts the combination of sensor signal features in the grinding process to realize the prediction of the surface roughness of precision grinding of the bearing outer ring. This improves the universality and accuracy of the prediction. The specific conclusions are as follows:

(1)

For the small sample dataset, the grinding signal data are preprocessed accordingly, and the prediction model of surface roughness of precision grinding of the bearing outer ring based on feature extraction is established. The vibration and acoustic emission signals during the grinding process are decomposed using the CEEMDAN algorithm, and the signal components are retained according to the size of the correlation coefficient. The retained signal components are subjected to multi-transform domain feature extraction in the time-, frequency- and entropy-domains, then fused and downgraded via principal component analysis. The grinding parameters and the downgraded signal features are combined to form a total feature set reflecting the surface roughness of the grinding surface, and the grinding surface roughness of the outer ring is predicted using the support vector extraction model based on the total feature set reflecting the surface roughness of the grinding surface. Based on the total feature set reflecting the grinding surface roughness, the support vector regression model is used to realize the monitoring of the grinding surface roughness of bearing rings with small samples and multiple working conditions.

(2)

Multi-case thin-walled bearing collar cylindrical grinding experiments are conducted to experimentally verify the established prediction method. In the small sample data set, the comparative experimental results show that, compared with simply extracting the signal time-domain and frequency-domain features, the introduction of entropy features proposed in this research can extract signals from the grinding vibration and acoustic emission that contain more information related to the surface roughness of the grinding surface. At the same time, the grinding parameters also contain information related to the surface roughness of the grinding surface, which in turn can be used to obtain regression prediction of a better degree of fit and higher accuracy. As a result, the R², MAE, RMSE, and MAPE values of the model are 0.9953, 0.0020, 0.0050, and 0.0187, respectively.

(3)

The experimental results show that support vector regression is more applicable to small sample sizes, which improves the accuracy of the prediction model of precision grinding surface roughness of the bearing outer ring. The values of R², MAE, RMSE, and MAPE of the model are improved from 0.9473, 0.0088, 0.0167, and 0.0612 to 0.9953, 0.0020, 0.0050, and 0.0187.

(4)

The prediction method of surface roughness of bearing outer ring precision grinding based on feature extraction proposed in this research can effectively carry out the prediction of surface roughness of bearing collar outer ring grinding with small samples and multiple working conditions, improve the accuracy of the prediction model, and provide a basis for the development of relevant optimization decisions for bearing collar precision grinding.

This paper employs feature selection and SVR for predicting surface roughness, achieving promising results. However, there remains significant room for improvement in specific algorithmic aspects such as feature selection methods, parameter optimization, and model architecture to enhance prediction accuracy and robustness, thereby further advancing surface roughness prediction technology. Additionally, the current research primarily focuses on surface roughness prediction techniques without systematically integrating the proposed prediction methods or developing an intelligent monitoring system. Although experimental data have validated the effectiveness of the approach, it cannot yet be directly applied to practical manufacturing processes. Future work will prioritize the systematic development and refinement of existing research to ensure this technology can be effectively implemented in real-world machining applications.

While this study successfully demonstrates the predictive capability of feature-fused sensor data, further research employing designed experiments with ANOVA would be valuable to isolate the individual effects of grinding parameters. Such work would require dedicated test protocols beyond typical production monitoring scenarios.