Phase Space Warping Research Articles

Degradation Tracking of Rolling Bearings Based on Local Polynomial Phase Space Warping

Degradation Tracking of Rolling Bearings Based on Local Polynomial Phase Space Warping

Geometry-informed phase space warping for reliable fatigue damage monitoring

This paper presents a new fatigue damage detection and monitoring approach using a geometry-informed implementation of phase space warping (PSW). The proposed method is based on continuous-time PSW theory and geometric constructs, which clarifies the relationship between the deformation of the reconstructed phase flow and the underlying damage evolution. A discrete-time approximation to the continuous-time theory is established for practical applications. The practical geometry-informed PSW (GIPSW) algorithm is developed with integrated geometry-informed heuristics and global sensitivity analysis to monitor fatigue damage evolution accurately. The proposed method is validated through numerical experiments simulating nonlinear systems with varying fatigue damage dynamics, exhibiting distinct response complexities. The results show that the GIPSW improves the monitoring accuracy by at least 41.4% and can achieve maximally four-orders-of-magnitude-lower monitoring error compared with the conventional PSW algorithm. The GIPSW is also applied in physical experiments exploring raster-angle-affected fatigue damage dynamics in 3D-printed materials. The estimated hidden-fatigue damage-time history reveals distinct crack propagation rates differentiated by the raster angles and can be used for damage prognosis and modeling the fatigue mechanisms. The critical inflection points identified in the incremental damage-time histories detect the crack growth phase transitions as early as 0.17 of the total time to failure, which can be used for early damage awareness.

Structural Health Monitoring via Phase Space Warping and Time-Delay Embedding

The state of a structure’s health can be determined by investigating its vibration characteristics. Vibration-based structural health monitoring (SHM) enables early detection and diagnosis of damage as well as extension of service life. Herein, linear and nonlinear tracking metrics are proposed to track deterioration in the condition of multistory structures and assess their instantaneous health in real-time via measurement of floor accelerations. The linear metrics, the amplitude damage index ADIn(t¯) and the frequency damage index FDIn(t¯), are based on tracking the power spectra of floor accelerations. The nonlinear metric e(t¯) is based on a novel implementation of the phase-space warping method and obtained from the orbits representing the floors’ motions in pseudo phase-space. A scaled-down model of a four-floor moment-resisting frame building is designed and fabricated to demonstrate and to compare the capabilities of the three damage indices. Structural damage is introduced to individual columns, to mimic damage initiation, by cutting two notches on opposite sides of a column cross-section at midheight. The ADIn(t¯) detected large damage events, fast deterioration beyond them, and the onset of failure using any of the four floor accelerations. The FDIn(t¯) proved insensitive to damage compared with the other two metrics. The nonlinear metric e(t¯) detected gradual (fatigue-induced) deterioration in the building’s health before introduction of damage, large damage events, fast deterioration beyond them, and the onset of failure, using any of the four floor accelerations. The e(t¯) metric varied slowly and continuously with gradual deterioration and exhibited larger discontinuous jumps with discrete damage events. This was true for all three damage experiments undertaken on the model building. The nonlinear e(t¯) metric was also found to be more efficient, in terms of signal utilization, in comparison with the linear ADIn(t¯) metric.

Remaining Useful Life Prediction of Rolling Bearings Based on Segmented Relative Phase Space Warping and Particle Filter

Predictive maintenance plays a crucial role in the field of intelligent machinery fault diagnosis, which improves the efficiency of maintenance. This paper focuses on the extraction of real-time damage feature and the prediction of remaining useful life (RUL) in predictive maintenance of rolling bearings. Some RUL prediction approaches lack dynamic foundations and require large amounts of data and prior knowledge. This paper proposes the algorithm of segmented relative phase space warping (SRPSW) and a strategy combining double exponential model (DEM) and particle filter (PF) to predict the RUL. SRPSW provides a dynamic basis for real-time RUL prediction in different stages. The DEM-based PF reduces the need for prior knowledge and improves accuracy. The analysis results from normal and accelerated degradation experiments show that the proposed SRPSW overcomes the failure of the original PSW in depicting the later operating stage of bearings. Further, the relative damage indicators (RDIs) extracted by SRPSW are more accurately than commonly used indicators. The predicted results show that the DEM-based PF does not require professional and prior information whilst ensuring the accuracy of RUL prediction. The proposed approach in this paper provides a new avenue for predictive maintenance of bearings.

Bearing Remaining Useful Life Prediction Based on a Scaled Health Indicator and a LSTM Model with Attention Mechanism

Rotor systems are of considerable importance in most modern industrial machinery, and the evaluation of the working conditions and longevity of their core component—the rolling bearing—has gained considerable research interest. In this study, a scale-normalized bearing health indicator based on the improved phase space warping (PSW) and hidden Markov model regression was established. This indicator was then used as the input for the encoder–decoder LSTM neural network with an attention mechanism to predict the rolling bearing RUL. Experiments show that compared with traditional health indicators such as kurtosis and root mean square (RMS), this scale-normalized bearing health indicator directly indicates the actual damage degree of the bearing, thereby enabling the LSTM model to predict RUL of the bearing more accurately.

Tracking and Identifying CRDM Degradation State Based on SWT and Phase Space Warping

The control rod drive mechanism (CRDM) is important for ensuring the normal operation and safety of reactors. Considering the low speed, latent performance degradation signs, and susceptibility to interference from other signal sources of the CRDM, this paper tracks and identifies the CRDM degradation state by a combination of the semi-soft wavelet threshold (SWT) and phase space warping (PSW). By comparison with the method based on the root mean square (RMS) of vibration amplitude, the method combining the SWT with PSW, unlike the traditional linear signal processing theory with poor performance in tracking and identifying the latent performance degradation state, can track and identify adequately the performance degradation state of rotating machinery with high background noise.

Improved phase space warping method for degradation tracking of rotating machinery under variable working conditions

Health Indicators (HI) that can effectively track the degradation state and are insensitive to the change of working conditions are of great significance to the process of Prognostic and Health Management (PHM). Based on the rapidly changing one-dimensional vibration signal to reconstruct the phase space, the Phase Space Warping (PSW) method was used to quantify the small curvature of the phase space trajectory caused by the damage, and then the HI that can represent the evolution process of the slow-changing damage could be obtained with the curvature. However, there are still several problems with the PSW method that need to be solved urgently. Firstly, it does not provide clear optimization criteria and adaptive determination strategies for the two key parameters of phase space reconstruction, which are, time delay and embedded dimension. Secondly, the HI extracted based on the PSW is greatly affected by noise and lacks a proper feature fusion strategy. Aiming at these problems, the Improved Phase Space Warping (IPSW) method is proposed in this paper. The contribution of IPSW is as follows: (1) IPSW considers the degree of comprehensive mutual information between multi-dimensional components, and uses comprehensive mutual information entropy as an optimization index, and gives clear guidelines on how to select optimal parameters; (2) IPSW introduces the correlation analysis method to fuse the acquired multi-modal trend components and then obtains the HI that can characterize the damage degradation trend. The simulation and experimental signal verification results of bearing fault degradation under the condition of variable speed show that the HI extracted based on the IPSW method can reduce the influence of variable speed and track the bearing damage degradation trend effectively.

Prediction of Bearings Remaining Useful Life Across Working Conditions Based on Transfer Learning and Time Series Clustering

Recently, data-driven remaining useful life (RUL) prediction has become a promising tool in prognostics and health management for rolling bearings. In many actual applications, however, it is not easy to collect whole-life degradation data of bearings, while training with an insufficient amount of data would result in a biased RUL prediction model. If the monitoring data historically accumulated under different working conditions are introduced to facilitate model training, it probably tends to get inadequate prediction performance due to the violation of the precondition of independent and identical distribution ( ${i.i.d.}$ ). To solve this problem, a new bearing RUL prediction approach is proposed by utilizing the transfer learning strategy. First, a new time series clustering algorithm is proposed to exploit clusters of different degradation series. By integrating phase space warping and dynamic time warping, this algorithm is able to measure the similarity by using global degradation information and then get better clustering performance. Second, a new temporal domain adaptation method is proposed to obtain the pivot feature set of different bearings based on meta-degradation information which is represented by the principal curve of each cluster. Finally, support vector machine is run to establish the prediction model by means of these features, and RUL of the target bearing can be predicted via the same feature adaptation. A theoretical analysis is also provided to prove that the proposed approach has an upper bound of information loss in the transfer process. Experimental results on the IEEE PHM challenge 2012 dataset demonstrate the effectiveness of the proposed approach.

A novel method for bone fatigue monitoring and prediction

Bone fatigue, often manifest as stress fractures, is a common injury that plagues many individuals, adversely affect quality of life, and is an obstacle to extended human spaceflight. This manuscript details a pilot study that was conducted to determine if the Phase Space Warping (PSW) methodology could be used to monitor/predict fatigue failure in bone tissue. A Moon's beam experimental apparatus was used to perform variable amplitude fatigue tests on bovine bone specimens. Scanning electron microscopy was used to evaluate the fracture surface and identify the fracture type. The PSW method allowed for successful identification of the various damage modes and may lead to the development of a viable tool for predicting the health and fatigue life of bone.

Fatigue Detection Using Phase-Space Warping.

A novel application of phase-space warping (PSW) method to detect fatigue in the musculoskeletal system is presented. Experimental kinematic, force, and physiological signals are used to produce a fatigue metric. The metric is produced using time-delay embedding and PSW methods. The results showed that by using force and kinematic signals, an overall estimate of the muscle group state can be achieved. Further, when using electromyography (EMG) signals the fatigue metric can be used as a tool to evaluate muscles activation and load sharing patterns for individual muscles. The presented method will allow for fatigue evolution measurement outside a laboratory environment, which open doors to applications such as tracking the physical state of players during competition, workers in a plant, and patients undergoing in-home rehabilitation.

A multi-time scale approach to remaining useful life prediction in rolling bearing

This paper presents a novel multi-time scale approach to bearing defect tracking and remaining useful life (RUL) prediction, which integrates enhanced phase space warping (PSW) with a modified Paris crack growth model. As a data-driven method, PSW describes the dynamical behavior of the bearing being tested on a fast-time scale, whereas the Paris crack growth model, as a physics-based model, characterizes the bearing's defect propagation on a slow-time scale. Theoretically, PSW constructs a tracking metric by evaluating the phase space trajectory warping of the bearing vibration data, and establishes a correlation between measurement on a fast-time scale and defect growth variables on a slow-time scale. Furthermore, PSW is enhanced by a multi-dimensional auto-regression (AR) model for improved accuracy in defect tracking. Also, the Paris crack growth model is modified by a time-piecewise algorithm for real-time RUL prediction. Case studies performed on two run-to-failure experiments indicate that the developed technique is effective in tracking the evolution of bearing defects and accurately predict the bearing RUL, thus contributing to the literature of bearing prognosis .

New invariant measures to track slow parameter drifts in fast dynamical systems

Estimates of quantitative characteristics of nonlinear dynamics, e.g., correlation dimension or Lyapunov exponents, require long time series and are sensitive to noise. Other measures (e.g., phase space warping or sensitivity vector fields) are relatively difficult to implement and computationally intensive. In this paper, we propose a new class of features based on Birkhoff ergodic theorem, which are fast and easy to calculate. They are robust to noise and do not require large data or computational resources. Application of these metrics in conjunction with the smooth orthogonal decomposition to identify/track slowly changing parameters in nonlinear dynamical systems is demonstrated using both synthetic and experimental data.

Fault tracking of rotating machinery under variable operation based on phase space warping

For fault prognosis of rotating machinery under variable operation, a fault tracking method based on phase space warping and smooth orthogonal decomposition (SOD) is presented to describe the degradation process of rotating machinery. Firstly, phase space is reconstructed using vibration time-series, and a tracking function of damage evolution is built by quantifying phase space warping. To compensate for cumulative model error and the error caused by variation of local probability distribution of the reference phase space points, the original time-series is partitioned into several data segments and the phase space is partitioned into several subspaces correspondingly. Several feature vectors are concatenated into tracking matrix. Secondly, the different trends caused by actual damage degradation and operation variety in the tracking matrix are separated by smooth orthogonal decomposition. According to smooth orthogonal values, dominant smooth orthogonal coordinates which reflect actual fault degradation trends are extracted. Finally, fault degradation process of bearing out-race is simulated. Rotating speed is varied during the degradation process. Applying the presented method to the degradation process tracking, the tracking matrix is built and decomposed by SOD, and the results show that the proposed method can track the evolution trend of the rotating machinery fault without the influence of operation condition variety.

Application of Phase Space Warping on Damage Tracking for Bearing Fault

Nowadays, the significance of keeping equipment function properly each time is obvious. If equipment fails during its use, it may have disastrous consequences. Estimating remaining useful life (RUL) of equipment is a key to prevent such calamities, improve its reliability, provide security and reduce unnecessary maintenance and operational cost. The evolution and tracking of damage is the foundation of RUL predicting, and also is one of the most important content of mechanical fault diagnosis. Slow-time variable process of mechanical damage would lead the phase space reconstructed by fast-time variable vibrate signals warping. Search the dynamics characteristic law of damage evolution analysis in the phase space, and build the relationship between fast-time variable signals and slow-time variable damage, and then damage evolution tracking is possible. To validate the theory, simulation model of bearing damage evolution is built, the outer-race fault evolution signals is obtained, and the trend of evolution of degradation of bearing fault is described with Phase Space Warping (PSW) theory and Smooth Orthogonal Decomposition (SOD). The results proved the feasibility of the methodology of PSW in damage evolution tracking.

Nonlinear smooth orthogonal decomposition of kinematic features of sawing reconstructs muscle fatigue evolution as indicated by electromyography.

Tracking or predicting physiological fatigue is important for developing more robust training protocols and better energy supplements and/or reducing muscle injuries. Current methodologies are usually impractical and/or invasive and may not be realizable outside of laboratory settings. It was recently demonstrated that smooth orthogonal decomposition (SOD) of phase space warping (PSW) features of motion kinematics can identify fatigue in individual muscle groups. We hypothesize that a nonlinear extension of SOD will identify more optimal fatigue coordinates and provide a lower-dimensional reconstruction of local fatigue dynamics than the linear SOD. Both linear and nonlinear SODs were applied to PSW features estimated from measured kinematics to reconstruct muscle fatigue dynamics in subjects performing a sawing motion. Ten healthy young right-handed subjects pushed a weighted handle back and forth until voluntary exhaustion. Three sets of joint kinematic angles were measured from the right upper extremity in addition to surface electromyography (EMG) recordings. The SOD coordinates of kinematic PSW features were compared against independently measured fatigue markers (i.e., mean and median EMG spectrum frequencies of individual muscle groups). This comparison was based on a least-squares linear fit of a fixed number of the dominant SOD coordinates to the appropriate local fatigue markers. Between subject variability showed that at most four to five nonlinear SOD coordinates were needed to reconstruct fatigue in local muscle groups, while on average 15 coordinates were needed for the linear SOD. Thus, the nonlinear coordinates provided a one-order-of-magnitude improvement over the linear ones.

Identifying invariant manifold using phase space warping and stochastic interrogation

Advances in the generalization of invariant manifolds to finite time, experimental (or observational) flows have stimulated many recent developments in the approximation of invariant manifolds and Lagrangian coherent structures. This paper explores the identification of invariant manifold like structures in experimental settings, where knowledge of a flow field is absent, but phase space trajectories can be experimentally measured. Several existing methods for the approximation of these structures modified for application when only unstructured trajectory data is available. We find the recently proposed method, based on the concept of phase space warping, to outperform other methods as data becomes limited and show it to extend the finite-time Lyapunov exponent method. This finding is based on a comparison of methods for various data quantities and in the presence of both measurement and dynamic noise.

Nonlinear analysis of gait kinematics to track changes in oxygen consumption in prolonged load carriage walking: A pilot study

Linking human mechanical work to physiological work for the purpose of developing a model of physical fatigue is a complex problem that cannot be solved easily by conventional biomechanical analysis. The purpose of the study was to determine if two nonlinear analysis methods can address the fundamental issue of utilizing kinematic data to track oxygen consumption from a prolonged walking trial: we evaluated the effectiveness of dynamical systems and fractal analysis in this study. Further, we selected, oxygen consumption as a measure to represent the underlying physiological measure of fatigue. Three male US Army Soldier volunteers (means: 23.3 yr; 1.80 m; 77.3 kg) walked for 120 min at 1.34 m/s with a 40-kg load on a level treadmill. Gait kinematic data and oxygen consumption (VO 2) data were collected over the 120-min period. For the fractal analysis, utilizing stride interval data, we calculated fractal dimension. For the dynamical systems analysis, kinematic angle time series were used to estimate phase space warping based features at uniform time intervals: smooth orthogonal decomposition (SOD) was used to extract slowly time-varying trends from these features. Estimated fractal dimensions showed no apparent trend or correlation with independently measured VO 2. While inter-individual difference did exist in the VO 2 data, dominant SOD time trends tracked and correlated with the VO 2 for all volunteers. Thus, dynamical systems analysis using gait kinematics may be suitable to develop a model to predict physiologic fatigue based on biomechanical work.

Slow-Time Changes in Human EMG Muscle Fatigue States Are Fully Represented in Movement Kinematics

The ability to identify physiologic fatigue and related changes in kinematics can provide an important tool for diagnosing fatigue-related injuries. This study examined an exhaustive cycling task to demonstrate how changes in movement kinematics and variability reflect underlying changes in local muscle states. Motion kinematics data were used to construct fatigue features. Their multivariate analysis, based on smooth orthogonal decomposition, was used to reconstruct physiological fatigue. Two different features composed of (1) standard statistical metrics (SSM), which were a collection of standard long-time measures, and (2) phase space warping (PSW)-based metrics, which characterized short-time variations in the phase space trajectories, were considered. Movement kinematics and surface electromyography (EMG) signals were measured from the lower extremities of seven highly trained cyclists as they cycled to voluntary exhaustion on a stationary bicycle. Mean and median frequencies from the EMG time series were computed to measure the local fatigue dynamics of individual muscles independent of the SSM- and PSW-based features, which were extracted solely from the kinematics data. A nonlinear analysis of kinematic features was shown to be essential for capturing full multidimensional fatigue dynamics. A four-dimensional fatigue manifold identified using a nonlinear PSW-based analysis of kinematics data was shown to adequately predict all EMG-based individual muscle fatigue trends. While SSM-based analyses showed similar dominant global fatigue trends, they failed to capture individual muscle activities in a low-dimensional manifold. Therefore, the nonlinear PSW-based analysis of strictly kinematic time series data directly predicted all of the local muscle fatigue trends in a low-dimensional systemic fatigue trajectory. These results provide the first direct quantitative link between changes in muscle fatigue dynamics and resulting changes in movement kinematics.

Identifying damage using local flow variation method

In this paper, a damage identification method called local flow variation is introduced. Itis a practical implementation of a phase space warping concept. A hierarchicaldynamical system is considered where a slow-time damage process causes drifts inthe parameters of a fast-time system describing the measurable response of astructure. The method is based on a hypothesis that the probability distributionfunction of the fast-time trajectory in its phase space is a function of a damagestate. In this method, an ensemble of estimated expectations of a trajectory indifferent locations of the reconstructed phase space is used as a damage featurevector. Using these feature vectors, damage identification is realized by a smoothorthogonal decomposition. An experiment is conducted to validate the method. Atwo-dimensional slow-time damage process is identified from experimental fast-time data.Although damage identification results from the local flow variation are not asaccurate as those from the direct application of phase space warping trackingfunctions, the required computation time is about two orders of magnitude shorter.

Multidimensional Damage Identification Based on Phase Space Warping: An Experimental Study

A multidimensional damage identification scheme developed in previous work is modified and investigated experi- mentally. An experimental apparatus consists of a driven two-well magneto-elastic oscillator, where a cantilever beam vibrates in a magnetic potential field perturbed by two electromagnets. These electromagnets are activated by a computer controlled power supply and their terminal voltages are considered a two-dimensional damage variable. The effect of total change in the supply voltage of the electromagnets is approximately 4% shift in the experimentally measured natural frequencies of small oscillations in each well of the potential. Experimental runs are started in a nominally chaotic regime. The battery voltages are altered on specific trajectories in the damage (voltage) phase space. Damage identification is accomplished based on the elastic vibration data collected using laser vibrometers and a accelerometer. The phase space warping based damage tracking feature vectors are estimated using a new phase space partitioning scheme. The damage identification is achieved by applying smooth orthogonal decomposition to the obtained statistics. The effect of the data record size on the quality of reconstructed damage trajectory is investigated in a series of experiments. It is also demonstrated that the new partitioning scheme considerably improves signal-to-noise ration of the identified damage states.