E-booster transmission fault diagnosis via feature fusion

Abstract

Considering the characteristics of an E-Booster, such as its short working time and sealed internal components, a multi-domain feature fusion method was proposed to diagnose the fault of E-Booster by analyzing the vibration signal. The root mean square(RMS), crest factor and kurtosis of the vibration signal were extracted as time domain features. Using short-time Fourier transform (STFT), the vibration signal was transformed into frequency-domain. The maximum energy in the spectrogram and the extremum of energy ridge were extracted as frequency-domain features. The vibration signal was resampled with equal angles, and the resampled signal was transformed by Fourier transform to obtain the order spectrum, and the order information was extracted by segmenting and taking the extreme values, serving as angular-domain features; then, kernel principal component analysis (KPCA) was used to fuse multi-domain features; finally, the support vector machine (SVM) and eXtreme Gradient Boosting(XGBoost) were used to classify the multi-domain feature fusion information of the samples. Experimental verification shows that the fault diagnosis accuracy rate under multi-domain feature fusion information is 99.04%. Results shows that multi-domain feature fusion information is better than that of single-domain feature information and multi-domain feature information on the fault diagnosis accuracy rate, and can effectively detect the E-Booster transmission faults online.

Keywords

fault diagnosis short time Fourier transform vibration signal order analysis information fusion kernel principal component analysis

Introduction

Amidst the shift toward intelligent and electric vehicle development, automotive braking systems are increasingly adopting Electro-Hydraulic Braking (EHB). Consequently, traditional vacuum boosters are being replaced by electric brake boosters (E-Boosters) as the core power-assist component.^1,2

The electric power-assist device comprises a brushless DC motor, a single-stage gear, a ball screw transmission mechanism, a pedal travel sensor, and an electronic control unit (ECU), as illustrated in Figure 1. When the brake pedal is pressed, the sensor detects pedal displacement and transmits this signal to the ECU. The ECU subsequently sends commands to the motor, which drives the gear assembly to actuate the ball screw and generate braking force. Gear defects such as tooth fracture or wear induce disturbances at meshing frequencies, resulting in abnormal vibration and noise.

Figure 1.

The structural diagram of the electric power-assist device.

Currently, transmission faults in domestically manufactured E-Boosters are primarily identified through manual auditory inspections by technicians or offline acoustic testing in anechoic chambers. These methods lack efficiency, objectivity, and scalability, compromising end-of-line quality assurance. As the critical power source for electro-hydraulic braking systems, E-Booster reliability directly impacts driver safety and property protection. Consequently, implementing 100% online fault diagnostics during final assembly - prior to factory release - is imperative to prevent safety hazards and property losses. However, most current studies on partial domain fault diagnosis^3–6 focus exclusively on single-source to single-target domain scenarios. To address category distribution shifts in cross-domain applications, Feng et al.⁷ proposed a global-local multi-source domain adaptation framework. Separately, Li and Yu⁸ designed a feature selection module that identifies transferable features from both global and local representations across source domains, aiming to eliminate negative transfer caused by irrelevant information while mitigating partial domain-induced transfer interference.

Advances in vibration signal processing have driven the increasing adoption of vibration-based analysis for fault diagnosis across diverse engineering fields, including aero-engines,⁹ cable-stayed bridges,¹⁰ and engineering structures.¹¹ However, extending these methods to sealed electromechanical systems like E-Boosters faces two fundamental limitations: (1) Encapsulation prevents tachometer installation, eliminating conventional order tracking due to signal acquisition constraints; (2) Transient dynamics from ultra-short operational windows (<10% of standard bearing test cycles) violate the stationarity assumptions underpinning Fourier-based analysis. To overcome these constraints, we propose a multi-domain feature fusion framework that reconstructs angular signals via displacement-guided resampling while integrating time-frequency-order features to characterize transient fault signatures.

Venkata and Rao proposed an SVM-based vibration classification method for flow control valve fault diagnosis.¹² Zhang et al. developed a probabilistic neural network (PNN) approach for vehicle gearbox fault diagnosis using frequency-domain features from gear vibration signals.¹³ To address weak early-fault signals with obscure features in high-noise environments, Cao et al. employed an improved wavelet threshold function for denoising, successfully extracting rolling bearing characteristic frequencies.¹⁴ Concurrently, Gao et al. introduced a time-frequency squeezing method for rotation frequency calculation, enabling bearing fault diagnosis through time-frequency compression and order resampling,¹⁵ while Wang and Chu established a diagnostic approach via fault characteristic order (FCO) index demodulation and stepwise resampling in the fault phase angle (FPA) domain.¹⁶ Wang et al.¹⁷ developed the LFIgram method, constructing Log Envelope Auto-correlation Bispectrum (LEAB) slices and the LEAB Feature Index (LFI) to achieve precise compound fault identification in rolling bearings. Complementarily, Liu et al.¹⁸ proposed ISEANet – an Interpretable Subdomain Enhanced Adaptive Network – which improves cross-domain bearing diagnosis under noise-shift conditions through: sparse spectral noise reduction (SSNR), local multi-feature extraction modules (LMFEMod), and an information-theoretic discrepancy metric (ILMMD), demonstrating superior transfer learning performance.

Unlike conventional rotational or linear-motion components (e.g. engines,⁹ bearings,¹⁹ motors²⁰) that permit continuous vibration signal acquisition across speeds with favorable SNR—enabling single-domain fault diagnosis—the E-Booster presents unique challenges. During braking, it converts rotational to linear motion via a ball screw for force output, but its limited stroke restricts operation to transient cycles, preventing extended vibration monitoring. Concurrently, hybrid rotational-linear motion causes fault information overlap, rendering single-domain features inadequate for comprehensive condition assessment. To address this, we extract time-domain features, time-frequency features, and order-domain features to form an eight-dimensional feature vector. Kernel PCA fuses these into a four-dimensional representation, enabling effective binary classification of component health (normal vs faulty) via SVM and XGBoost classifiers.

Vibration signal acquisition

Given the E-Booster’s gear-rolling bearing transmission architecture, a box measurement approach is employed to effectively identify transmission mechanism faults. To preserve clamping integrity while enabling rapid sensor deployment, the test fixture utilizes a slide table cylinder (8) to position the vibration testing assembly, with a test cylinder (9) driving the vibration sensor (10) into contact with the E-Booster casing surface.²¹ This non-invasive configuration ensures unimpeded brake function while securing measurement stability, as illustrated in Figure 2.

Figure 2.

E-booster clamping tooling diagram. 1-hydraulic cylinder, 2-vibration pad, 3-hydraulic cylinder mounting plate, 4-product clamping block, 5-clamping block slider, 6-product positioning block, 7-product support plate, 8-slide table cylinder, 9-test cylinder, 10-vibration sensor, 11-E-booster test subject.

Owing to the E-Booster’s sealed integrated electronic servo design, direct rotational speed measurement is infeasible. To circumvent this limitation, a displacement sensor is installed at the output push rod to capture linear motion kinematics. For production-line compatibility, hydraulic circuit delays are mitigated through a shared master cylinder system requiring only single oil injection during product changeovers, coupled with an automated docking mechanism for rapid master cylinder engagement. Vehicle operational realism is maintained via a two-stage spring-coupling simulated loading device adaptable across eBooster variants. Complementary mechanical fixturing enables instantaneous product clamping and sensor deployment. Electrically, a brake-by-wire DMA acquisition system employing NI-4462 (data acquisition) and NI-8512 (CAN communication) cards achieves high-fidelity vibration capture, while NI high-speed hardware (10 kHz sampling) ensures real-time processing. The integrated test bench is shown in Figure 3.

Figure 3.

Physical diagram of E-booster transmission fault diagnosis test bench.

As evidenced in Figure 4(a) and (b), E-Booster vibration and displacement signals exhibit three distinct operational phases: (1) motor braking (outbound phase), (2) motor release (return phase), and (3) an intermediate stop phase. Significant disparities in vibration characteristics and intensity between these phases preclude joint analysis. Crucially, stop-phase vibrations constitute environmental noise with negligible diagnostic value. Consequently, raw signals undergo preprocessing to isolate outbound and return phase vibrations, as demonstrated in Figure 5(a) and (b), ensuring subsequent analysis focuses exclusively on operationally relevant signatures.

Figure 4.

Vibration signal and displacement signal. (a) Full vibration signal image. (b) Full displacement signal image.

Figure 5.

Preprocessed vibration signal. (a) outbound phase. (b) return phase.

Multi-domain feature extraction

The E-Booster’s transient power assistance duration during braking precludes extended vibration signal acquisition, while inherent non-stationarity and nonlinearity further complicate analysis. Single-domain feature diagnosis consequently yields insufficient accuracy. To address this, our methodology extracts time-domain features before progressively transforming signals: short-time Fourier conversion enables time-frequency feature extraction, while order tracking generates angular-domain signatures. Ultimately, multi-domain features are synthesized by integrating these complementary representations.

Time domain feature extraction

Time-domain E-Booster vibration signals contain fundamental features reflecting operational health. RMS quantifies overall vibration energy intensity, while Crest Factor detects abnormal transient impacts, and Kurtosis identifies subtle faults like gear surface damage by characterizing signal impulsivity. These diagnostically critical parameters are mathematically defined in Table 1.

Table 1.

Statistical feature index calculation formula.

Term	Definition
RMS	$RMS = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {x_{i}}^{2}}$
Crest factor	$C = \frac{{\| x_{i} \|}_{\max}}{RMS}$
Kurtosis	$K = \frac{1}{N} \sum_{i = 1}^{n} (\frac{x_{i} - \bar{x}}{σ})$

In which, $N$ represents the number of test points; $x_{i}$ represents the vibration signal collected at each point, $\bar{x}$ and $σ$ represent the mean and standard deviation of $x_{i}$ , respectively. The RMS, C, and K were employed as time-domain features for analyzing the vibration signals. The distributions of these time-domain features across 30 normal components and 30 faulty components are illustrated in Figure 6. As shown in Figure 6, there exists a distinct difference in the distribution of time-domain feature values between normal and faulty components: the feature values of faulty components are generally larger, whereas those of normal components tend to be smaller. However, the value ranges of the two groups exhibit substantial overlap, resulting in limited discriminative capability and rendering it infeasible to differentiate between normal and faulty components via simple threshold segmentation.

Figure 6.

Time domain feature distribution. (a) RMS distribution. (b) Crest factor distribution. (c) Kurtosis distribution.

Time-frequency domain feature extraction

In response to the non-stationary and nonlinear characteristics of the E-Booster’s vibration signals, this paper employs the STFT for time-frequency feature analysis: First, the vibration signal $x (t)$ is multiplied by a window function $w (t - τ)$ centered at $τ$ , dividing the vibration signal into several small segments, then applying FFT transformation, with the calculation formula as shown in equation (1). Finally, the spectra of each segment are stitched together to obtain a spectrogram that reflects both temporal and frequency domain information.

X (t, f) = \int_{- \infty}^{\infty} w (t - τ) x (t) e^{- j 2 π f τ} d τ

(1)

During the implementation of the STFT, critical factors affecting the resulting spectrogram include the type of window function, window size, window overlap ratio, and the volume of data subjected to Fourier transform. The window width directly influences both the time resolution and frequency resolution of the STFT. Specifically, time resolution T and frequency resolution F are inversely and directly proportional to the window width, respectively. The computational formulas for time resolution T and frequency resolution F are presented in equations (2) and (3).

T = [\frac{N_{x} - N_{0}}{N - N_{0}}]

(2)

F = {\begin{matrix} \frac{N_{f}}{2} + 1, N_{f} is even \\ \frac{N_{f} + 1}{2}, N_{f} is odd \end{matrix}

(3)

In the formula, $N_{x}$ represents the width of the data involved in the STFT, $N$ represents the width of window, $N_{0}$ represents the window overlap width, and $N_{f}$ represents the amount of data undergoing Fourier transformation. [•] indicates rounding downwards. According to the definition of image resolution, the resolution of the spectrogram is defined as in equation (4).

P = T \times F

(4)

To find the optimal window width and overlapping window width using an exhaustive search method, the optimality criteria are as follows: The 214 Hz maximum energy ridge data of normal samples should exhibit good clustering ( $σ_{1}$ low variance), the 107 Hz maximum energy ridge of faulty samples should also show good clustering ( $σ_{2}$ low variance), and the 107 Hz maximum energy ridge data between normal and faulty samples should have poor clustering ( $σ_{ab}$ high variance). The optimal window width and overlapping window width are determined based on maximizing these criteria $σ_{ab} = σ_{12} / (σ_{1} + σ_{2})$ . $σ_{12}$ represents the statistical variance value between the maximum energy ridge at 107 Hz for normal components and the maximum energy ridge at 107 Hz for faulty components. Considering the significant frequency variations in the vibration data, the initial window width is set to 200, and the final window width is set to 1000.

After multiple cross-validation experiments, the Hanning window was selected, with a window width of 550 and an overlap of 370, yielding a relatively optimal STFT spectrogram (Figure 7).

Figure 7.

Time spectrum of STFT. (a) Time spectrum of normal products. (b) Time spectrum of faulty products.

In Figure 7, the horizontal axis denotes time, the vertical axis denotes frequency, and the color scale represents energy values. From the frequency perspective, both Figure 7(a) and (b) exhibit an energy ridge near 107 Hz, with the intensity of this energy ridge being higher for the faulty component than for the normal one. An overall analysis of the spectrograms reveals that the maximum energy of the faulty component is significantly greater than that of the normal component. To differentiate the time-frequency signals of normal and faulty components, two features are extracted from the time-frequency domain of the vibration signal: the maximum energy in the spectrogram and the maximum energy on the energy ridge (at 107 Hz).

Angular-frequency domain feature extraction

Although the STFT enables the analysis of non-stationary signals and the extraction of their instantaneous frequencies, it is constrained by the uncertainty principle, failing to achieve high resolution in both the time and frequency domains simultaneously. To mitigate this limitation, this paper adopts order analysis to extract angular frequency domain features from the vibration signals. When the rotational frequency of the drive shaft is defined as the fundamental frequency, the order is physically interpreted as the number of vibrations occurring per rotation of the drive wheel. Mathematically, this equals the ratio of vibration count to rotation count—that is, the ratio of vibration frequency to rotational frequency—as presented in equation (5).

O = \frac{f}{f_{r}}

(5)

In the formula, $O$ represents the order of the analyzed rotating body, $f$ represents the vibration frequency, $f_{r}$ represents the rotational frequency.

Angular resampling of vibration signals

Order analysis relies on angular frequency domain analysis and necessitates equiangular sampling of vibration signals. The E-Booster is a sealed, integrated electronic servo power steering system, precluding the direct measurement of rotational speed signals. Furthermore, the motion of its internal mechanisms is non-stationary, which prevents direct equiangular sampling in the time domain.

However, the E-Booster converts rotational motion to linear motion via a ball screw, with a linear relationship existing between the rotation angle of the internal gear and the output displacement. Sampling the vibration signal at fixed displacement intervals enables equiangular sampling. The specific process is as follows: First, the time-displacement signal is fitted to yield a continuous time-displacement signal. Next, time is sampled at specific displacement intervals to generate a phase demodulation moment sequence. Subsequently, the vibration data corresponding to this phase demodulation moment sequence is extracted, and any missing data at specific moments is interpolated to acquire equiangular vibration signals. Finally, a Fourier transform is performed to derive the characteristic order spectrum of the vibration signal, as illustrated in the flowchart in Figure 8.

Figure 8.

Vibration signal angle resampling flowchart.

In this paper, with a sampling interval of 0.1 ms, within this extremely short time frame, the acceleration of the output plunger can be assumed constant (uniform acceleration), and the output displacement of the E-Booster is a quadratic relationship with time. The fitting formula is shown in equation (6).

S_{i} (t) = a_{i} + b_{i} t + c_{i} t^{2}

(6)

In which, $S_{i} (t)$ represents the function of the segment displacement curve, $i$ represents the serial number of the segment displacement, $a_{i}$ , $b_{i}$ , $c_{i}$ are the polynomial coefficients obtained from the piecewise fitting. Verification shows that the coefficient of determination reaches 0.999316, satisfying the precision requirement.

The E-Booster converts rotational motion to linear motion via a ball screw, and its output displacement is directly proportional to the cumulative rotations of the internal rotating component, where the proportionality constant corresponds to the gear ratio of the ball screw. The relationship between the output displacement and the resampled angular interval is given by:

2 π L S_{i} (t_{n}) = n Δ θ

(7)

In which, $L$ represents the gear ratio of ball screw, $n$ represents time index of resampling, $Δ θ$ represents interval of equiangular sampling, $t_{n}$ is the phase demodulation moment sequence when the time index is $n$ . Substituting equation (7) into equation (6), the result is shown in equation (8).

2 π L (a_{i} + b_{i} t_{n} + c_{i} t_{n}^{2}) = n Δ θ

(8)

Solving the above equation yields the phase demodulation moment for equiangular resampling.

For the E-Booster, the motor drives the pinion gear (16 teeth), which in turn actuates the large gear (71 teeth); the large gear then drives the ball screw. With the motor’s rotational speed defined as the fundamental frequency, each rotation of the motor yields 16 gear meshes, such that the meshing frequency corresponds to the 16th order. Since the vibration energy of the E-Booster primarily originates from gear meshing, the vibration energy at the 16th order is comparatively high.

The vibration signal of the E-Booster comprises discrete data points. When resampling the signal based on phase demodulation moments, missing values arise, requiring interpolation. To avoid the Runge phenomenon that may occur in polynomial interpolation at equidistant nodes, this paper employs Lagrange linear interpolation to obtain a steady-state, equiangularly resampled vibration signal.

Order spectrum analysis

After performing Fourier transform on the vibration signals resampled at equiangular intervals, an order spectrum is derived. The order spectrum of normal components is presented in Figure 9, which exhibits three local extrema. Analysis of the order spectra of 30 normal components reveals that these three local extrema are predominantly distributed around the 16th, 32nd, and 64th orders.

Figure 9.

Order spectrum of normal products.

The order spectrum of faulty components is illustrated in Figure 10, which also contains local extrema with magnitudes greater than those of normal components. Since a fault introduces new order components into the signal, the orders corresponding to the local extrema exhibit a shift. In Figure 10, the order of the second local extremum has shifted from the original 32nd order to approximately the 41st order. Based on this analysis, this paper adopts a segmented approach to extract order features: extremum extraction is performed for three order ranges (15th–25th, 25th to 50th, and 50th to 70th orders), and these three local extrema are employed as angular frequency domain features for fault diagnosis.

Figure 10.

Order spectrum of the faulty product.

Multi-domain feature fusion

Combining the features of time domain, time-frequency domain and angular-frequency domain into a multi-domain feature information vector $x = {(x_{1}, x_{2}, \dots, x_{8})}^{T}$ , $x_{1} ~ x_{8}$ are defined in Table 2.

Table 2.

Multi-domain feature list.

Variable	Domain	Definition	Variable	Domain	Definition
$x_{1}$	Time-domain	RMS	$x_{5}$	Time-frequency	Max energy of energy ridge
$x_{2}$	Time-domain	Crest factor	$x_{6}$	Angular-frequency	Max of 15th–25th order
$x_{3}$	Time-domain	Kurtosis	$x_{7}$	Angular-frequency	Max of 25th–50th order
$x_{4}$	Frequency-domain	Max energy of spectrum	$x_{8}$	Angular-frequency	Max of 50th–70th order

To address redundancy and conflicts among features, KPCA is utilized for fusing multi-domain features, thereby reducing the complexity of data processing and minimizing information redundancy.^22,23 For nonlinear fault characteristics, KPCA yields higher accuracy than Principal Component Analysis (PCA).²⁴

The fundamental steps for multi-domain feature fusion are as follows.

1) By applying a kernel function, the feature sample set of $n$ individual samples $X = {x_{1}, x_{2}, \dots, x_{n}}$ is transformed from the input space to a new feature space to obtain a matrix $K = {[K_{ij}]}_{n \times n}$ , with the calculation of elements $k_{ij}$ as shown in equation (9).

k_{ij} = (Φ (x_{i}) • Φ (x_{j})) = G (x_{i}, x_{j})

(9)

In the formula, $x_{i} \in X$ , $x_{j} \in X$ , $Φ (•)$ represent non-linear transformations, $G (•, •)$ represents kernel function. Common kernel functions include the linear kernel, polynomial kernel, Gaussian radial basis function (RBF) kernel, and multilayer perceptron kernel.²⁵ This paper employs grid search for hyperparameter optimization, selecting the polynomial (poly) kernel function, as shown in equation (10).

G (x_{i}, x_{j}) = {(a (x_{i}^{T} * x_{j}) + b)}^{d}

(10)

In the equation, $a$ is kernel coefficient, $b$ is independent term (coef0), $d$ is Polynomial degree. Through grid search-based hyperparameter optimization, parameters $a$ , $b$ , and $d$ are set to 0.06, 2.5, and 3, respectively.

2) Solve the Characteristic equation of the eigenspace matrix $K$ , as shown in equation (11).

\frac{1}{n} K α = λ α

(11)

The obtained eigenvalues $λ_{1} \geq λ_{2} \geq \cdot \cdot \cdot \geq λ_{n}$ and their corresponding eigenvectors $α_{1}, α_{2}, \dots, α_{n}$ . In which, $α^{l} = [a_{1}^{l}, a_{2}^{l}, \dots, a_{n}^{l}]$ . The l-th kernel principal component is calculated as shown in equation (12).

v^{l} = \sum_{i = 1}^{n} α_{i}^{l} Φ (x_{i})

(12)

3) Compute the projection of the samples onto the nonlinear principal components. The projection of a sample $x_{j}$ onto the l-th nonlinear principal component is calculated as shown in equation (13).

Z^{l} (x_{j}) = (v^{l} • Φ (x_{j})) = \sum_{i = 1}^{n} α_{i}^{l} G (x_{i}, x_{j})

(13)

4) Calculate the contribution rate of the kernel principal components to determine the number of kernel principal components to retain. The contribution rate of the k-th kernel principal component is calculated as shown in equation (14).

C o n t r (λ_{k}) = \frac{λ_{k}}{\sum_{i = 1}^{n} λ_{i}} \times 100 %

(14)

The number of retained principal components is also determined via grid search-based hyperparameter optimization. Results indicate that selecting four components yields the optimal performance and can effectively eliminate redundant features. The projection of the sample $x_{j}$ in the new feature space onto the kernel principal components is $[z^{1} (x_{j}), z^{2} (x_{j}), z^{3} (x_{j}), z^{4} (x_{j})]$ .

Experimental validation

To validate the efficacy of multi-domain feature fusion for E-Booster transmission fault diagnosis, this study employs SVM and XGBoost classifiers—both established methods in fault diagnosis.^26–30 While time-domain feature extraction, STFT time-frequency analysis, and angular-domain order analysis revealed distinctive fault characteristics, these techniques cannot directly evaluate component health. For production-line binary classification (normal vs faulty units), pattern recognition offers an optimal solution. SVM demonstrates particular effectiveness in mechanical fault contexts,³¹ outperforming ANN and PCA classifiers especially with limited, imbalanced training data—a critical advantage given the scarcity of fault samples (significantly outnumbered by normal units). Its algorithmic efficiency enables robust pattern recognition even with small datasets and high-dimensional nonlinear inputs. Complementarily, XGBoost provides computational advantages as the fastest open-source boosting toolkit (>10× acceleration over conventional methods) with industrial-scale capability through distributed operation across Kubernetes, Hadoop, and related environments. Hyperparameters for both classifiers were optimized via grid search, with final configurations detailed in Table 3.

Table 3.

Configurations of hyperparameters.

Model	Parameter	Definition	Value
SVM	kernel	Maps data to higher-dimensional space to enable linear separation	Poly
	gamma	Scales input similarity for non-linear kernels	0.1
	coef0	Adjusts polynomial complexity	2.0
	C	Controls overfitting tolerance	1000
	degree	Adjusts polynomial complexity	2
XGBoost	n_estimators	Number of boosting rounds/trees in the ensemble	200
	max_depth	Maximum depth per tree	7
	learning_rate	Step size shrinkage for weight updates	0.01
	subsample	Fraction of training data sampled per tree	1.0
	colsample_bytree	Fraction of features sampled per tree	1.0

Classifier performance was rigorously evaluated using accuracy, error rate, and critically prioritized sensitivity metrics, with formal definitions provided in Table 4. Within this framework, True Positives (TP) represent correct identifications of normal components, True Negatives (TN) denote accurate detections of faulty units, while False Positives (FP) constitute dangerous misclassifications where defective components are erroneously passed as normal - representing the highest-risk error category in automotive safety systems. Conversely, False Negatives (FN) indicate conservative misclassifications where functional components are incorrectly flagged as faulty.

Table 4.

List of evaluation indicators.

Evaluation metrics	Definition
Accuracy	(TP + TN)/(TP + TN + FP + FN)
Error rate	(FP + FN)/(TP + TN + FP + FN)
Sensitivity	TP/(TP + FN)
Specificity	TN/(TN + FP)
Precision	TP/(TP + FP)

A balanced dataset comprising 416 E-Booster components was utilized, with 208 normal and 208 faulty units allocated for training, and an additional 52 normal and 52 faulty units reserved for testing. Normal components were labeled as ‘1’ while faulty components received ‘0’ classification. Both SVM and XGBoost classifiers were systematically applied to three distinct feature configurations: single-domain features, multi-domain features, and KPCA-fused multi-domain features. This comparative assessment evaluated diagnostic performance across progressively sophisticated feature representations, with comprehensive results for SVM and XGBoost presented in Tables 5 and 6 respectively. The experimental design specifically addressed real-world constraints through intentional training data imbalance while maintaining balanced test sets for unbiased metric evaluation.

Table 5.

Fault diagnosis results under different characteristic information (SVM).

Feature information	Accuracy	Error rate	Sensitivity	Specificity	Precision
Time domain feature information	93.27%	6.73%	100%	86.54%	88.14%
Time-frequency domain feature information	77.88%	22.12%	80.77%	75.00%	76.36%
Angular-frequency domain feature information	75.96%	24.04%	96.15%	55.77%	68.49%
Multi-domain feature information	93.27%	6.73%	96.15%	90.38%	90.91%
Multi-domain feature fusion information	98.08%	1.92%	100%	96.15%	96.30%

Table 6.

Fault diagnosis results under different characteristic information (XGBoost).

Feature information	Accuracy	Error rate	Sensitivity	Specificity	Precision
Time domain feature information	96.15%	3.85%	100%	92.31%	92.86%
Time-frequency domain feature information	83.65%	16.35%	90.38%	s76.92%	79.66%
Angular-frequency domain feature information	81.73%	18.27%	80.77%	82.69%	82.35%
Multi-domain feature information	97.12%	2.88%	98.08%	96.15%	96.23%
Multi-domain feature fusion information	99.04%	0.96%	100%	98.08%	98.11%

Tables 5 and 6 demonstrate that both SVM and XGBoost classifiers achieve sub-95% diagnostic accuracy when processing any single-domain feature set (time, time-frequency, or angular domain) in isolation. However, integrating these features into multi-domain representations significantly enhances all evaluation metrics, with KPCA fusion proving particularly impactful. By eliminating feature redundancies while preserving discriminative information, the KPCA-reduced 4-dimensional feature space elevates diagnostic accuracy to 99.04%—surpassing both single-domain and unfused multi-domain approaches. This 4+ percentage point improvement validates feature fusion as essential for reliable E-Booster fault detection, especially critical given the safety implications of braking system failures.

Conclusion

In response to the E-Booster’s ultra-transient operation and sealed architecture—which preclude extended vibration monitoring and direct rotational sensing—conventional single-domain feature analysis yields unacceptably low diagnostic accuracy. Our multi-domain feature fusion methodology overcomes these constraints by synergistically integrating time, time-frequency, and angular-domain characteristics, enabling 99.04% accurate online detection of transmission faults. This approach meets stringent industrial testing requirements while preventing safety-critical components with latent defects from entering automotive supply chains.

The study acknowledges limitations in sample diversity and volume, which influence model generalizability. Future work will expand datasets to encompass varied failure modes while algorithmically exploring deep architectures like temporal transformers or 3D-CNNs that may enhance diagnostic granularity given sufficient training data.

Notably, this framework demonstrates unique efficacy for electromechanical systems exhibiting coupled rotational-linear motion, with real-time capability suitable for high-throughput production environments. Its principles show significant promise for fault diagnosis in more complex industrial scenarios including robotic drivetrains, aerospace actuators, and precision mechatronic systems where multi-physics signal interactions challenge conventional monitoring approaches.

Footnotes

Acknowledgements

Thank you to every teacher for their guidance and resource provision, and thank you to all funding units

ORCID iD

Weijun Fan

Ethical considerations

This study did not involve any human or animal participants.

Consent to participate

Informed consent was obtained from all participants in written or electronic form after the nature of the study had been fully explained.

Author contributions

Xixiang Wu conducted the experiments, analyzed the data, and wrote the manuscript. Weijun Fan supervised the research and guided the writing. Bin Guo provided the experimental equipment.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: National Natural Science Foundation of China (52075511); Zhejiang Province Science and Technology Plan Project Provincial Key R&D Plan (2021C01136); Zhejiang Province Public Welfare Technology Application Research Program (LGG21E050019).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available on reasonable request.

References

Liu Yu

Zheng

, et al. Research on redundant anti-lock braking algorithm based on e-booster. Automot Eng 2022; 44(1): 82–93.

Zhao

Deng

Zhu

, et al. Sliding mode control based on RBF network for hydraulic pressure in electric power-assisted Brake System. J Mech Eng 2020; 56(24): 106–114.

Qin

Qian

Wang

, et al. Adaptive manifold partial domain adaptation for fault transfer diagnosis of rotating machinery. Eng Appl Artif Intell 2023; 126(C): 107082.

Huang

Wen

Han

, et al. Intelligent fault identification for industrial internet of things via prototype-guided partial domain adaptation with momentum weight. IEEE Internet Things J 2023; 10(18): 16381–16391.

Dong

, et al. Instance weighting-based partial domain adaptation for intelligent fault diagnosis of rotating machinery. IEEE Trans Instrum Meas 2023; 72: 1–14.

Wang

She

Shi

, et al. Partial adversarial domain adaptation by dual-domain alignment for fault diagnosis of rotating machines. ISA Trans 2023; 136: 455–467.

Feng

Chen

, et al. Globally localized multisource domain adaptation for cross-domain fault diagnosis with category shift. IEEE Trans Neural Netw Learn Syst 2023; 34(6): 3082–3096.

A multisource domain adaptation network for process fault diagnosis under different working conditions. IEEE Trans Ind Electron 2023; 70(6): 6272–6283.

Tang

Wei

, et al. Fault diagnosis for aero-engine accessory gearbox by adaptive graph convolutional networks under intense background noise conditions. Chin J Sci Instrum 2021; 41(8): 78–86.

10.

Gao

Wang

, et al. Performance enhancement of the viscous damper combined with the negative stiffness device for cable vibration control. J Vib Eng 2022; 35(2): 255–263.

11.

Liu

LC.

Adoption of vibration response covariance and data fusion for damage identification. J Vib Eng 2021; 34(1): 141–149.

12.

Venkata

Rao

Fault detection of a flow control valve using vibration analysis and support vector machine. Electronics 2019; 8(10): 1062.

13.

Zhang

Jia

, et al. Application of probabilistic neural network to typical fault diagnosis of vehicle gearbox. Automot Eng 2020; 42(7): 972–977.

14.

Cao

Peng

, et al. Rolling bearing fault diagnosis method based on improved wavelet threshold denoising. J Vib Eng 2022; 35(2): 454–463.

15.

Gao

Huang

Fault detection method for varying rotating speed bearings based on time-frequency squeeze and order analysis. J Vib Shock 2020; 39(3): 205–210. +226.

16.

Wang

Chu

Bearing fault diagnosis under time-varying rotational speed via the fault characteristic order (FCO) index based demodulation and the stepwise resampling in the fault phase angle (FPA) domain. ISA Trans 2019; 94(2019): 391–400.

17.

Wang

Yan

Liu

, et al. The LFIgram: A targeted method of optimal demodulation band selection for compound faults diagnosis of rolling bearing. IEEE Sens J 2024; 24(5): 6687–6699.

18.

Liu

Yan

Liu

, et al. ISEANet: an interpretable subdomain enhanced adaptive network for unsupervised cross-domain fault diagnosis of rolling bearing. Adv Eng Inform 2024; 62(A): 102610.

19.

Zhang

Feng

Huang

, et al. Rolling bearing state recognition under variable condition using part-based representation of nonnegativity constrained autoencoder networks. Chin J Sci Instrum 2020; 41(4): 77–85.

20.

Gong

Chen

Zhang

, et al. Intelligent diagnosis method for incipient fault of motor bearing based on deep learning. Chin J Sci Instrum 2020; 41(1): 195–205.

21.

Fan

Zhai

, et al. Design of on-line vibration detection system for transmission faults in electric booster. Instrum Tech Sens 2020; 3: 78–83.

22.

Neffati

Ben Abdellafou

Taouali

, et al. Enhanced SVM–KPCA method for brain MR image classification. Comput J 2020; 63(3): 383–394.

23.

Qin

Zhang

, et al. Fault diagnosis based on weighted extreme learning machine with wavelet packet decomposition and KPCA. IEEE Sens J 2018; 18(20): 8472–8483.

24.

Sun

Yuan

Wang

. Research on fault diagnosis of energy storage power stations based on the KPCA method. Auto Appl 2025; 66(7): 141–143.

25.

Yang

Zhuge

, et al. DNA clustering algorithm for tea plants based on machine learning. J Guangxi Univ 2024; 49(2): 386–399.

26.

Shao

Tan

, et al. Fault diagnosis for rolling bearings based on composite multiscale fine-sorted dispersion entropy and SVM with hybrid mutation SCA-HHO algorithm optimization. IEEE Access 2020; 8(99): 13086–13104.

27.

Panda

Rapur

Tiwari

Prediction of flow blockages and impending cavitation in centrifugal pumps using support vector machine (SVM) algorithms based on vibration measurements. Measurement 2018; 130(2018): 44–56.

28.

Dong

Zheng

Huang

, et al. Time-shift multi-scale weighted permutation entropy and GWO-SVM based fault diagnosis approach for rolling bearing. Entropy 2019; 21(6): 621.

29.

Trizoglou

Liu

Lin

Fault detection by an ensemble framework of extreme gradient boosting (XGBoost) in the operation of offshore wind turbines. Renew Energy 2021; 179: 945–962.

30.

Xie

Zhou

, et al. A novel bearing fault classification method based on XGBoost: the fusion of deep learning-based features and empirical features. IEEE Trans Instrum Meas 2021; 70: 1–9.

31.

Pan

Yan

Zheng

, et al. Fatigue detection method for UAV remote pilot based on multi feature fusion. Electron Res Arch 2023; 31(1): 442–466.