The rotary vector reducer stands as one of the three core components in robotics, with its reliability and stability directly impacting the safety and operational integrity of the entire robotic system. The inherent complexity of the rotary vector reducer, comprising numerous interacting components, makes its vibration signals highly susceptible to significant noise interference during operation. This noise often masks the characteristic signatures of incipient faults, rendering accurate fault localization and diagnosis a formidable challenge. Traditional spectral analysis techniques frequently fail under such conditions due to phenomena like frequency modulation and strong background noise, leading to missed or false diagnoses. Therefore, developing robust signal processing methodologies capable of extracting faint fault features from noisy vibration data is of paramount importance for ensuring the health and longevity of rotary vector reducers.
This article addresses this critical issue by proposing a novel hybrid fault diagnosis methodology. Our approach synergistically combines an improved Variational Mode Decomposition (VMD) technique with Maximum Correlated Kurtosis Deconvolution (MCKD). The core innovation lies in employing a Particle Swarm Optimization (PSO) algorithm to intelligently determine the optimal parameters for both signal processing blocks, thereby overcoming the subjective and often suboptimal manual parameter selection. We validate the efficacy of our method using experimental vibration data collected from a rotary vector reducer with a known fault, demonstrating its superior capability in isolating fault characteristics amidst substantial noise.
Methodological Framework: PSO-VMD-MCKD
The proposed methodology is built upon a two-stage optimization and processing pipeline. The first stage involves denoising the raw vibration signal via an adaptively parameterized VMD. The second stage then amplifies the periodic fault impulses within the denoised signal using an optimally tuned MCKD filter. A PSO algorithm orchestrates this entire process by finding the best parameters for each stage, guided by an appropriate fitness function.
1. Particle Swarm Optimization (PSO) as the Parameter Tuner
Particle Swarm Optimization is a heuristic, population-based optimization algorithm inspired by the social foraging behavior of birds or fish. In our diagnostic framework, the PSO algorithm’s role is to search for the parameter combination that yields the most effective signal decomposition and deconvolution. We define a “particle” whose position in the search space represents a candidate set of parameters. Each particle adjusts its trajectory based on its own historical best position and the best position discovered by its neighbors.
The position update equations for particle \(i\) in dimension \(d\) are:
$$
v_{id}^{t+1} = \omega \cdot v_{id}^{t} + c_1 \cdot r_1 \cdot (pbest_{id} – x_{id}^{t}) + c_2 \cdot r_2 \cdot (gbest_{d} – x_{id}^{t})
$$
$$
x_{id}^{t+1} = x_{id}^{t} + v_{id}^{t+1}
$$
where \(v_{id}\) and \(x_{id}\) are the velocity and position of the particle, \(\omega\) is the inertia weight, \(c_1\) and \(c_2\) are acceleration coefficients, \(r_1\) and \(r_2\) are random numbers, \(pbest\) is the particle’s best-known position, and \(gbest\) is the swarm’s global best-known position.
2. Stage 1: Optimized Variational Mode Decomposition (PSO-VMD)
Variational Mode Decomposition is a fully adaptive, non-recursive signal decomposition technique. It aims to decompose a real-valued input signal \(f(t)\) into a discrete number of band-limited Intrinsic Mode Functions (IMFs) \(u_c(t)\), each with a specific center frequency \(\omega_c\). The VMD algorithm solves a constrained variational problem:
$$
\min_{\{u_c\},\{\omega_c\}} \left\{ \sum_{c=1}^{C} \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_c(t) \right] e^{-j\omega_c t} \right\|_2^2 \right\}
$$
subject to: \(\sum_{c=1}^{C} u_c = f(t)\)
Here, \(C\) is the total number of modes to be decomposed, \(\partial_t\) denotes the partial derivative with respect to time, \(\delta(t)\) is the Dirac delta, \(*\) represents convolution, and \(j\) is the imaginary unit.
The performance of VMD is critically dependent on two key parameters: the number of modes \(C\) and the penalty factor \(\alpha\) which controls the bandwidth of each IMF. Suboptimal choices lead to either mode mixing (under-decomposition) or spurious, meaningless modes (over-decomposition). We employ the PSO algorithm to find the optimal pair \([C_{opt}, \alpha_{opt}]\). The fitness function for this optimization is chosen to be the Multi-scale Permutation Entropy (MPE) of the first IMF. MPE measures signal complexity across scales; a lower MPE often indicates a cleaner, more regular signal component. Thus, minimizing the MPE of the first IMF guides the PSO towards parameters that best separate noise from the primary signal components.
3. Signal Reconstruction via Kurtosis Criterion
After decomposing the signal using the optimal VMD parameters, we obtain \(C\) IMFs. To reconstruct a signal focused on fault information, we calculate the kurtosis value for each IMF. Kurtosis is a fourth-order statistical moment highly sensitive to transient impulses—the hallmark of mechanical faults. The IMF with the highest kurtosis value is selected, as it contains the most significant impulse-like features related to the fault. This selected IMF, denoted as \(IMF_{fault}\), is then passed to the next stage for further enhancement.
4. Stage 2: Optimized Maximum Correlated Kurtosis Deconvolution (PSO-MCKD)
Maximum Correlated Kurtosis Deconvolution is a powerful blind deconvolution filter designed to highlight periodic impulse trains buried in noise. Given a signal \(y(n)\), MCKD finds a Finite Impulse Response (FIR) filter \(f(l)\) of length \(L\) that produces an output signal \(x(n)\) which maximizes the Correlated Kurtosis (CK):
$$
CK_M(T) = \frac{\sum_{n=1}^{N} \left( \prod_{m=0}^{M} x_{n-mT} \right)^2}{\left( \sum_{n=1}^{N} x_n^2 \right)^{M+1}}
$$
where \(T\) is the period of the fault impulses (in samples), \(M\) is the shift order (number of periods considered), and \(N\) is the signal length. Maximizing \(CK_M(T)\) forces the filter to recover the periodic impulse sequence.
The effectiveness of MCKD is governed by the filter length \(L\) and the shift parameter \(M\). An inappropriate \(L\) cannot capture the impulse structure, while an unsuitable \(M\) may not effectively exploit the periodicity. We again deploy the PSO algorithm to search for the optimal pair \([L_{opt}, M_{opt}]\). The fitness function for this stage is the negated CK value itself; thus, PSO minimizes \(-CK_M(T)\) to find the parameters that maximize the periodic impulse characteristics in the deconvolved signal \(x(n)\).
5. Overall Diagnostic Procedure
The complete workflow for diagnosing a rotary vector reducer using the PSO-VMD-MCKD method is summarized as follows:
- Data Acquisition & Preprocessing: Collect vibration data from the rotary vector reducer under test. Calculate the theoretical fault characteristic frequency \(f_{fault}\) based on kinematic parameters and operating speed.
- PSO-VMD Optimization: Use the PSO algorithm to find the optimal VMD parameters \([C_{opt}, \alpha_{opt}]\) by minimizing the MPE of the first IMF of the raw signal.
- Signal Decomposition & Reconstruction: Decompose the raw signal using VMD with \([C_{opt}, \alpha_{opt}]\). Calculate the kurtosis of each IMF and select the component \(IMF_{fault}\) with the maximum value.
- PSO-MCKD Optimization: Use the PSO algorithm to find the optimal MCKD parameters \([L_{opt}, M_{opt}]\) for the signal \(IMF_{fault}\), using the known fault period \(T = f_s / f_{fault}\) (where \(f_s\) is the sampling frequency). The objective is to maximize CK.
- Fault Feature Enhancement: Apply MCKD with parameters \([L_{opt}, M_{opt}]\) and period \(T\) to the \(IMF_{fault}\) signal, producing the enhanced signal \(x(n)\).
- Feature Extraction & Diagnosis: Compute the envelope spectrum of \(x(n)\). Identify spectral peaks at \(f_{fault}\) and its harmonics to confirm the presence and type of fault in the rotary vector reducer.
Experimental Validation on a Rotary Vector Reducer
To validate the proposed method, we conducted experiments on a dedicated rotary vector reducer comprehensive performance test rig. The test subject was a rotary vector reducer with a confirmed fault on its needle roller components.
1. Experimental Setup and Parameters
The vibration signal was acquired using a piezoelectric accelerometer mounted on the reducer housing in the X-direction. The data was sampled at a frequency \(f_s = 25.6\) kHz for a duration yielding 100,000 data points. The output speed of the rotary vector reducer was set to 18 rpm. To rigorously test the noise immunity of our method, we intentionally added -5 dB of Gaussian white noise to the measured vibration signal. The structural parameters of the tested rotary vector reducer are listed in the table below.
| Component | Parameter | Value |
|---|---|---|
| Gearing | Input Pinion Teeth (\(z_1\)) | 12 |
| Planetary Gear Teeth (\(z_2\)) | 36 | |
| Cycloid Wheel Lobes (\(z_3\)) | 39 | |
| Needle Wheel Pins (\(z_4\)) | 40 | |
| Overall | Reduction Ratio | 121:1 |
The theoretical fault characteristic frequency for the needle roller assembly, \(f_{needle}\), can be derived from the kinematics. With the input speed \(n_1\) derived from the output speed and reduction ratio, the calculation is as follows:
$$
f_1 = \frac{n_1}{60}, \quad f_{needle} = -\frac{(z_4 – z_3) \cdot z_1}{z_2 \cdot z_4} \cdot f_1
$$
For the given parameters and operating condition, the calculated \(f_{needle} \approx 30\) Hz.
2. PSO Optimization and Signal Processing
We configured the PSO with a population size of 30 and a maximum of 20 iterations. The search boundaries for the parameters were set as: \(C \in [2, 20]\), \(\alpha \in [100, 4000]\), \(L \in [100, 500]\), and \(M \in [1, 8]\). The PSO algorithm successfully converged to the following optimal parameter sets:
| Algorithm | Optimized Parameters | Optimal Value |
|---|---|---|
| VMD | Number of Modes (\(C\)), Penalty (\(\alpha\)) | [10, 1533] |
| MCKD | Filter Length (\(L\)), Shift (\(M\)) | [432, 7] |
The raw, noise-added vibration signal and its direct frequency spectrum are shown below (conceptual description). The time-domain waveform exhibited no clear periodic impacts, and the frequency spectrum was dominated by noise and mesh frequencies, with no distinct peak observable at 30 Hz or its harmonics, making diagnosis impossible from the raw data.
Applying PSO-VMD with \([C=10, \alpha=1533]\) decomposed the signal into 10 IMFs. The kurtosis values for these IMFs were computed to identify the most fault-informative component.
| IMF Component | Kurtosis Value |
|---|---|
| IMF1 | 2.873 |
| IMF2 | 3.844 |
| IMF3 | 2.818 |
| IMF4 | 2.519 |
| IMF5 | 2.347 |
| IMF6 | 2.163 |
| IMF7 | 1.964 |
| IMF8 | 1.756 |
| IMF9 | 1.258 |
| IMF10 | 1.097 |
As evident, IMF2 possessed the highest kurtosis and was therefore selected as \(IMF_{fault}\). The frequency spectrum of IMF2 showed a slight prominence near 30 Hz but remained cluttered with interfering frequencies and indistinct harmonics.
3. Fault Feature Extraction via PSO-MCKD
The selected IMF2 signal was then processed using the MCKD algorithm with the PSO-optimized parameters \([L=432, M=7]\) and the fault period \(T = f_s / f_{needle}\). The deconvolved output signal \(x(n)\) displayed a clear, periodic impulse train in the time domain. Subsequent envelope analysis on \(x(n)\) yielded a decisive result.
The envelope spectrum revealed prominent peaks at precisely 30.21 Hz, 60.42 Hz, and 89.60 Hz. These frequencies correspond almost exactly to the theoretical needle roller fault frequency (\(f_{needle} \approx 30\) Hz) and its second and third harmonics (\(2f_{needle} \approx 60\) Hz, \(3f_{needle} \approx 90\) Hz). The clear presence of this harmonic series in the envelope spectrum provides unambiguous evidence of a fault in the needle roller assembly of the rotary vector reducer.
Comparative Analysis and Discussion
To underscore the superiority of the integrated PSO-VMD-MCKD approach, we performed a comparative experiment. We applied a standalone PSO-optimized MCKD method (PSO-MCKD) directly to the original noisy vibration signal. The PSO found optimal parameters \([L=458, M=7]\) for this direct application.
While the envelope spectrum from the direct PSO-MCKD processing showed a peak at the fundamental fault frequency (30.21 Hz) and a less distinct peak at the second harmonic (60.42 Hz), the third harmonic (89.60 Hz) was completely buried in the noise floor and was not discernible. This contrast highlights the critical role of the preliminary denoising stage.
The integrated method’s strength is two-fold: First, the PSO-VMD stage acts as an intelligent, adaptive noise filter, isolating the fault-related signal component (IMF2) from the majority of the background noise and other vibration sources within the complex rotary vector reducer. Second, the subsequent PSO-MCKD stage operates on this cleaner signal, allowing it to more effectively tune its filter to the specific periodic impulse structure, resulting in a much clearer enhancement of the entire fault harmonic series. The standalone MCKD, forced to deal with the raw, noisy signal, allocates a significant portion of its effort to suppressing broad-band noise rather than perfectly recovering the weak, periodic impulses, leading to a less definitive diagnostic outcome.
Conclusion
This article has presented a robust and intelligent fault diagnosis methodology for rotary vector reducers operating under significant noise interference. The proposed PSO-VMD-MCKD method successfully addresses the key challenges in this domain through a structured, optimized pipeline.
The Particle Swarm Optimization algorithm serves as the intelligent core, automatically determining the optimal parameters for both the Variational Mode Decomposition and the Maximum Correlated Kurtosis Deconvolution stages. This eliminates the guesswork and empirical tuning often associated with these powerful techniques. The VMD stage, with its optimized parameters, effectively separates noise from the fault signal within the complex vibration signature of the rotary vector reducer. The subsequent MCKD stage, finely tuned by PSO, then powerfully amplifies the periodic impulse train characteristic of a mechanical fault.
Experimental results on a rotary vector reducer with a seeded needle roller fault conclusively demonstrate the method’s effectiveness. Even with added strong noise, the integrated approach successfully extracted a clear harmonic series in the envelope spectrum at the exact theoretical fault frequency and its multiples, enabling a definitive diagnosis. The comparative study confirmed that the synergistic combination of denoising (VMD) and impulse enhancement (MCKD), both guided by PSO, yields significantly superior results than applying advanced deconvolution techniques alone on noisy data. Therefore, the PSO-VMD-MCKD framework provides a highly reliable and automated solution for the early and accurate fault diagnosis of critical components like the rotary vector reducer, contributing directly to improved robotic system reliability and predictive maintenance strategies.