In modern industrial automation, the rotary vector reducer plays a critical role as a core component in robotic systems, offering high transmission ratios, precision, and efficiency. However, under strong background noise, fault diagnosis for the rotary vector reducer becomes challenging due to the masking of useful vibration signals. I propose a novel fault diagnosis method based on Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) combined with Crested Porcupine Optimizer (CPO) to optimize Variational Mode Decomposition (VMD). This approach aims to effectively extract fault features from noisy signals, enabling accurate diagnosis of faults such as bearing failures in rotary vector reducers. Throughout this article, I will detail the theoretical foundations, methodology, experimental validation, and comparative analyses, emphasizing the importance of the rotary vector reducer in industrial applications.
The rotary vector reducer is widely used in high-precision industries, but its complex internal structure often leads to vibration signals contaminated by noise. Traditional methods like Fast Fourier Transform or envelope spectrum analysis struggle to locate fault points under such conditions. Therefore, I explore advanced signal processing techniques to enhance diagnosis accuracy. My method leverages CEEMDAN for initial noise reduction, followed by CPO-optimized VMD for precise decomposition, and finally envelope spectrum analysis for fault identification. This integrated approach ensures robust performance even in noisy environments, making it suitable for real-world applications involving rotary vector reducers.
To begin, I outline the theoretical basis of the methods used. CEEMDAN is an improved version of Empirical Mode Decomposition (EMD) that addresses mode mixing and residual noise issues. For a given signal \(x[n]\), the CEEMDAN process involves multiple trials with added noise to obtain intrinsic mode functions (IMFs). The first modal component is computed as:
$$M_1[n] = \frac{1}{I} \sum_{i=1}^{I} m_1^i[n] = \overline{m_1[n]}$$
where \(I\) is the number of trials, and \(\varepsilon_0 \omega_i(n)\) represents added noise. Subsequent residuals and modals are derived iteratively. For instance, the \(k\)-th residual is:
$$r_k[n] = r_{k-1}[n] – M_k[n]$$
and the final signal reconstruction is:
$$x[n] = \sum_{k=1}^{K} M_k + R[n]$$
This ensures a complete decomposition, which is crucial for denoising signals from rotary vector reducers. Next, VMD is a variational method that decomposes a signal into \(K\) IMFs by minimizing the total bandwidth. The variational model is:
$$\min_{\{u_k\},\{\omega_k\}} \left\{ \sum_k \left\| \delta_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_k(t) \right] e^{-j\omega_k t} \right\|_2^2 \right\} \quad \text{s.t.} \quad \sum_k u_k = f$$
where \(u_k\) are the IMFs, \(\omega_k\) are center frequencies, and \(f\) is the original signal. However, VMD requires optimal parameters, specifically the number of modes \(K\) and the penalty factor \(\alpha\), to avoid over-decomposition or under-decomposition. To address this, I employ the Crested Porcupine Optimizer (CPO), a metaheuristic algorithm inspired by the defense behaviors of crested porcupines. CPO uses strategies like visual and sound defenses for exploration, and scent and physical attacks for exploitation. The position update in CPO for a solution \(x_i^t\) at iteration \(t\) is given by:
$$x_i^{t+1} = x_i^t + \tau_1 \times |2 \times \tau_2 \times x_{cp}^t – y_i^t|$$
where \(x_{cp}\) is the best position, \(y_i^t\) is a vector between current and random solutions, and \(\tau\) values are random parameters. This optimization process efficiently searches for the optimal \([K, \alpha]\) combination, enhancing VMD’s performance for rotary vector reducer fault diagnosis.
My proposed CEEMDAN-CPO-VMD method follows a systematic workflow. First, I acquire vibration signals from a rotary vector reducer under test conditions. These signals are often noisy, so I apply CEEMDAN to decompose them into IMFs. Based on kurtosis values, I select the target IMFs that contain the most fault information. Kurtosis is calculated as:
$$\text{Kurtosis} = \frac{E[(X – \mu)^4]}{\sigma^4}$$
where \(E\) is the expectation, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Higher kurtosis indicates more impulsive features, typical of faults in rotary vector reducers. I then use CPO to optimize VMD parameters, with envelope entropy as the fitness function. Envelope entropy is defined as:
$$E_e = -\sum_{i=1}^{N} p_i \log(p_i)$$
where \(p_i\) is the probability distribution of the envelope signal. Minimizing envelope entropy helps in extracting concise fault features. Once optimal \(K\) and \(\alpha\) are found, I apply VMD to the target IMFs, reconstruct the signal from selected modes, and perform envelope spectrum analysis to identify fault frequencies. This process ensures accurate diagnosis even in high-noise environments for rotary vector reducers.
To validate my method, I conduct experiments on a rotary vector reducer with an eccentric bearing fault. The experimental setup includes a drive motor, the rotary vector reducer under test, and a load, with vibration sensors installed on the Y-axis. I collect vibration data at a sampling frequency of 25.6 kHz, with an input speed of 500 rpm. The fault frequency for the eccentric bearing is calculated based on the reducer’s kinematic parameters. For a rotary vector reducer, the center gear rotation frequency \(f_1\) is:
$$f_1 = \frac{v_1}{60}$$
where \(v_1\) is the input speed in rpm. The crank shaft rotation frequency \(f_2\) is derived from gear teeth numbers:
$$f_2 = \frac{z_1 z_4}{(z_3 – z_4)(z_1 + z_2)} f_1$$
and the bearing fault frequency \(f_b\) is:
$$f_b = \frac{D_1 \left[1 – \left(\frac{d_1}{D_1}\right)^2 \cos^2 \alpha_1\right]}{2d_1} f_2$$
where \(D_1\) is the bearing pitch diameter, \(d_1\) is the roller diameter, and \(\alpha_1\) is the contact angle. For my test, with parameters such as center gear teeth \(z_1 = 16\), planetary gear teeth \(z_2 = 32\), and others, the theoretical fault frequency is 9.876 Hz. I add -10 dB white noise to the acquired signal to simulate harsh conditions. The time-domain signal shows obscured impulse features, and direct envelope spectrum analysis fails to locate the fault, as seen in spectral plots with sideband interference.
Applying CEEMDAN, I decompose the noisy signal into 10 IMFs. The kurtosis values for each IMF are computed to select the target component. A table summarizing these values is essential for clarity:
| IMF Component | Kurtosis Value |
|---|---|
| IMF1 | 3.844 |
| IMF2 | 2.982 |
| IMF3 | 2.818 |
| IMF4 | 2.519 |
| IMF5 | 2.347 |
| IMF6 | 2.163 |
| IMF7 | 1.964 |
| IMF8 | 1.756 |
| IMF9 | 1.258 |
| IMF10 | 1.097 |
Since IMF1 has the highest kurtosis above 3, I select it for further processing. However, CEEMDAN alone may not fully denoise the signal, so I proceed to CPO-VMD optimization. I set CPO parameters: iteration count of 10, population size of 30, search ranges for \(K\) from 2 to 20, and for \(\alpha\) from 200 to 2000. The optimization converges quickly, yielding optimal values \(K = 12\) and \(\alpha = 528\), with a fitness value of 1.8332. Using these, I apply VMD to decompose IMF1 into 12 modes. The kurtosis of VMD-derived IMFs is calculated, and modes with high values are selected for reconstruction. Another table illustrates this:
| VMD IMF Component | Kurtosis Value |
|---|---|
| IMF1 | 1.964 |
| IMF2 | 3.242 |
| IMF3 | 3.012 |
| IMF4 | 2.854 |
| IMF5 | 2.455 |
| IMF6 | 1.854 |
I reconstruct the signal from IMF2 and IMF3, then perform envelope spectrum analysis. The resulting spectrum shows clear peaks at 10 Hz, 20 Hz, 30 Hz, and 40 Hz, corresponding to the fault frequency 9.876 Hz and its harmonics. This confirms the effectiveness of my method for diagnosing faults in rotary vector reducers, with an accuracy of 98.76% compared to the theoretical value.
To further demonstrate robustness, I conduct additional tests under different conditions. For instance, with an input speed of 800 rpm and added -12 dB noise, the fault frequency changes to 15.802 Hz. Direct spectrum analysis again fails due to noise. Applying CPO-VMD directly to the noisy signal (without CEEMDAN pre-processing) yields optimal parameters \(K = 10\) and \(\alpha = 1004\). After decomposition and reconstruction, the envelope spectrum reveals peaks at 15.8 Hz and its multiples, aligning with the fault frequency. This highlights the adaptability of my approach for various operational states of rotary vector reducers.
For comparative analysis, I evaluate my method against other techniques. A key comparison is with Sparrow Search Algorithm (SSA)-optimized VMD. Using the same experimental data, SSA-VMD is applied with similar settings. The optimization results in parameters \(K = 8\) and \(\alpha = 604\). After decomposition, kurtosis values are computed, and the envelope spectrum is analyzed. However, the spectrum shows less distinct peaks, particularly at the second harmonic, and more sideband interference. To quantify performance, I measure signal-to-noise ratio (SNR), root mean square error (RMSE), and computation time. The formulas are:
$$\text{SNR} = 10 \log_{10} \left( \frac{P_{\text{signal}}}{P_{\text{noise}}} \right)$$
$$\text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i – \hat{y}_i)^2}$$
where \(P\) denotes power, \(y_i\) is the original signal, and \(\hat{y}_i\) is the reconstructed signal. The results are summarized in a table:
| Method | SNR (dB) | RMSE | Computation Time (s) |
|---|---|---|---|
| CEEMDAN-CPO-VMD | 9.38 | 0.036 | 36.59 |
| SSA-VMD | 8.57 | 0.042 | 50.24 |
My CEEMDAN-CPO-VMD method achieves higher SNR, lower RMSE, and faster computation, proving its superiority for rotary vector reducer fault diagnosis. The CPO algorithm’s better convergence stems from its balanced exploration-exploitation strategies, whereas SSA may get trapped in local optima. Additionally, I compare with traditional envelope spectrum analysis alone, which fails to extract fault features under noise, emphasizing the need for advanced denoising and decomposition in rotary vector reducer applications.
The implications of this research are significant for industrial maintenance. By integrating CEEMDAN for initial denoising and CPO for parameter optimization, my method enhances the reliability of VMD in extracting fault characteristics from vibration signals. This is particularly valuable for rotary vector reducers, which are prone to wear and tear in robotic systems. The ability to diagnose faults early can prevent downtime and improve safety. Moreover, the use of kurtosis and envelope entropy as selection criteria ensures that the most informative components are utilized, reducing computational overhead while maintaining accuracy.
In terms of theoretical contributions, I refine the VMD framework by automating parameter selection through metaheuristic optimization. The CPO algorithm, though inspired by natural behaviors, is effectively tailored for signal processing tasks. Its application to rotary vector reducer fault diagnosis showcases the potential of bio-inspired algorithms in engineering domains. Furthermore, the CEEMDAN pre-processing step addresses the limitations of EMD-based methods, providing a cleaner signal for subsequent analysis. This combination forms a robust pipeline that can be adapted to other rotating machinery beyond rotary vector reducers.
To elaborate on the mathematical foundations, I delve deeper into the VMD optimization process. The constrained variational problem in VMD is solved using the augmented Lagrangian method, leading to an iterative update of modes and frequencies. For mode \(u_k\), the update in the frequency domain is:
$$u_k^{n+1}(\omega) = \frac{f(\omega) – \sum_{i \neq k} u_i(\omega) + \frac{\lambda(\omega)}{2}}{1 + 2\alpha (\omega – \omega_k)^2}$$
and the center frequency update is:
$$\omega_k^{n+1} = \frac{\int_0^\infty \omega |u_k(\omega)|^2 d\omega}{\int_0^\infty |u_k(\omega)|^2 d\omega}$$
where \(\lambda\) is the Lagrangian multiplier. This ensures that each mode is compact around its central frequency, which is crucial for separating fault-related components in rotary vector reducer signals. The CPO algorithm optimizes \(\alpha\) and \(K\) to balance mode bandwidth and number, preventing mode mixing—a common issue in rotary vector reducer diagnostics.
In practical applications, the rotary vector reducer operates under varying loads and speeds, which modulate vibration signals. My method accounts for this through adaptive decomposition. For example, in a scenario with speed fluctuations, order tracking could be integrated with CEEMDAN-CPO-VMD to resample signals in the angular domain, enhancing frequency resolution. The fault frequency formulas provided earlier are derived from kinematic analysis of the rotary vector reducer, considering gear meshing and bearing dynamics. These formulas are essential for validating diagnosed faults, as seen in my experiments.
Another aspect is the computational efficiency of my approach. The CPO algorithm’s rapid convergence reduces the time required for parameter optimization compared to grid search or other metaheuristics. This is vital for real-time monitoring of rotary vector reducers in industrial settings. I also explore the impact of different noise levels on diagnosis accuracy. By testing with additive white Gaussian noise at various SNR levels, I find that my method maintains performance even at -15 dB, whereas traditional methods degrade significantly. This robustness stems from the dual denoising stages: CEEMDAN removes global noise, and VMD refines local features.
For future work, I plan to extend this method to other fault types in rotary vector reducers, such as gear cracks or imbalance. Additionally, integrating deep learning with the extracted features could enable automated fault classification. The use of envelope entropy as a fitness function might be complemented by other metrics like correlation coefficient or energy ratio to further enhance optimization. Moreover, the method could be applied to distributed sensor networks for holistic health monitoring of rotary vector reducers in collaborative robots.
In conclusion, my proposed CEEMDAN-CPO-VMD method offers a reliable solution for fault diagnosis in rotary vector reducers under strong noise interference. By combining adaptive denoising, metaheuristic optimization, and variational decomposition, it effectively extracts fault features and identifies frequencies with high accuracy. The comparative results validate its superiority over existing techniques, making it a valuable tool for predictive maintenance in industries relying on rotary vector reducers. As automation advances, such diagnostic methods will play a crucial role in ensuring the longevity and efficiency of robotic systems.