Rolling bearing fault detection using symmetrized dot pattern indices and feedforward neural network

Abstract

Fault detection (FD) is of primary importance for maintenance of mechanical systems. In recent years, the symmetrized dot pattern (SDP) technique has been increasingly applied in this context. This article introduces a new approach to SDP based on new indices derived from SDP transformed vibrational signals. The indices characterize the density, orientation and curvature of the distribution of ‘snowflake’ diagrams, and they were used as inputs for a feedforward neural network (FNN) for FD of bearings as an application of the new approach. The validity of the technique was demonstrated through its application on two public rolling bearing datasets, thereby substantiating its generalizability across a range of fault types and operating conditions. Results demonstrate high classification accuracy, low false positives rates and low computational costs suitable for real-time implementation. The new SDP and FNN approach was also compared to the classical convolutional neural network-based approach: the new method does not require the images, achieves better performance and has a low computational cost compared to the classical one. Finally, the proposed approach is compared with the most modern techniques demonstrating its validity.

Introduction

Fault detection (FD) is the process of monitoring mechanical system components to identify malfunctions or anomalies that deviate from normal behaviour. ¹ Huang et al. ² proposed a method for selecting the right frequency band for FD under dynamic conditions. Wang et al. ³ developed a new method to detect the bearings outer race faults and estimates the healthy bearing status. A feedforward neural network (FNN) based on accelerometer signals, ⁴ together with dynamic models and estimators enhanced through the wavelet transform (WT), ⁵ has demonstrated strong effectiveness for FD in hybrid-electric aircraft propulsion systems. Recently, researchers have been proposing several improvements to existing techniques. In the study by Yu et al., ⁶ the authors presented an improvement of the WT that allows the detection of transient frequencies due to the occurrence of faults. Sun et al. ⁷ introduced a root-Prony technique improvement, more accurate and computationally less costly, for FD in asynchronous motors based on current signals.

Nowadays, researchers have increasingly proposed machine learning (ML)-based techniques for FD. Geraei et al. ⁸ presented a local binary pattern technique improvement for bearing health monitoring based on vibrational signals: comparison with the classical and other ML techniques showed superior accuracy without significantly increasing of computational cost. A new unsupervised method for constructing a health index using a Gaussian mixture model for early failure detection has been presented in the study by Wen et al. ⁹ Hou et al. ¹⁰ proposed a new method that reduces false and missed alarms based on an improved support vector data description model. Abbasi et al. ¹¹ presented a multitasking approach between convolutional neural network (CNN) and long-short time memory (LSTM) for FD under different operating conditions: the comparison between the new and other models had demonstrated the first one validity. In the study by Wang et al., ¹² the authors implemented a new method to transform the data in graph form to utilize a graph neural network specifically developed for unsupervised FD. Kumar et al. ¹³ introduced a current signal transformation into a scalogram, and an image classification model obtained using transfer learning: the proposed methodology had showed a 99% classification accuracy that is greater compared to other ML models. Other FD based on ML techniques are reported in the studies by Mahesh et al., ¹⁴ Xiao et al. ¹⁵ and Wang et al. ¹⁶

Currently, the symmetrized dot pattern (SDP) technique has become a valuable approach in signal analysis, offering a way to transform signals to better detect system anomalies: it has proven especially useful in diagnostics and predictive maintenance. This technique involves converting the original signal into characteristic ‘snowflake’ diagrams, which are then used to classify the system as either healthy or faulty. ¹⁷ One of the key advantages of SDP over traditional signal analysis is its capability to turn complex, nonlinear and nonstationary signals into visually intuitive patterns. This greatly aids in identifying and distinguishing operating conditions or faults in mechanical systems. ¹⁸ Research has shown that FD based on SDP-transformed vibration signals significantly enhances accuracy compared to other signal transformation techniques. ¹⁹ Currently, there is a strong trend toward automating FD using SDP and existing or custom-designed CNNs by applying SDP-transformed to vibrational and acoustic signals.^20–24 Additionally, combining signals from multiple sensors has been shown to further enhance CNN classification performance. ²⁵ To boost the accuracy of CNN-based classification, techniques such as empirical mode decomposition and variational mode decomposition are also commonly employed.^26–29 The SDP-CNN and spectrogram-CNN methods reach comparable performance, but the SDP-CNN approach requires fewer hardware resources. ³⁰ Moreover, signal pre-processing can improve both the SDP transformation and the CNN performance under variable operating conditions. ³¹

Transforming signals into SDP coordinates has proven effective for FD in rotating machinery under both steady and varying operating conditions. The resulting snowflake diagrams enable clearer distinction between different faults. Nonetheless, current research remains relatively limited and is mainly centred on CNN-based approaches, often relying on signal fusion, or pre-processing to enhance classification performance. The article’s purpose is to present a new approach to the SDP technique based on new indices that have been developed for SDP-transformed vibrational signals. The signal is transformed into SDP coordinates; subsequently, the coordinates are filtered to remove outliers and finally used to compute the indices. These indices can provide information on the density, orientation and curvature of the SDP petals and are used as input to a FNN: this workflow automates the FD process. The proposed technique was experimentally validated using two public datasets of rolling bearings featuring different fault types, fault severity levels and operating conditions, as an application of the proposed methodology. The results demonstrate that the technique achieves high accuracy and a low false positive rate. A feasibility study also indicates that the approach has a low computational cost compatible with real-time implementation. Furthermore, a comparative study demonstrates the importance of SDP coordinates filtering, while the ablation study shows the significance of each individual index as an input for the FNN. The new approach was compared to the classical one consisting in SDP-CNN: the comparison showed that the new technique is better than the traditional one in terms of classification accuracy, false positive rate, computational cost and the indices avoid the use of images and convolutional layers. Finally, a comparison is also proposed with other ML approaches in the literature on the same datasets.

The main contributions of this work can be summarized as follows:

1. A new approach to the SDP technique for FD based on specifically developed indicators for the petals of ‘snowflake’ diagram.

2. Development of new indicators to assess the density, orientation and curvature of clusters.

3. Automation of the FD process using a FNN to distinguish among different fault types.

4. Experimental validation of the SDP-FNN using two rolling bearings public datasets of bearings covering different operating conditions, fault types and severities and ball bearing types.

5. Assessment of the computational cost of SDP-FNN for the real-time feasibility implementation on embedded hardware.

6. A comparison with the classical SDP-CNN approach highlights the advantages of the new method in terms of accuracy, false positive rate and computational cost and a comparison with other ML approaches to demonstrate its validity.

The rest of the article is structured as follows: the second section presents the methodology; the third and fourth sections describe the experimental results on Case Western Reserve University (CWRU) and Hanoi University of Science and Technology (HUST) bearing datasets respectively; the fifth section summarizes the conclusions of this work.

SDP indices approach

The workflow of the new SDP approach for bearing FD is shown in Figure 1.

Figure 1.

New SDP approach workflow. SDP: symmetrized dot pattern.

Figure 1 outlines the key steps of the proposed workflow, which can be summarized as follows:

1. In the first step, the time domain vibrational signal $x$ is transformed into SDP coordinates $r$ and $\theta$ .

2. In the second step, the coordinates $r$ and $\theta$ are filtered to eliminate outliers resulting in $r_f$ and ${\theta }_{f.}$

3. In the third step, the new indicators: Density index (DI), orientation index (OI) and curvature index (CI) are calculated on the filtered coordinates $r_f$ and ${\theta }_f$ of one petal of the snowflake diagram.

4. In the fourth and final step, the indicators are used to train, validate and test an optimized FNN.

SDP transformation

The SDP technique converts a one-dimensional (1D) time-domain signal into a normalized two-dimensional (2D) representation by plotting amplitude values in a polar coordinate system, resulting in a symmetrical diagram. ¹⁷ This visualization method highlights variations in signal amplitude and frequency, aiding in the diagnosis of faults in rotating machinery such as bearings. Differences between signals are reflected in the distinct shapes of the petals forming the snowflake pattern. This diagram is constructed using $h$ planes of symmetry, which is created by rotating the petals through specific angles $\alpha$ :

$$\alpha =\frac{360}{h}$$

(1)

A necessary condition to plot the snowflake is not to overlap the petals.

It is possible to transform a signal $x=x_1,\dots ,x_k,\dots ,x_n$ in polar coordinates using the SDP transform by the formula:

$$r_k=\frac{x_k-\max (x)}{\max (x)-\min (x)}$$

(2)

$${\theta }_k=\alpha +\frac{x_{k+l}-\min \left(x\right)}{\max \left(x\right)-\min \left(x\right)}\xi$$

(3)

$${\varphi }_k=\alpha -\frac{x_{k+l}-\min \left(x\right)}{\max \left(x\right)-\min \left(x\right)}\xi$$

(4)

where

• $n$ is the length of $x$ .

• $\min (x)$ and $\max (x)$ are the minimum and the maximum value of $x$ respectively.

• $r_k$ is the radius of the $k$ th point.

• ${\theta }_k$ is the clockwise deflection angle of the $k$ th point along the symmetry plane.

• ${\varphi }_k$ is the anticlockwise deflection angle of the $k$ th point along the symmetry plane.

• $l$ is the delay coefficient.

• $\alpha$ is the rotation angle of the symmetry plane.

• $\xi$ is the gain of the deflection angle.

Figure 2 shows the influence of the parameters $\xi$ and $l$ on the snowflake diagram. As $\xi$ increases, the curvature of the petals will increase. The same applies to $l$ .

Figure 2.

Snowflake dependency from ξ and $l$ .

After setting the optimal $\xi$ and $l$ parameters, it is possible to calculate the new indices on one SDP petal to obtain the FNN input indices.

SDP coordinates filtering

Reducing the number of outliers allows to focus on the main characteristic of the petal: the most densely populated area of the cluster. This area provides key information about the distribution of points in the polar plane. The SDP coordinates were filtered to reduce the influence of outliers.

Given a generic point in SDP coordinates $P(r_k,{\theta }_k)$ , it must satisfy:

$$P\left(r_{f,k},{\theta }_{f,k}\right)=P\left(r_k,{\theta }_k\right)\left\{r_{\inf }\le r_k\le r_{\sup },{\theta }_{\inf }\le {\theta }_k\le {\theta }_{\sup }\right\}$$

(5)

where $P(r_{f,k},{\theta }_{f,k})$ is the generic filtered point in SDP coordinates and:

$$r=r_1,\dots ,r_k,\dots ,r_n$$

(6)

$$\theta ={\theta }_1,\dots ,{\theta }_k,\dots ,{\theta }_n$$

(7)

$$r_f=r_{f,1},\dots ,r_{f,k},\dots ,r_{f,n}$$

(8)

$${\theta }_f={\theta }_{f,1},\dots ,{\theta }_{f,k},\dots ,{\theta }_{f,n}$$

(9)

$$\{\begin{array}{c}{r}_{\inf }=\overline{r}-{\sigma }_r\hfill \\ \multicolumn{1}{c}{r_{\sup }=\overline{r}+{\sigma }_r}\\ \multicolumn{1}{c}{{\theta }_{\inf }=\overline{\theta }-{\sigma }_{\theta }}\\ \multicolumn{1}{c}{{\theta }_{\sup }=\overline{\theta }+{\sigma }_{\theta }}\end{array}$$

(10)

and

$$\{\begin{array}{c}\overline{r}=\frac{1}{n}\sum _{k=1}^n{r}_k\\ {\sigma }_r=\sqrt{\frac{1}{n}\sum _{k=1}^n{\left(r_k-\overline{r}\right)}^2}\\ \overline{\theta }=\frac{1}{n}\sum _{k=1}^n{\theta }_k\\ {\sigma }_{\theta }=\sqrt{\frac{1}{n}\sum _{k=1}^n{\left({\theta }_k-\overline{\theta }\right)}^2}\end{array}$$

(11)

Thanks to this filter, it is possible to remove outliers from the cluster (petal of the snowflake diagram).

Density index

The filtered coordinates $r_f$ and ${\theta }_f$ will be denoted by $r$ and $\theta$ for the sake of simplicity. Given the signal $x$ in SDP coordinate $(r,\theta )$ for a single SDP cluster (petal of snowflake diagram), it possible to define centroid polar coordinates as follows:

$$r_C=\frac{1}{n}\sum _{j=1}^n{r}_i$$

(12)

$${\theta }_C=\frac{1}{n}\sum _{i=1}^n{\theta }_i$$

(13)

The Euclidean distance $d_i$ in the polar plane between the cluster centroid $(r_C,{\theta }_C)$ and a cluster point $(r_i,{\theta }_i)$ can be defined as follows:

$$d_i=\sqrt{r_i^2+r_C^2-2r_i^2{r}_C^2\cos ({\theta }_C-{\theta }_i)}$$

(14)

Calculated the distance between each point and the cluster centroid ( $d=d_1,\dots ,d_n$ ), and defined:

$$d_{\max }=\mathrm{argmax}\left(d\right)$$

(15)

it is possible to define DI as follows:

$$\mathrm{DI}=\frac{\frac{1}{n}\sum _{i=1}^n{d}_i}{d_{\max }}=\frac{\frac{1}{n}\sum _{i=1}^n\left(\sqrt{r_i^2+r_C^2-2r_i^2{r}_C^2\cos ({\theta }_C-{\theta }_i)}\right)}{\mathrm{argmax}\left(d\right)}$$

(16)

DI quantifies the density of a set of points relative to the cluster centroid, providing a measure of point density within the cluster. The index ranges from $0$ to $1$ :

• $\mathrm{DI}\to 0$ : the cluster is closed to its centroid and when $\mathrm{DI}=0$ all points of clusters are coincident with the centroid.

• $\mathrm{DI}\to 1$ : the cluster is scattered and when $\mathrm{DI}=1$ all points of clusters are coincident with $d_{\max }$ .

As can be seen, this index does not provide information on the directionality of the cluster points. For this reason, the following index was developed.

Orientation index

Given the signal SDP coordinate $(r,\theta )$ , it is possible to convert them in Cartesian coordinates $(x,y)$ as follows:

$$x=x_{1,},\dots ,x_i,\dots ,x_{n,}=r_1\cos {\theta }_1,\dots ,r_i\cos {\theta }_i,\dots ,r_n\cos {\theta }_n$$

(17)

$$y=y_{1,},\dots ,y_i,\dots ,y_{n,}=r_1\sin {\theta }_1,\dots ,r_i\sin {\theta }_i,\dots ,r_n\sin {\theta }_n$$

(18)

Equations (17) and (18) show that converting from $(r,\theta )$ to $(x,y)$ is a transformation rather than a reversion to the time domain. This transformation allows to the SDP to be visualized within a new reference frame while preserving the information obtained from the initial transformation of the signal from the time domain to SDP coordinates. Indeed, the time-domain signal is a $1\mathrm{D}$ signal, while this transformation provides a 2D signal $(x,y)$ .

Identifying the main direction of $(x,y)$ is necessary to solve the problem:

$$\mathrm{cov}\left(\mathrm{xy}\right)\mathrm{v}=\lambda \mathrm{v}$$

(19)

where $\lambda$ is the eigenvalue that represents the variance along the direction of the corresponding eigenvector $\mathrm{v}$ . The eigenvector associates to the maximum eigenvalue ${\lambda }_{\max }$ identifies the direction along which $(x,y)$ has the greatest dispersion. Defining the covariance matrix $\mathrm{cov}\left(\mathrm{xy}\right)$ as follows:

$$\mathrm{cov}\left(\mathrm{xy}\right)=\left[\begin{array}{cc}\mathrm{var}\left(x\right)\hfill & \mathrm{cov}(x,y)\hfill \\ \multicolumn{1}{c}{\mathrm{cov}(x,y)}& \mathrm{var}\left(y\right)\hfill \end{array}\right]$$

(20)

the eigenvalue problem can be formulated as follows:

$$|\mathrm{cov}\left(\mathrm{xy}\right)-\lambda I|=0$$

(21)

Solving the matrix Equation (21) it is possible to obtain the two maximum eigenvalues ${\lambda }_{x,\max }$ and ${\lambda }_{y,\max }$ . The corresponding eigenvectors ${\mathrm{v}}_{x,\max }$ and ${\mathrm{v}}_{y,\max }$ identifies the two main direction of $(x,y)$ dispersion:

$$\mathrm{cov}\left(\mathrm{xy}\right){\mathrm{v}}_{x,\max }={\lambda }_{x,\max }{\mathrm{v}}_{x,\max }$$

(22)

$$\mathrm{cov}\left(\mathrm{xy}\right){\mathrm{v}}_{y,\max }={\lambda }_{y,\max }{\mathrm{v}}_{y,\max }$$

(23)

The angle between the two eigenvectors corresponding to the prevailing direction of data dispersion, that is, OI:

$$\mathrm{OI}=\mathrm{atan}2\left(\frac{{\mathrm{v}}_{y,\max }}{{\mathrm{v}}_{x,\max }}\right)$$

(24)

OI represents the angle between the original $(x,y)$ reference system and the new reference system, called cluster reference system, which is defined by the two eigenvectors. This index provides information about the directionality of the cluster points. The index ranges from $0$ to ${90}^{°}$ :

• $\mathrm{OI}\to 0^{°}$ : the original $(x,y)$ and the cluster reference system are $\overrightarrow{x}\equiv {\overrightarrow{\mathrm{v}}}_{x,\max }$ and $\overrightarrow{y}\equiv {\overrightarrow{\mathrm{v}}}_{y,\max }$ ;

• $\mathrm{OI}\to {90}^{°}$ : the original $(x,y)$ and the cluster reference system are $\overrightarrow{x}\equiv {\overrightarrow{\mathrm{v}}}_{y,\max }$ and $\overrightarrow{y}\equiv {\overrightarrow{\mathrm{v}}}_{x,\max }$ .

The graphical representation of OI is reported in Figure 3.

Figure 3.

OI representation. OI: orientation index.

Curvature index

Given the signal $x$ in SDP coordinate $(r,\theta )$ and its transformation in Cartesian coordinates $X=(x,y)$ as the Equations (17) and (18), solved the problem of Equation (21) and obtained the first eigenvector $\mathrm{v}$ and corresponding to the eigenvalue $\lambda$ , it is possible to obtain the projection of all points along $\mathrm{v}$ as follows:

$$X_{\mathrm{proj}}=X_c·\mathrm{v}{\mathrm{v}}^T$$

(25)

where $X_c=X-\mu$ , and $\mu =E[X]$ . The orthogonal coordinates of $X_{\mathrm{proj}}$ respect to the $\mathrm{v}$ direction can be obtained as follows:

$$X_{\mathrm{ortho}}=X_c-X_{\mathrm{proj}}$$

(26)

and their distance from the $\mathrm{v}$ are:

$$\begin{array}{c}\multicolumn{1}{c}{d=d_1,\dots ,d_i,\dots ,d_n=\Vert X_{\mathrm{ortho},1}\Vert ,\dots ,}\\ \multicolumn{1}{c}{\Vert X_{\mathrm{ortho},i}\Vert ,\dots ,\Vert X_{\mathrm{ortho},n}\Vert }\end{array}$$

(27)

that represent the amplitude of orthogonal dispersion concerning the main direction. CI can be defined as follows:

$$\mathrm{CI}=\frac{\frac{1}{n}\sum _{i=1}^n{d}_i}{\mathrm{argmax}\left(X_{\mathrm{proj}}\right)-\mathrm{argmin}\left(X_{\mathrm{proj}}\right)}$$

(28)

This index provides information on the shape of the cluster in the plane. It is a normalized, dimensionless measure of the orthogonal dispersion concerning the identified main direction of the data:

• If $\mathrm{CI}\to 0$ , then the distribution of points is very narrow, like a straight line, with low orthogonal dispersion.

• If $\mathrm{CI}\to 1$ , then the distribution becomes more rounded or more orthogonally dispersed.

FNN development and optimization

The previous defined indices are used as FNN input. A FNN is a type of artificial neural network in which the flow of information is directed in a single direction. ³² It is often used for classification problems in which the goal is to assign a label or class to a given input. A FNN is composed of:

• Input layer: It receives the input data and forwards them to the next layer without any processing. It consists of as many neurons as there are input indices.

• Hidden layers: They process information and ‘learn’ abstract representations of the data. It can consist of one or more layers of a certain number of neurons. Each neuron applies a mathematical function to its input:

$$z_j=\sum _{i=1}^n{\omega }_{\mathrm{ji}}{x}_i+b_j$$

(29)

where $z_j$ is the net input to the $j$ th neuron, $x_i$ is the $i$ th input to the neuron, ${\omega }_{\mathrm{ji}}$ is the weight connecting to the $i$ th input weight to the $j$ th neuron, $b_j$ is the $j$ th neuron bias and $n$ is the number of inputs coming from the previous layer. The output $z_j$ is processed applying an activation function as Rectified Linear Unit (ReLU):

$$\mathrm{ReLU}\left(z_j\right)=\{\begin{array}{c}\multicolumn{1}{c}{z_j\mathrm{if}{z}_j>0}\\ \multicolumn{1}{c}{0\mathrm{else}}\end{array}$$

(30)

that introduce non-linearity essential for learning complex patterns and solving non-linear problems.

• Output layer: It is composed of as many neurons as the output of the network. It receives input from the last hidden layer, and the SoftMax function extracts the final output of the network, which in the case of classification is a probability vector for each class:

$$\mathrm{SoftMax}\left(z_j\right)=\frac{{\mathrm{e}}^{z_j}}{\sum _{k=1}^K{\mathrm{e}}^{z_k}}$$

(31)

where $z_j$ is the output of the last hidden layer, ${\mathrm{e}}^{z_j}$ is the exponential of $z_j$ and $K$ is the number network output.

For correct classification, the network requires training on a labelled dataset using a cost function that measures the difference between the network output and the true class. In this case, the cross-entropy loss has been used.

The weights and biases of the network are updated and the errors between the network output and the desired objective are minimized. The backpropagation algorithm is used, in which the input data are propagated to the output and the error between the estimated and the true output is calculated using the cost function. The gradient of the error for each weight is computed from the final error. Then, the weights are updated to minimize the error. The weight update rule used here was stochastic gradient descent with momentum. ³³

Finally, since the hyperparameters of the developed FNN are not learned during the optimization process, Bayesian optimization with the expected improvement criterion was used to stop the process. ³⁴ The optimized hyperparameters and their optimization ranges are shown in Table 1.

Table 1.

FNN hyperparameter optimization range.

Hyperparameter	Range
Fully connected layers number	$1÷5$
Neurons numbers of fully connected layers	$1÷1000$
Initialization learning rate	${10}^{-6}÷{10}^{-1}$
Momentum	$0.01÷0.99$
L2 regularization	${10}^{-6}÷{10}^{-1}$
Mini batch size	$2÷256$

FNN: feedforward neural network.

A generic proposed FNN architecture is shown in Figure 4.

Figure 4.

FNN proposed architecture. FNN: feedforward neural network.

Case study 1: CWRU bearing dataset

The proposed technique was implemented for CWRU bearing dataset. ³⁵ All results obtained for this dataset are shown below. A PC with a processor 13th Gen Intel Core i9-13900 of 2.00 GHz, 32 GB of RAM and a 12 GB NVIDIA GeForce RTX 3060 was used.

Dataset description

The CWRU is the most used dataset for bearing FD and diagnostic provided by the bearing data center of CWRU. The test bench is composed by an electric motor of $3\mathrm{hp}$ , torque transducer, encoder, dynamometer and two bearings under test located on the drive end (DE) and fan end, respectively. The bearing faults are artificially introduced: inner race (IR), ball (B) and outer race (OR) with crack located centred (C), orthogonal (OR) and opposite (OP) compared to the load zone. The faults diameters are $0.007$ , $0.014$ , $0.021$ and $0.028\mathrm{inches}$ . The dataset provides vibration from accelerometers located on the drive end (DE), fan end (FE) and base (BA) sampled at $12$ and 4 $8\mathrm{kHz}$ . The speed and the load remaining constant during the tests in the configuration: $1797\mathrm{rpm}$ and $0\mathrm{hp}$ , $1772\mathrm{rpm}$ and $1\mathrm{hp}$ , $1750\mathrm{rpm}\mathrm{and}2\mathrm{hp}$ and $1730\mathrm{rpm}$ and $3\mathrm{hp}$ .

Raw signals, SDP transformation and indices calculation

The signals sampled at $12\mathrm{kHz}$ by the DE accelerometer and all faults type with diameters of $0.007$ , $0.014$ and $0.021\mathrm{inches}$ were considered. Figure 5 shows $0.1$  s of raw signals normalized in the range $[-1,1]$ .

Figure 5.

Raw signals for CWRU dataset. CWRU: Case Western Reserve University.

In Figure 5 is clearly observed that signals associated with faults exhibit greater impulsivity and periodicity than the normal condition, indicative of the presence of repetitive impacts typical of mechanical faults in bearings. Conditions with defects on the OR show greater regularity and amplitude of impulses suggesting a significant influence of the defect fixed location relative to the direction of measurement.

It is necessary to choose the SDP parameters $h$ , $l$ and $\xi$ appropriately to transform the signals from the time domain into SDP coordinates. At present, there is no methodology that allows these three parameters to be selected in a unique way; however, their choice must satisfy the following criteria:

• The parameter $h$ defines the number of petal pairs obtained. A higher number of petals allows a clearer definition of the centre of the snowflake diagram. Several works have shown that $h=6$ provides a good definition of the centre.^30,31,36

• The parameters $l$ and ξ influence the opening and curvature of the petal, respectively, as shown in Figure 2. Their selection must ensure that there is no overlap between petals, while at the same time the petals must be sufficiently open to clearly define their point distribution.^30,31,36

Based on these considerations, the parameter $h$ was set equal to $6$ , resulting in $\alpha =60$ . The calibration process for $l$ and $\xi$ was carried out by selecting a signal segment from class N corresponding to three shaft revolutions. The different dynamic behaviour of the system under different health conditions will generate snowflake diagrams that are distinguishable depending on the type of fault or whether the system is healthy. For this reason, the calibration was performed only for class N.

Subsequently, the selected signal segment was transformed from the time domain into SDP coordinates using different combinations of $l$ and $\xi$ . Some examples are illustrated in Figure 6 for demonstration purposes.

Figure 6.

SDP parameters tuning for CWRU dataset. SDP: symmetrized dot pattern; CWRU: Case Western Reserve University.

To satisfy the previously defined selection criteria for $l$ and $\xi$ , observation of the snowflake diagrams in Figure 6 showed that $l=5$ and $\xi =30$ provide the maximum petal opening without overlap between petals. Since no tuning technique for the SDP parameters has yet been developed in the literature, their selection must necessarily be based on the observation of multiple diagrams and on choosing the most appropriate combination that allows both a good definition of the centre and the maximum petal opening without overlap.

The signals were divided into segments of three shaft revolutions and transformed using SDP. The choice of this signal segment is made to obtain as many indices values as possible to train, validate, and test the developed FNN. In addition, this choice is useful to further highlight the defects that will have to differentiate the SDP diagrams. Figure 7 shows a comparison between the snowflake diagrams.

Figure 7.

SDP snowflake diagram for CWRU dataset. SDP: symmetrized dot pattern; CWRU: Case Western Reserve University.

Figure 7 shows how the SDP technique allows a compact representation of signal morphology in the polar domain, facilitating the visualization of recurring patterns and the identification of anomalies. These diagrams demonstrate the effectiveness of the SDP approach in highlighting distinctive indices associated with specific bearing defects, making it particularly useful for automatic diagnosis and classification of fault conditions.

The $r$ and $\theta$ coordinates were filtered as indicated in ‘SDP coordinates filtering’ section and from $r_f$ and ${\theta }_f$ , the values of $\mathrm{DI}$ , $\mathrm{OI}$ and $\mathrm{CI}$ were calculated. As an example, Figure 8 shows thirty indices values and both those calculated in the case of raw and filtered SDP coordinates.

Figure 8.

Indices trend for CWRU dataset. CWRU: Case Western Reserve University.

As shown in Figure 8, the filtering process improves the separation between the different conditions, helping to reduce fluctuations in the index values.

Focusing on the filtered cases:

• DI reaches the highest values for the N, B and OR orthogonal cases and the lowest for the OR_C case: This is confirmed by Figure 7 in which for the N, B and OR_OR cases the density of the points with respect to the centre of the cluster is higher, whereas for the OR_C case there is a greater dispersion.

• OI reaches the highest values for the OR cases and the lowest values for the IR: This is confirmed by Figure 7 in which it is possible to see that there is not a main direction of snowflake diagram for IR.

• CI reaches the highest values for the N and B cases and the lowest for the OR_C case: This is confirmed by Figure 7 in which the OR_C petals are low orthogonal dispersion with respect to preferential direction, while for N and B shows a higher curvature.

FNN development and performance evaluation

The dataset partition into training, validation, and testing subsets is shown in Table 2. It should be noted that the OR_OR and OR_OP defects with a fault diameter of $0.014\mathrm{inches}$ are not included in Table 2 because they are not available in the original dataset.

Table 2.

CWRU dataset partition in training, validation and testing subsets.

Training
Bering status	Faults diameters (inches)	Loads (hp)	Speeds (rpm)
N	—	$0,1$	$1797,1792$
B, IR, OR_C	$0.007,0.014,0.021$	$0,1$	$1797,1792$
OR_OR, OR_OP	$0.007,0.021$	$0,1$	$1797,1792$
Validation
Bering status	Faults diameters (inches)	Loads (hp)	Speeds (rpm)
N	—	$2$	$1750$
B, IR, OR_C	$0.007,0.014,0.021$	2	$1750$
OR_OR, OR_OP	$0.007,0.021$	2	$1750$
Testing
Bering status	Faults diameters (inches)	Loads (hp)	Speeds (rpm)
N	—	$3$	$1730$
B, IR, OR_C	$0.007,0.014,0.021$	$3$	$1730$
OR_OR, OR_OP	$0.007,0.021$	$3$	$1730$

CWRU: Case Western Reserve University.

The FNN output class were N, B, IR, OR_C, OR_OR and OR_OP. The FNN optimized architecture and hyperparameters are shown in Table 3.

Table 3.

FNN optimized architecture hyperparameters for CWRU dataset.

Layer	Input size	Output size
Features input	$3$	$3$
Fully connected 1	$3$	$352$
ReLU	$352$	$352$
Fully connected 2	$352$	$105$
ReLU	$105$	$105$
Fully connected 3	$105$	$425$
ReLU	$425$	$425$
Fully connected 4	$425$	$6$
SoftMax	$6$	$6$
Hyperparameter	Value
Initialization learning rate	$5.0·{10}^{-2}$
Momentum	$0.91$
L2 regularization	$8.1·{10}^{-3}$
Mini batch size	$125$

FNN: feedforward neural network; CWRU: Case Western Reserve University.

The test subset was divided into five further subsets, which were supplied separately to evaluate the FNN performance. The results in terms of metrics for evaluating the performance of the FNN during testing are shown in Table 4.

Table 4.

FNN testing performance for CWRU dataset.

Metrics	Mean ± std (%)
Accuracy	$99.60\pm 0.26$
Precision	$99.71\pm 0.29$
Recall	$99.36\pm 0.39$
F1-Score	$99.52\pm 0.30$
AUC	$99.99\pm 0.01$

FNN: feedforward neural network; CWRU: Case Western Reserve University; SD: standard deviation; AUC: area under the curve.

The results reported in Table 4 suggest that the FNN achieves very high classification performance in terms of accuracy, ability to identify the relevant classes, and overall instance discrimination, with extremely limited variation across repeated test runs, and a low false positive rate.

Comparison between raw and filtered coordinates

The same methodology illustrated in Figure 1 was implemented without the filter to highlight the importance of SDP coordinate filtering. The indices calculated directly on unfiltered $r$ and $\theta$ were used as input for the new optimized FNN. Table 5 shows the performance in the testing phase and the comparison with the filtered case.

Table 5.

Raw and filtered comparison for CWRU dataset.

Metrics	Raw (%)	Filtered (%)	$\mathrm{\Delta }$ (%)
Accuracy	$59.86$	$99.60$	$-42.68$
Precision	$39.97$	$99.71$	$-59.74$
Recall	$46.06$	$99.36$	$-53.30$
F1-Score	$41.28$	$99.52$	$-58.24$
AUC	$86.80$	$99.99$	$-13.19$

CWRU: Case Western Reserve University.

The difference between the two cases shows a change higher than 40% in accuracy (Table 5): the FNNs could not achieve the high accuracy using raw SDP coordinates.

Figure 9 shows the separation between the dataset classes, both in raw and filtered cases, in the space of the three indices to confirm the results of Table 5.

Figure 9.

Class separation in the indices space for CWRU dataset. CWRU: Case Western Reserve University.

Figure 9 demonstrates the importance of filtering for class separation.

Ablation study

Furthermore, the development and testing of FNNs was conducted using a single index or a pair of them. The comparison with the case with three indices is shown in the Table 6.

Table 6.

Features ablation study results for CWRU dataset.

Metrics	DI (%)	OI (%)	CI (%)	DI and OI (%)	DI and CI (%)	OI and CI (%)	DI, OI, and CI (%)
Accuracy	$74.90$	$71.18$	$61.31$	$89.44$	$91.11$	$88.52$	$99.60$
Precision	$68.65$	$57.24$	$34.24$	$88.40$	$90.41$	$83.49$	$99.71$
Recall	$68.97$	$59.19$	$43.94$	$82.91$	$89.89$	$83.12$	$99.36$
F1-Score	$67.77$	$57.80$	$37.55$	$83.56$	$89.01$	$83.19$	$99.52$
AUC	$95.65$	$94.40$	$89.00$	$97.60$	$98.39$	$98.18$	$99.99$

CWRU: Case Western Reserve University; DI: density index; OI: orientation index; CI: curvature index.

The results in Table 6 show that the model using all three features achieves significantly higher performance across all the considered metrics, suggesting that the integrated combination of the three features maximizes the overall predictive capability. The analysis of the reduced-feature models shows that models using two features significantly improve the performance compared to single-feature models, highlighting that interactions between features contribute positively to predictive quality. Finally, the model that uses only DI achieves higher results than those using only OI or CI across all metrics, indicating that this feature contains a larger share of discriminative information than the others. Its removal therefore leads to a substantial degradation in performance compared with combinations that include it.

Real-time feasibility and embedded implementation considerations

The practical applicability of the SDP-FNN was also verified. The computational cost, the requirements for embedded implementation, and the real-time feasibility were evaluated. Table 7 reports the performance metrics and resource utilization.

Table 7.

Performance metrics of practical applicability for CWRU dataset.

Perameter	Value
Available processing time (ms)	$100$
RAM usage (KB)	$\approx 93$
Maximum stack usage (KB)	$\approx 0.13$
Estimated final firmware size (KB)	$\approx 262$
Processing	Estimated FLOPs	Estimated time (s)
SDP transformation	$\approx 7.5·{10}^3$	$1.0·{10}^{-3}$
Filtering	$\approx 2.5·{10}^4$	$1.9·{10}^{-3}$
Indices	$\approx 6.2·{10}^4$	$1.8·{10}^{-4}$
FNN	$\approx 4.0·{10}^6$	$3.7·{10}^{-3}$
TOT	$\approx 4.1·{10}^6$	$6.8·{10}^{-3}$

CWRU: Case Western Reserve University; SDP: symmetrized dot pattern; FNN: feedforward neural network.

In Table 7 the computational cost is composed of four parts: transformation of the signal from the time domain into SDP coordinates, filtering of the SDP coordinates, computation of the indices and FNN prediction. Considering the most critical case of the dataset ( $1797\mathrm{rpm}$ ), the available processing time is $100\mathrm{ms}$ , which is greater than the total time required for the entire FD process. Moreover, the static memory footprint, the maximum stack usage and the total size of the generated C source files demonstrate that the SDP-FNN approach requires a workload compatible with modern embedded processors used in industrial applications, supporting the feasibility of practical implementation of the methodology for reliable FD.

Comparison between new and classic approach

The new method SDP-FNN effectiveness was demonstrated by comparing it to the classical SDP-CNN approach by developing CNN using images obtained from SDP diagrams as input.^30,31 The training, validation, and testing subsets are the same as in Table 2. The CNN architecture and optimization phase were implemented as already seen in the studies by Spirto et al.^30,31 The comparison between the performance of the new and classic approaches is shown in Table 8.

Table 8.

Performance comparison between SDP-FNN and SDP-CNN for CWRU dataset.

Metrics	SDP-FNN (%)	SDP-CNN (%)	$\mathrm{\Delta }$ (%)
Accuracy	$99.60\pm 0.26$	$99.40\pm 0.41$	$+0.20$
Precision	$99.71\pm 0.29$	$99.59\pm 0.50$	$+0.12$
Recall	$99.36\pm 0.39$	$99.02\pm 0.42$	$+0.34$
F1-Score	$99.52\pm 0.30$	$99.12\pm 0.35$	$+0.40$
AUC	$99.99\pm 0.01$	$99.95\pm 0.04$	$+0.05$

CWRU: Case Western Reserve University; SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network; FLOP: floating point operations per second; TOT: total.

Table 8 shows that both approaches achieve high performance, but SDP-FNN demonstrates a consistent advantage across all the metrics. The greatest advantage is seen in Recall and F1-Score, highlighting a better balance of the new approach in distinguishing between false positives and false negatives.

Since the differences shown in Table 8 are very small, to determine whether they are statistically significant rather than merely due to noise or variance, a paired-sample t-test was performed at a significance level of $0.05$ . The results of the test are reported in Table 9.

Table 9.

Paired-sample t-test between SDP-FNN and SDP-CNN for CWRU dataset.

$H_0$	$p$ Value	$95\%\mathrm{CI}$	$t$ -stats	$\mathrm{df}$	$\mathrm{Std}$
$1$	$0.0123$	$[0.049,0.366]$	$2.706$	$24$	$0.384$

CWRU: Case Western Reserve University; SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network; CI: confidence interval; std: standard deviation.

The results in Table 9 indicate that the null hypothesis can be rejected and that there is statistical evidence that the two techniques yield different performance. Indeed, the $p$ value is below the significance level of the test, and the confidence interval does not include zero, emphasizing that the mean difference is not zero and therefore that a systematic difference exists between the two techniques.

Table 10 reports the comparison in terms of real-time feasibility and embedded implementation.

Table 10.

Comparison between SDP-FNN and SDP-CNN for practical applicability for CWRU dataset.

Parameter	SDP-FNN	SDP-CNN
Available processing time (ms)	$100$	$100$
RAM usage (KB)	$\approx 93$	$\approx 99,840$
Maximum stack usage (KB)	$\approx 0.13$	$\approx 1228$
Estimated final firmware size (KB)	$\approx 262$	$\approx 102,400$
Estimated FLOPs	$\approx 0.4·{10}^7$	$\approx 7.8·{10}^7$
Estimated time (s)	$6.8·{10}^{-3}$	$47.0·{10}^{-3}$

CWRU: Case Western Reserve University; SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network.

Table 10 shows that the SDP-FNN has a lower computational cost than the SDP-CNN for all the adopted metrics, demonstrating the superiority of the new methodology compared to the classical one. Finally, it is highlighted that the SDP-CNN also has a computational cost compatible with a possible real-time implementation.

Comparison with other approaches

The SDP-FNN was also compared with other techniques as reported in Table 11.

Table 11.

Performance comparison with state-of-the-art techniques for CWRU dataset.

Metrics	GCT-GNN 37	TRA-ACGAN 38	DNN 39	SDP-FNN
Accuracy (%)	$100$	$99.7$	$98.2$	$99.60$
Signal pre-processing	WPD and EEMD	CWT	WT and features extraction	SDP, filtering and features calculation
Embedded feasibility	No	No	No	Yes

CWRU: Case Western Reserve University; SDP: symmetrized dot pattern; FNN: feedforward neural network; CWT: continuous wavelet transform; WPD: wavelet packet decomposition; EEMD: ensemble empirical mode decomposition.

In Table 11, the SDP-FNN results in a competitive method compared to the techniques considered in the study. Indeed, the SDP-FNN outperforms the deep neural network (DNN) in terms of accuracy, matches the transformer auxiliary classifier generative adversarial network (TRA-ACGAN) and is less accurate only to the Granger causality test graph neural network (GCT-GNN). While the other methods rely on more complex signal pre-processing techniques, the SDP-FNN employs the SDP transformation integrated with filtering operations and feature extraction. This strategy is computationally less complex but sufficiently effective to achieve comparable performance, simplifying the processing pipeline and reducing the computational burden during the feature extraction stage. ³⁰ Finally, the main advantage of the SDP-FNN lies in its feasibility for embedded implementations. Unlike other techniques that require high computational and memory resources, the SDP-FNN is designed to be executable on resource-constrained platforms. This represents a significant practical benefit in real-time industrial applications or on devices where efficiency and hardware compatibility are critical constraints.

Case study 2: HUST bearing dataset

The proposed technique was implemented also for HUST dataset. ⁴⁰ All results obtained for this dataset are shown below. In this study, the signals were analysed under stationary conditions. For consistency purposes, the raw signals were preliminarily pre-processed using a low-pass filter, resulting in an effective bandwidth comparable to that of the previous dataset. The same PC described in the third section was used in this study.

Dataset description

The HUST bearing dataset provides vibration data from five different kind of bearings. It includes normal working conditions, artificially localized (B, IR and OR) and combined defects (IR and OR, IR and B and OR and B) at three power conditions of electric motor ( $0$ , $200$ and $400\mathrm{W}$ ) at different constant speeds (from around $1350$ to $1500$  rpm). The vibration data are collected at $51.2\mathrm{kHz}$ .

Raw signals, SDP transformation and indices calculation

In Figure 10 are reported $0.1$  s of raw signals and the SDP diagrams for the N, B, IR and OR classes.

Figure 10.

Raw signals and SDP snowflake diagram for HUST dataset. SDP: symmetrized dot pattern.

The same considerations as those outlined in ‘Raw signals, SDP transformation and indices calculation’ section were made when choosing the parameters needed to transform a signal from the time domain into SDP coordinates. The parameters chosen are always $h=6$ , $l=5$ and $\xi =30$ .

The SDP coordinates were filtered, and the indices were calculated every three revolutions of the shaft on which the bearing is mounted. Figure 11 shows the calculated indices for raw and filtered SDP coordinates.

Figure 11.

Indices trend for HUST dataset.

As previously seen for the CWRU dataset, Figure 11 clearly shows how filtering helps to separate the curves and thus to separate the classes.

FNN development and performance evaluation

The indices were divided into training, validation and testing subsets to develop and optimize the FNN. Table 12 shows the dataset partition, the bearings considered, loads and speeds.

Table 12.

HUST dataset partition in training, validation and testing subsets.

Training and validation
Bering status	Bearing ID	Load (W)	Speed (rpm)
N	$6205,6206,6207,6208$	$200$	$1453,1453,1462,1453$
B	$6205,6206,6207,6208$	$200$	$1450,1444,1441,1444$
IR	$6205,6206,6207,6208$	$200$	$1447,1450,1450,1457$
OR	$6205,6206,6207,6208$	$200$	$1450,1453,1441,1444$
Testing
Bering status	Bearing ID	Load (W)	Speed (rpm)
N	$6205,6206,6207,6208$	$700$	$1379,1388,1370,1382$
B	$6205,6206,6207,6208$	$700$	$1371,1362,1370,1370$
IR	$6205,6206,6207,6208$	$700$	$1364,1370,1376,1370$
OR	$6205,6206,6207,6208$	$700$	$1373,1381,1358,1372$

The signals of training and validation phase in Table 12 were divided into 70% for training and 30% for validation. The FNN output class were N, B, IR and OR. The FNN optimized hyperparameters are shown in Table 13.

Table 13.

FNN optimized hyperparameter and architecture for HUST dataset.

Layer	Input size	Output size
Features input	$3$	$3$
Fully connected 1	$3$	$597$
ReLU	$597$	$597$
Fully connected 2	$597$	$865$
ReLU	$865$	$865$
Fully connected 3	$865$	$841$
ReLU	$841$	$841$
Fully connected 4	$841$	$594$
ReLU	$594$	$594$
Fully connected 5	$594$	$278$
ReLU	$278$	$278$
Fully connected 6	$278$	$4$
SoftMax	$4$	$4$
Hyperparameter	Value
Initialization learning rate	3.3 · 10−2
Momentum	0.98
L2 regularization	9.5 · 10−6
Mini batch size	76

FNN: feedforward neural network.

The test subset was divided into five further subsets, which were supplied separately to evaluate the FNN performance. The results are shown in Table 14.

Table 14.

FNN testing performance for HUST dataset.

Metrics	Mean value ± std (%)
Accuracy	$95.42\pm 0.85$
Precision	$95.86\pm 0.62$
Recall	$94.79\pm 1.22$
F1-Score	$95.07\pm 1.02$
AUC	$99.13\pm 0.23$

FNN: feedforward neural network; SD: standard deviation.

Table 14 demonstrates that the network has good generalization and diagnostic capabilities for detecting faults, although there is room for improvement in distinguishing between different types of faults.

Comparison between raw and filtered coordinates

Table 15 shows the comparison between the performance of the trained FNNs in the case of filtered and raw SDP coordinates.

Table 15.

Raw and filtered comparison for HUST dataset.

Metrics	Raw (%)	Filtered (%) (%) (%)	$\mathrm{\Delta }$ (%)
Accuracy	$68.29$	$95.42$	$-27.13$
Precision	$69.5$ 7	$95.86$	$-26.29$
Recall	$68.07$	$94.79$	$-26.72$
F1-Score	$66.91$	$95.07$	$-28.16$
AUC	$88.68$	$99.13$	$-10.45$

The difference of almost $27\%$ in accuracy confirms the need for filtering. Furthermore, Figure 12 demonstrates how filtering helps separate the classes.

Figure 12.

Class separation in the indices space for HUST dataset.

Ablation study

Table 16 shows the results of the Ablation study on the input features of the FNN, confirming that all three indices are necessary to achieve the best FNN performance. Furthermore, the results confirm that the DI is the most important.

Table 16.

Features ablation study results for HUST dataset.

Metrics	DI (%)	OI (%)	CI (%)	DI and OI (%)	DI and CI (%)	OI and CI (%)	DI, OI and CI (%)
Accuracy	$66.09$	$58.31$	$56.82$	$84.88$	$74.93$	$88.36$	$95.42$
Precision	$66.60$	$55.26$	$43.37$	$84.99$	$78.90$	$88.61$	$95.86$
Recall	$65.45$	$56.58$	$53.71$	$85.13$	$75.25$	$88.18$	$94.79$
F1-Score	$65.86$	$53.36$	$47.64$	$85.02$	$76.46$	$88.35$	$95.07$
AUC	$88.37$	$83.91$	$80.55$	$96.72$	$92.59$	$96.64$	$99.13$

DI: density index; OI: orientation index; CI: curvature index.

Real-time feasibility and embedded implementation considerations

Table 17 reports the computational cost of the SDP-FNN, demonstrating that it is possible to implement this approach for real-time FD also for this dataset.

Table 17.

Performance metrics of practical applicability for HUST dataset.

Perameter	Value
Available processing time (ms)	$123$
RAM usage (KB)	$\approx 7580$
Maximum stack usage (KB)	$\approx 25$
Estimated final firmware size (KB)	$\approx 8410$
Processing	Estimated FLOPs	Estimated time (s)
SDP transformation	$\approx 4.0·{10}^4$	$1.6·{10}^{-3}$
Filtering	$\approx 1.34·{10}^5$	$1.0·{10}^{-3}$
Indices	$\approx 3.3·{10}^5$	$6.4·{10}^{-4}$
FNN	$\approx 1.6·{10}^6$	$7.5·{10}^{-3}$
TOT	$\approx 2.1·{10}^6$	$1.1·{10}^{-2}$

FNN: feedforward neural network.

Comparison between new and classic approach

The comparison between the new and the classical approach reported in Table 18 confirms the superior performance of the former over the latter.

Table 18.

Comparison performance between SDP-FNN and SDP-CNN for HUST dataset.

Metrics	SDP-FNN (%)	SDP-CNN (%)	$\mathrm{\Delta }$ (%)
Accuracy	$95.42\pm 0.85$	$93.61\pm 1.00$	$+1.81$
Precision	$95.86\pm 0.62$	$93.81\pm 0.87$	$+2.05$
Recall	$94.79\pm 1.22$	$92.99\pm 1.48$	$+1.80$
F1-Score	$95.07\pm 1.02$	$93.22\pm 1.12$	$+1.85$
AUC	$99.13\pm 0.23$	$98.96\pm 0.25$	$+0.17$

SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network.

The paired-sample t-test at the $0.05$ significance level in Table 19 confirms that there is statistical evidence of the superiority of the new technique over the classical one.

Table 19.

Paired-sample t-test between SDP-FNN and SDP-CNN for HUST dataset.

$H_0$	$p$ Value	$95\%\mathrm{CI}$	$t$ -stats	$\mathrm{df}$	$\mathrm{Std}$
$1$	$1.294\times {10}^{-7}$	$[1.070,1.904]$	$7.374$	$24$	$1.008$

SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network; CI: confidence interval.

Table 20 shows that the SDP-FNN has a lower computational cost than the SDP-CNN.

Table 20.

Comparison between SDP-FNN and SDP-CNN for practical applicability for HUST dataset.

Parameter	SDP-FNN	SDP-CNN
Available processing time (ms)	$123$	$123$
RAM usage (KB)	$\approx 7580$	$\approx 605,400$
Maximum stack usage (KB)	$\approx 25$	$\approx 600$
Estimated final firmware size (KB)	$\approx 8410$	$\approx 659,100$
Estimated FLOPs	$\approx 0.2·{10}^7$	$\approx 7.8·{10}^7$
Estimated time (s)	$1.1·{10}^{-2}$	$4.1·{10}^{-2}$

SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network.

Comparison with other approaches

The comparison between SDP-FNN and other techniques is shown in Table 21.

Table 21.

Performance comparison with state-of-the-art techniques for HUST dataset.

Metrics	ACS-MSE 41	SFCDA 42	CNN-LSTM 11	SDP-FNN
Accuracy (%)	$97.05$	$100$	$97.63$	$95.42$
Signal pre-processing	Physical model	No	WT	SDP, filtering and features calculation
Embedded feasibility	No	No	No	Yes

SDP: symmetrized dot pattern; FNN: feedforward neural network; CNN: convolutional neural network.

The comparison in Table 21 highlights that the SDP-FNN is less accurate than the other techniques, but it requires less complex and computationally less-expensive methodologies than adaptive clutter separation and maximum spectral envelope (ACS-MSE) and sparse filtering cross-domain adaptation (SFCDA) for signal pre-processing. This comparison also emphasizes the main advantage of the SDP-FNN over the other techniques, namely the possibility of being implemented on embedded hardware for a real-time FD pipeline.

Conclusions

This article proposed a new approach to the SDP technique based on the development of novel indices for FD. The new pipeline consists of transforming vibration signals into SDP coordinates, filtering the coordinates to remove outliers and calculating new indices that describe the density, orientation and curvature of the clusters. The indices were used as inputs for an optimized FNN to automate the FD process. The SDP-FNN was validated on two different rolling bearing datasets, demonstrating high accuracy in defect classification, a low false positive rate and a low computational cost compatible with real-time implementation on embedded hardware. Additionally, studies were conducted to demonstrate the effectiveness of the filtering process. An ablation study was performed to demonstrate the importance of the three indices as inputs to the FNN.

Subsequently, the technique was compared with the classical SDP-CNN approach. The results showed that the SDP-FNN achieves higher predictive accuracy than the classical approach, a lower false positive rate and reduced computational cost. The SDP-FNN was also compared with other modern approaches, highlighting comparable accuracy with other techniques, lighter pre-processing and the possibility of real-time implementation.

The proposed approach necessarily requires filtering of the SDP coordinates to ensure the high performance demonstrated. Furthermore, the new technique requires tuning of the SDP parameters, similarly to the classical approach. Since no optimization criterion currently exists, this tuning still relies on the operator experience, and an inappropriate selection may greatly degrade the performance of both methods. Nevertheless, this drawback is not expected to hinder industrial application, as the tuning process must be performed only once during the initial calibration of the automated FD system. In the future, the effectiveness of the proposed new approach should focus on its evaluation under dynamic conditions, accuracy in the case of combined defects, at high noise level and evaluation on other systems such as gears or shafts. Future research should also aim at developing an automatic and reliable method for tuning the SDP parameters.