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Hidden Markov Model-Based Remaining Useful Life Estimation of Rolling Bearings Using Vibration Signals: A feasibility study
Ioannis Ksoulos, Dimitrios M. Bourdalos, John S. Sakellariou, Sonia Malefaki
TL;DR
Rolling-bearing RUL estimation is challenged by differing degradation patterns across bearings, and many approaches require target-bearing data or have other practical constraints. This study uses residual-based vibration features and a statistical HMM trained on two run-to-failure bearings to estimate RUL for a third, unseen bearing. The method provides adequate but conservative estimates without training on target-bearing data, while further assessment is needed beyond controlled laboratory conditions.
Problem
RUL estimation across bearings is difficult because degradation patterns can differ, and domain-adaptation approaches often require some data from the unseen target.
Source: §1 · p. 2Method
The method builds a degradation indicator from AR-residual vibration features and models degradation states with an HMM trained on two run-to-failure bearings.
Source: §1 · p. 2, Step 2: Condition Indicator Selection · p. 3, Step 3: Health State Separation · p. 4Results
Without target-bearing training data, the method provided adequate, albeit conservative, RUL estimates for an unseen bearing despite substantial differences in training-bearing degradation behavior.
Source: §5 · p. 12Takeaways & Limitations
The findings support the feasibility of HMM-based RUL estimation on an unseen bearing under the study's controlled experimental conditions.
Source: §5 · p. 12The method is validated at TRL 4 in a controlled laboratory setup; higher-level validation requires more complex configurations, varying operating conditions, and further performance improvement.
Source: §5 · p. 12
Abstract
from arXiv · showhide
This feasibility study presents a Hidden Markov Model (HMM)-based methodology for the Remaining Useful Life (RUL) estimation of rolling element bearings when the training vibration signals differ significantly from those of the target bearing. An AutoRegressive (AR) model is first employed to capture the machinery dynamics using vibration acceleration measurements from an initial operating period where all components including the considered bearing are still under healthy condition. The AR model is subsequently used to filter a newly acquired vibration signal from the faulty machinery, and typical Envelope Analysis is performed on the residual signal to identify fault-related repetition frequencies. The amplitudes of these frequencies, combined with statistical features of the AR residuals are fused through Principal Component Analysis to construct a sensitive to bearing degradation Condition Indicator (CI). Based on training data from the healthy state, a threshold is established, and a fault is declared when the CI exceeds this threshold. Once a fault is detected, its progression is modelled as a sequence of consecutive, distinct states, which are identified using K-Means clustering. A left-right HMM with continuous observation densities is estimated to represent the fault progression and thus predict the RUL. The HMM-based RUL estimation methodology is trained using vibration measurements from a limited number of two run-to-failure experiments with two nominally identical bearings, while RUL estimation performance is assessed on a third bearing of the same type. Despite the significant differences between the vibration data used for training and the life cycles of the training bearings with respect to the target bearing, the RUL estimation results indicate an adequate but conservative performance of the postulated HMM-based methodology.
1. Introduction
Rolling-bearing RUL methods face a transfer challenge because degradation patterns differ across bearings, while some existing approaches need data from an unseen target. This study investigates statistical HMM-based estimation without target-bearing training data.
1. Introduction
- Physics-based models rely on assumptions and expertise, while neural approaches can require large datasets, often use same-bearing evaluation, and offer limited transparency.Data-driven statistical models are presented as more explainable and able to describe degradation uncertainty.Source: §1 · p. 2
- Models often struggle to transfer across bearings with differing degradation patterns; earlier unseen-bearing approaches performed satisfactorily only near end of life, while domain adaptation often needs target data.The study addresses this challenge using an HMM-based method without data from the target bearing.Source: §1 · p. 2
- The study investigates HMM-based RUL estimation using vibration measurements from two run-to-failure bearings and assesses performance on a third unseen bearing without target-bearing training data.The training bearings show significantly different wear patterns from the target, and the experiments use a single accelerometer.Source: §1 · p. 2
- The proposed pipeline models healthy machinery dynamics with an AR model, extracts residual-based time- and frequency-domain features, and combines them using PCA to construct a degradation indicator.The introduction describes defining a healthy threshold and dividing the bearing lifetime into states using k-means clustering.Source: §1 · p. 2
- Standard HMM approaches face computational and data demands, hidden-state selection difficulties, and a first-order Markov assumption that may miss long-term degradation dependencies; this study uses time-dependent RUL estimation.The cited discussion also identifies sensitivity to the first predicting time.Source: §1 · p. 3
2. Methodology
The methodology processes vibration signals into a degradation-sensitive condition indicator, separates bearing health states, and trains a left-right HMM to estimate RUL. It evaluates both state-transition-based and sojourn-time-aware RUL predictions.
Step 1: Signal Pre-processing
- AR modeling captures healthy machinery dynamics and filters later signals; a kurtogram selects the envelope-analysis band, and the residual envelope spectrum supplies frequency-domain features.The kurtogram selects the band maximizing spectral kurtosis before filtering and Hilbert-transform demodulation.Source: Step 1: Signal Pre-processing · p. 3
Step 2: Condition Indicator Selection
- PCA fuses standardized AR-residual time-domain features and envelope-spectrum amplitudes into candidate one-dimensional condition indicators, with the highest average prognostic-metric score selecting the final CI.Candidate indicators are assessed using trendability, prognosability, and robustness; the first principal component typically explains 85–90% of variance.Source: Step 2: Condition Indicator Selection · pp. 3–4
Step 3: Health State Separation
- A healthy-state threshold of μ + 3σ marks fault onset, after which one-dimensional K-means partitions the CI into Degradation and Critical states.The three states are Healthy, Degradation, and Critical; the latter captures rapid defect propagation and damage growth before observable failure.Source: Step 3: Health State Separation · p. 4
Step 4: Hidden Markov Model Training
- The degradation sequence is modeled with a three-state left-right HMM, allowing self-transitions or transitions only to the next state, with continuous Gaussian observation densities estimated iteratively by EM.Training uses multiple independent observation sequences and maximizes their joint probability under the model.Source: Step 4: Hidden Markov Model Training · p. 4
Step 5: Remaining Useful Life Estimation
- The HMM estimates RUL after Viterbi identifies the current state, using either the number of transitions to failure or a prediction that accounts for time already spent in a state.The time-dependent approach models state duration geometrically and combines durations of future states until failure.Source: Step 5: Remaining Useful Life Estimation · pp. 5–6
3. Experimental Setup
Three accelerated run-to-failure tests used the same bearing type and operating conditions. Two tests trained the methodology, and the third evaluated RUL estimates.
Vibration measurements
- Two accelerometers were mounted at 90° on the bearing housing, but only horizontal measurements were used.The tests used PCB 352C33 accelerometers.Source: §3 · pp. 6–7
Test duration and acquisition
- Each test ran until the maximum horizontal or vertical vibration amplitude exceeded 10 × A_h, its normal-operation maximum.Signals were sampled at 25.6 kHz in 1.28-second segments acquired every 60 seconds.Source: §3 · pp. 6–7
Experimental conditions
- All three tests used LDK UER204 bearings at 2,100 rpm and 12 kN, and each ended with an outer-race defect.The shaft speed was held constant during each experiment.Source: §3 · pp. 6–7
Training and evaluation
- Bearings No. 2 and No. 3 trained the methodology, while Bearing No. 1 served as the evaluation bearing.The training split was chosen because Bearings No. 2 and No. 3 had comparable end-of-life durations.Source: §3 · p. 7
4. Performance assessment
The assessment applies the HMM trained on Bearings No. 2 and No. 3 to Bearing No. 1. It reports conservative RUL predictions during degradation that become closer to the true RUL in the critical state.
AR modelling and residuals
- AR filtering attenuated dominant low-frequency components and made degradation-related spectral content more apparent in residuals.The AR model was fitted to each bearing’s first segment; its order was selected using BIC.Source: §4 · p. 7, Figure 3 · p. 7
Kurtogram and envelope spectrum
- Envelope spectra showed a clear component at the theoretical BPFO for all three bearings.Residuals were bandpass-filtered using bandwidths selected from representative end-of-life kurtograms.Source: §4 · p. 8, Figure 5 · p. 8, Figure 4 · p. 8
Condition indicator selection
- The first principal component of EnvelopeMax and IQR achieved the highest average prognostic score and was selected as the CI.Despite identical bearing type and operating conditions, the selected CI trajectories varied across experiments.Source: §4 · p. 9, Figure 6 · p. 9
Health-state separation
- A global threshold based on the initial 20 segments separated Healthy samples from post-threshold Degradation and Critical clusters.The threshold was μ + 3σ per training bearing, with the larger of the two thresholds used globally.Source: §4 · p. 9, Figure 7 · p. 10
HMM training
- For all three health states, Q–Q plots and Kolmogorov–Smirnov tests supported the adequacy of the fitted GMM observation distributions.The null hypothesis was not rejected; the reported p-values were 0.7242, 0.8030, and 0.7480, respectively.Source: §4 · p. 10, Figure 8 · p. 10, Figure 9 · p. 11
RUL estimation
- During degradation (10–80 min), predictions were conservative as Bearing No. 1 transitioned states earlier than the training bearings; in the critical state (80–120 min), they moved closer to true RUL.The time-independent method produced stepwise estimates; the time-dependent method used state and sojourn time to update at every prediction step.Source: §4 · pp. 11–12, Figure 10 · p. 11
5. Concluding Remarks
The HMM method was trained on two same-type bearings and assessed on a third unseen bearing; despite substantial degradation differences, it gave adequate but conservative estimates, with further validation and accuracy refinement needed.
5. Concluding Remarks
- Training used single-axis measurements from two run-to-failure bearings of the same type under constant conditions, and assessment used a third previously unseen bearing.The method also introduced a new condition indicator to track degradation.Source: §5 · p. 12
- Despite substantial differences in training-bearing degradation, the method provided adequate but conservative RUL estimates without target-bearing training.The conservative predictions indicate that further accuracy refinement is needed.Source: §5 · p. 12
- Validation was limited to controlled laboratory experiments (TRL 4); progress toward higher readiness requires assessment under more complex tests and varying operating conditions.The authors also call for continued performance improvement.Source: §5 · p. 12