Economic fatigue damage monitoring for vehicle fleets using the scattering transform
Abstract
Vehicle monitoring is an important prequisite for predictive maintenance applications. Virtual sensors can be deployed to establish relationships between fatigue related quantities of interest and readily available measurement data, which reduces the costs of monitoring for vehicle fleets. This work describes a data-driven virtual sensing approach using the scattering transform and principal component analysis. These data transformations are used to obtain a reduced representation of acceleration data, which is suitable for the identification of fatigue critical events during vehicle operation. Results of a previous study using an eBike demonstrator are summarized and the methodology is applied to experimental data of a sensor equipped light rail vehicle. In both applications, fictitious fatigue damage contributions are estimated accurately and physical interpretations of the reduced representation are found.
1 INTRODUCTION
Vehicle fleets play a central role in the modern mobility landscape, both in public transportation and shared mobility concepts. They are characterized by a significantly higher degree of utilization and shorter idle times than private vehicles. For both public and private mobility providers, mobility must be both affordable and reliable in order to be accepted. Fatigue monitoring and predictive maintenance strategies contribute to this goal, by predicting component failures ahead of time and enabling better utilization of remaining useful life.
Many promising approaches exist in the field of virtual sensing, where the aim is to approximate system responses of interest using available measurements of related sensor data. Generally, these methods can be subdivided into physics-based and data-driven approaches. Physics-based approaches often include finite-element models in conjunction with modal analysis or Kalman filters. Ugras et al. [1] deploys frequency based methods to monitor the fatigue life of trucks, while Tarpø et al. [2] and Hjelm et al. [3] provide structural health monitoring (SHM) approaches for offshore structures and lattice towers, respectively. Data-driven methods often use machine learning algorithms, which rely on measurement data of the physical quantity of interest for model parameterization. Example applications exist in system identification [4], accoustics [5] and SHM of wind turbines [6].
In the context of fleets, many similar vehicles need to be monitored, which can be leveraged to deploy sensor efficient strategies. This can be achieved using standard sensors, which are installed in all vehicles of the fleet, in conjunction with additional reference sensors, which are only required in a small number of vehicles. In a previous study, the authors presented a data driven strategy for fatigue monitoring [7], which used eBikes as a demonstrator example. It transforms the standard sensor measurement data into a reduced, low dimensional representation, which is suitable for the classification and regression of reference sensor information. Since this approach is very general and purely measurement based, it can be transferred and adapted to fleets of different vehicle types.
This paper summarizes the general monitoring strategy and important results of the eBike demonstrator example. Afterwards, measurement data of the Dresden Measuring Tram project [8] is used to showcase an application for light rail vehicles. First results are shown and challenges for further research are addressed in a short discussion.
2 METHODS
In this paper, a reduced representation is created from measurement time series data in order to enable a fatigue damage regression. This is mainly achieved by using the scattering transform in conjunction with principal component analysis (PCA). The general approach follows the workflow presented in the authors' previous work [7]. As a result, the required fundamentals are only briefly summarized here and the reader is referred to further literature for additional details.
The methods presented in this paper were implemented using the python packages Kymatio [9] and Scikit-learn [10]. In the following mathematical notation, single and double underscores are used to describe vectors 
and matrices 
, respectively.
2.1 Scattering transform
The scattering transform [11] is a transformation developed for signal classification. It applies cascaded wavelet convolutions, modulus operations and low-pass filter averaging to a time series in order to extract coefficients, which are invariant to translation in time and stable to time warping deformation. It has been used in audio and image classification, as well as seismic signal clustering [12].
Starting from a time series 
, the convolution with all members of a wavelet basis 
is computed. Here, ψ is a mother wavelet and 
denotes the set of dilation factors of the i-th layer, which in turn depend on the chosen number of wavelets per octave Q. The modulus operation is applied to the result and this process is repeated, leading to 
for the first layer, 
for the second layer, and so on. To achieve invariance to translation in time, the coefficients S are obtained by averaging the results of each layer using a low pass filter ϕ with an averaging scale of 2J, 
. As a result, the transform is defined by the chosen mother wavelet and the selected parameters J and Q. In this paper, the Kymatio [9] default implementation of two scattering layers and a complex valued Morlet mother wavelet are used.
2.2 PCA
PCA [13] is among the most commonly used algorithms for dimensionality reduction. It applies a linear transformation to a given dataset, in order to obtain a new representation, which is sorted by variance.
The dataset is represented as a data matrix 
, where each row 
represents a data sample and each column contains the values of a specific feature for each sample. By applying an orthogonal transformation 
, the principal component (PC) representation 
of the data matrix is obtained. The transformation matrix 
is composed of the eigenvectors 
of 
, ordered by the size of the corresponding eigenvalues, which can be computed using singular value decomposition. Individual data points 
can be transformed to PC space 
and reconstructed from it 
. The vector 
is the PC representation of 
, and its components are referred to as PC scores.
To reduce the dimensionality of a data sample, the transformation matrix 
is reduced by only including eigenvectors 
, whose eigenvalues are above a chosen variance threshold. While the quality of the reconstruction of 
from 
increases with an increasing number of PC scores used, relatively few PC scores are sufficient to reach small reconstruction errors in many practical applications.
2.3 Fatigue damage monitoring
Fatigue damage accumulates as a result of the dynamic load history, which individual components are subjected to. The aim of the proposed reference vehicle approach is to accurately record and classify all fatigue related events in the load history using the standard sensor equipment. If this is successful, it is sufficient to use a small number of reference sensor measurement in order to quantify the absolute fatigue damage contribution of each event. In the following examples, the standard measurements are collected using acceleration sensors, while strain gauges are used as reference sensors. In order to identify fatigue related events from acceleration data, all measurements are divided into small windows of duration T. For each such window, the scattering transform is used to extract signal invariant. This results in a high number of scattering coefficients S for each standard sensor channel. By applying a PCA, the dimensionality of the coefficient space is reduced to a small number of PC scores 
. This reduced representation contains highly condensed information about the dynamic state of the vehicle during the corresponding time window. The connection between the standard acceleration and reference strain sensor data is established on the time window level. Therefore, the strain data is also divided into windows of duration T, each corresponding to the same measurement period as an acceleration sequence. For each window, a fictitious fatigue damage computation is performed following the nominal stress concept [14] under the assumption that local stresses are proportional to local strains. This computation involves rainflow cycle counting using the 4-point algorithm [15], a fictitious Wöhler-curve of the form 
and the elementary Palmgren-Miner rule [16]. Here, N is the number of load cycles to failure for a given strain amplitude 
. This calculation results in a fictitious fatigue damage sum D, since fictitious values of 
and 
are used in the computation. As a result, component failure is not reached when 
, but has to be determined using a relative Miner type approach.
Once both the PC scores of the standard sensor data and the fictitious damage sums of the reference data are available, a simple regression approach can be deployed. This is done using a multivariate polynomial regression of degree two, but many other approaches would achieve similar results. Once the regression model is parameterized, fatigue damage predictions can be made for new time windows, using only standard sensor data. The entire fatigue damage monitoring approach is be parameterized using only unlabeled usage data, which is relatively simple to collect once all sensors are installed.
3 eBIKE DEMONSTRATOR APPLICATION
This section briefly reviews the results of the authors' previous study [7] using a sensor equipped eBike.
3.1 Dataset
The eBike dataset [17] used in this study contains data from 5 acceleration sensor and 16 strain gauges. Their positioning on the eBike is depicted in Figure 1. All sensor channels use a sampling rate of 1200 Hz and the only preprocessing applied to the data consists in the removal of idle times at the beginning and end of each ride.
The dataset is subdivided into unlabeled and labeled data. The unlabeled data consists of general eBike usage and contains data from three different riders, for which only a very broad description of the region of measurement is given. The individual measurement files contain a variety of undergrounds, ranging from asphalt streets and dirt roads to bike ways and short cobblestone sections. The individual files in the labeled data are far shorter, but here, the exact riding conditions are known for each measurement. The underground is either a very even or a cobblestone surface, two different riders are available and speeds are varied from 5 to 25 km/h on even surfaces and from 5 to 15 km/h on cobblestone.
3.2 Fatigue damage regression results
The fatigue damage regression follows the procedure outlined in Subsection 2.3. Here, the individual sequences each contain 4096 data points, corresponding to a duration of 
s. The scattering transform parameters were chosen as 
and 
through manual testing. In the PCA, the first nine PCs were considered.
In order to assess the prediction quality over longer periods of time, the sum over the fatigue damage contributions of all time windows in the test dataset was compared between prediction 
and fictitious fatigue damage sum from measurement D. The resulting fatigue damage ratio 
is visualized in Figure 2A. Here, only strain gauges which exhibited comparatively high strain amplitudes are shown. The predicted fatigue damage ratio is mostly between 
and 
, which relates to a margin of error of factor 2 or less.
3.3 Physical interpretation
While the fatigue damage regression approach can be parameterized using only unlabeled usage data, a physical interpretation of the PC scores can be obtained by examining the labeled dataset. Figure 2B and C show the first two PCs, while the color of the data points corresponds to the underground or riding speed, respectively. It can be observed that these first two components include information on both underground and riding speed, since the data clusters can be separated easily along different decision boundaries in PC 1 and PC 2.
4 LIGHT RAIL VEHICLE APPLICATION
During the project Dresden Measuring Tram [8], led by the Chair of Dynamics and Mechanism Design at TUD, a tram vehicle of type NGT D8 DD was equipped with sensors and monitored during regular operation between 2009 and 2021. During this time, measurement data was continuously collected to further the research and development light rail vehicles. Among many results, more realistic load assumptions were provided for the tram design process, advancements in vehicle dynamic simulation were achieved and infrastructure information was analyzed.
In this section, the presented fatigue monitoring approach is applied to measurement data of the Dresden Measuring Tram project, in order to assess its potential for predictive maintenance in the field of public transport.
4.1 Dataset
In this paper, measurement data is used, which was collected during three days of operation in 2014. The setup features a variety of sensors, including multiple three axial acceleration sensors with a sampling rate of 2 kHz, strain gauges with a sampling rate of 200 Hz and additional information from the vehicle bus system and GPS, sampled at 1 Hz. Figure 3 shows the locations of all sensors, whose data is used in the following evaluation.
4.2 Fatigue damage regression results
The data processing workflow for fatigue monitoring again follows Subsection 2.3. For this application, a sequence duration of 
s was chosen after initial testing. It should be noted that the prediction quality decreases drastically for a duration of 
s, which can most likely be attributed to the significant distance from the acceleration sensors to some strain gauges. For the scattering transform, an averaging scale of 
and the number of wavelets per octave is chosen as 
. In the following results, the reduced representation is parameterized using measurement data from the first measurement day, while the data of the second measurement day is used for testing. Further details on this procedure are given in the discussion in Subsection 4.4.
The sum of fictitious fatigue damage contributions is again compared between prediction 
and measurement result D, using the fatigue damage ratio 
introduced in Subsection 3.2. The results are visualized in Figure 4, sorted by the respective strain gauge position. It is observed, that the fatigue damage ratio is close to the ideal result of 1 for most strain gauges in car 1, while the prediction quality seems to decrease for strain gauges in car 2 and 3. This most likely results from the spacial distance between the acceleration sensors and strain gauges 14 to 18. Here, some highly damaging events occurring at the end of an acceleration time window might only be observed at the start of the subsequent strain measurement window, leading to asynchronous information corresponding windows.
is compared for various strain gauge locations. The colors and sensor numbers are chosen in accordance to Figure 3.4.3 Physical interpretation
The vehicle bus data of the Dresden Measuring Tram allows for an interpretation of the PC region of high fatigue damage contributions. Figure 5A visualizes the fatigue damage contributions of individual measurement windows as a function of PC 1 and PC 4. It can be observed, that the most damaging contributions are localized in a relatively small region of the PC space, highlighted in red. By mapping the vehicle speed information to PC space, shown in Figure 5B, it becomes evident that the highest fatigue damage is accumulated at relatively low vehicle speeds. Using the GPS information, measurement windows in the PC region highlighted in Figure 5A can be visualized in the street map of Dresden, shown in Figure 5C. The highly damaging regions result from the tramway switch crossing, which can be confirmed using infrastructure information on the location of switches. It can be concluded that the event of tramway switch crossing can be reliably detected using the PC 1 and PC 4 of the reduced representation of the acceleration sensor information.
4.4 Discussion
In the preparation of this work, multiple evaluation schemes of the available data were tested. When methods are parameterized and tested using varying combinations of data from measuring day 1 or 2, the observed quality of prediction results is very high. While data from measuring day 3 is still suitable for the qualitative detection of highly damaging events, a significant decrease in the quantitative prediction quality of fictitious fatigue damage sums is experienced. A variety of influences are assumed to be responsible for this behavior. While the route map of the tram is very similar between all three measurement days, certain differences exist, which could result in unknown data regions in the fatigue damage regression. Additionally, the average acceleration amplitudes at the axlebox seemed to be slightly increased on measurement day 3, which might have been caused by a wheel flat or similar out of round phenomenon. Lastly, day 3 included longer periods of standstill, though their effect on the regression results is assumed to be low.
While accumulated strains can be accurately predicted at multiple locations of the bogie, these locations are generally not critical for the prediction of system wide failures since most bogies are designed to be fatigue resistant. Therefore, the results of this paper are primarily intended to show the generality and adaptability of the presented method. High load cycles in the bogie are also transmitted to the car body and is therefore relevant for the fatigue assessment of further adjacent components.
5 CONCLUSION
This paper demonstrates that the combination of scattering transform and PCA provides very general approach to the identification of fatigue related events in vehicle monitoring. Regarding both experimental datasets, the reduced data representation was suitable for the detection of highly damaging events in the vehicle measurement history. A physical interpretation of this representation was also possible in both application cases, using small amounts of labeled data in the eBike example and auxiliary data from the vehicle bus in case of the Measuring Tram.
In future research, this methodology could also be applied to detect critical system states and malfunctions from different physical quantities. It should also be investigated how sensitive this monitoring approach reacts to sensor placement tolerances between different vehicles, as well as individual sensor failures, which are likely to occur over long monitoring intervals.
ACKNOWLEDGMENTS
The authors gratefully acknowledge funding from the German Federal Ministry for Digital and Transport within the mFUND research initiative, Funding code: 19FS2012A. This research is also funded by the European Regional Development Fund (ERDF) and co-financed by tax funds based on the budget approved by the members of the Saxon State Parliament. The authors would like to thank the GWK support for funding this project by providing computing time through the Center for Information Services and HPC (ZIH) at TU Dresden.
Open access funding enabled and organized by Projekt DEAL.
