Sliding Mode Coordinate Positioning-Based Friction Anomaly Monitoring of Multiple Wheelsets for Traction Drive System

Abstract

Accurately monitoring the wheelset–rail friction condition is crucial for ensuring the safety and operational efficiency of the traction drive system. However, the friction characteristics of wheelsets are easily influenced by factors such as ramp transitions and variable railway conditions in the complex environment. These factors significantly increase the difficulty of detecting friction anomalies and accurately locating faulty wheelsets in a timely manner. To address this issue, this paper proposes a sliding mode coordinate positioning–based friction anomaly monitoring scheme for multiple wheelsets in traction drive systems. First, a multi-sliding mode fusion-based friction characteristic observer is developed. Then, an friction coordinate analysis-based anomaly identification method is proposed. Finally, the proposed method is validated on a hardware-in-the-loop (HIL)-based experimental platform. Experimental results demonstrate that the proposed scheme can effectively detect friction anomalies and accurately locate abnormal wheelsets in multi-wheelset traction systems. Compared with traditional methods, the proposed scheme exhibits stronger robustness to varying railway conditions and does not require complex optimization mechanisms, making it suitable for practical on-board applications.

1. Introduction

The traction drive system serves as the core power unit of modern railway vehicles, with its operational reliability directly determining the safety and efficiency of train operation [1,2]. At the interface between the wheelset and the rail, the friction condition is fundamental for transmitting traction and braking forces. However, prolonged service of the train leads to varying degrees of wear between the wheelset and rail. Additionally, the wheelset–rail interface is a highly time-varying system. Particularly under conditions like rain, snow, and ice, abnormal friction fluctuations can occur between the wheelset and rail, leading to issues such as slipping or sliding friction faults [3,4]. If not detected and mitigated promptly, these anomalies not only degrade traction/braking performance but also cause severe damage to both wheelset treads and rails, significantly increasing maintenance costs and posing a threat to running safety [5]. Therefore, the rapid and accurate identification of wheelset–rail friction anomalies provides effective technical support for subsequent optimization control strategies, which is crucial for ensuring the reliable operation of trains.

In recent years, extensive research has been conducted on friction condition identification [6,7,8]. Traditional approaches to friction condition identification often rely on simple and easily measurable parameters such as angular acceleration or the difference in creep velocity between the wheelset and rail [9,10]. These methods offer a certain degree of simplicity and can be directly deployed in the traction drive system. However, this type of method has significant limitations. The most significant drawback is its poor adaptability to different operating conditions, meaning it struggles to adjust to friction fluctuations under different railway types or additional disturbances [11]. Moreover, these traditional methods typically respond only after the slip has already fully developed, which can lead to suboptimal performance and even damage to the railway or wheelset.

As the complexity of rail systems increases and the demand for more reliable and responsive monitoring systems grows, there is a need for more advanced techniques [12,13,14]. These newer methods aim to overcome the limitations of traditional approaches by incorporating real-time data from various sensors, leveraging machine learning algorithms, or utilizing advanced modeling techniques [15,16,17]. Such advancements are critical to ensuring safe and efficient rail operation, particularly in environments where friction conditions are constantly changing. The data-driven techniques have emerged to address these shortcomings by incorporating real-time data from a variety of sensors placed throughout the rail system. These methods leverage machine learning algorithms to continuously process data, enabling them to detect emerging patterns or anomalies in friction conditions [18,19]. While data-driven approaches offer significant improvements in terms of adaptability and precision, they also come with their own set of challenges. For instance, they often require substantial amounts of labeled training data, may struggle with generalization to unseen scenarios, and are highly dependent on the quality and availability of the data.

On the other hand, model-based methods offer distinct advantages that address some of the limitations of data-driven approaches [20,21]. By integrating known physical laws and system dynamics, model-based methods provide a deeper understanding of how different variables interact within the rail system [22]. This can lead to more robust fault detection. In particular, the observer techniques use system models to estimate unmeasurable states, offering a way to monitor the system performance even when direct measurements are unavailable or unreliable. Kalman-based multi-rate observers and Luenberger-based hybrid observers are among the more recent methods [23,24]. However, these approaches either overlook the fact that transitions between different railway line sections can easily lead to false friction anomaly detection, or they require complex optimization mechanisms that make them difficult to deploy on-board.

In addition, most existing methods focus on anomaly detection for a single wheelset. However, a train is a traction system composed of multiple wheelsets [25,26], and the precise localization of anomalous wheelsets is also a critical technical challenge that is urgently needed. In this paper, a sliding mode coordinate positioning-based friction anomaly monitoring scheme of multiple wheelsets for a traction drive system is proposed. The main contributions of this paper are listed as follows:

• A multi-sliding mode fusion (MSMF)-based friction characteristic observer (FCO) is proposed. MSMF can accommodate friction characteristic estimation under both flat and ramp railway line conditions, making the FCO a novel robust friction observer. Compared to traditional methods, the observer effectively improves the estimation accuracy of the friction coefficient on ramp railway lines.
• A friction coordinate analysis (FCA)-based multiple wheelsets anomaly identification method is proposed. The FCA performs anomaly detection at the carriage level and maps the friction characteristics of multiple wheelsets onto a coordinate system to locate the anomalous wheelsets. Compared to traditional methods, the method effectively reduces the misidentification rate when the train transitions to ramp railway lines and achieves precise localization of the abnormal wheelsets.
• In addition, compared to traditional methods, the proposed scheme does not require additional performance optimization mechanisms. The scheme is validated on the HIL-based platform. The results indicate that the proposed scheme can accurately identify the wheelset status of each carriage.

The rest of this article is organized as follows. The Section 2 introduces a multi-wheelset friction mechanism model and explains the urgency of abnormal monitoring strategies. Section 3 presents the proposed monitoring scheme based on the mechanism model. Section 4 validates the proposed scheme and discusses the dynamic data of the detection and localization functions within the scheme. Section 5 summarizes the proposed scheme.

2. Multiple Wheelset Model and Friction Anomaly Analysis

2.1. Multiple Wheelset Model

As shown in Figure 1, the driving force for train movement originates from the wheelset–rail friction force. The smooth operation of the train is achieved through the total friction force provided by multiple wheelsets. The traction dynamic model of multiple wheelsets is described as follows:

$$\left[\sum _{i=1}^{n}F{s}_i-f\left(v_t,M\right)\right]/M={\displaystyle \frac{d{v}_t}{dt}}$$

(1)

where $F{s}_i$ is the friction force of the $ith$ wheelset, $v_t$ is the train velocity, M is the train weight, and $f\left(v_t,M\right)$ is the running resistance function related to the train speed.

The friction force is expressed as follows:

$$F{s}_i=\mu \left(v_{si}\right)·W_i·g$$

(2)

where $v_{si}$ is the creep velocity of the $ith$ wheelset, $W_i$ is the axle load of the $ith$ wheelset, g is the acceleration of gravity, and $\mu \left(v_{si}\right)$ is the friction coefficient function related to the creep velocity.

2.2. Friction Anomaly Analysis

The friction state of the wheelset is strongly correlated with the wheelset–rail friction condition. The expressions of the friction coefficient under healthy friction conditions and abnormal friction conditions are as follows:

$$\mu \left(v_{si}\right)=\left\{\begin{array}{c}{c}_{hac\_0}·e^{a_{hac\_0}·v_{si}}-d_{hac\_0}·e^{b_{hac\_0}·v_{si}}\\ c_{aac\_j}·e^{a_{aac\_j}·v_{si}}-d_{aac\_j}·e^{b_{aac\_j}·v_{si}}\end{array}\right.$$

(3)

where $a_{hac\_0},b_{hac\_0},c_{hac\_0},d_{hac\_0}$ are the empirical coefficients of the friction function under healthy friction conditions, and $a_{aac\_j},b_{aac\_j},c_{aac\_j},d_{aac\_j}$ are the empirical coefficients of the friction function under abnormal friction conditions, $j=1,2,\cdots ,n$. This is an empirical model commonly used in current research to describe the friction characteristics between the wheelset and the rail [10,23].

As shown in Figure 2, the friction coefficient function between the wheelset and rail varies significantly under different friction conditions. Under the healthy friction condition (HFC), the friction coefficient consistently provides sufficient friction force to ensure the overall traction performance of the train. In contrast, under the abnormal friction condition (AFC), the friction coefficient exhibits a sudden decline, and the magnitude of this decrease varies depending on the severity of the abnormal condition.

It should be emphasized that each AFC corresponds to an optimal creep velocity. Under an AFC, when the friction coefficient drops abruptly, it will first gradually increase toward the optimal friction point. During this stage, the abnormal friction state of the wheelset remains relatively stable. Once the creep velocity of the wheelset exceeds the optimal friction point, the wheelset will rapidly enter a non-stationary state. Therefore, the ability to identify the operating condition of the wheelset at the very onset of an friction anomaly is crucial for ensuring the smooth and stable operation of the train.

3. Sliding Mode Coordinate Positioning-Based Friction Anomaly Identification

3.1. MSMF-Based Friction Characteristic Observer

Different operating railway lines lead to coupled variations in the friction characteristics of the wheelsets in each carriage. To address this issue, this section proposes the MSMF-based ACO scheme. Specifically, the ACO consists of the sliding mode-based load observer module, the sliding mode-based line gradient observer module, and the friction characteristic calculation module.

(1) Sliding mode-based load observer (SMLO)

The load information of the wheelsets in each carriage is an important component of the input to the reconstructed line gradient observer. The state equation of the $ith$ wheelset in each carriage is as follows:

$$\left[\begin{array}{c}{\displaystyle \frac{d}{dt}}{w}_i\hfill \\ {\displaystyle \frac{d}{dt}}{T}_{L_i}\hfill \end{array}\right]=\left[\begin{array}{cc}0& {\displaystyle \frac{-1}{G_i}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{w}_i\hfill \\ T_{L_i}\hfill \end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{G_i}}\hfill \\ 0\hfill \end{array}\right]T_{e_i}$$

(4)

where $w_i$ is the wheelset velocity of the $ith$ wheelset, $T_{L_i}$ is the load torque of the $ith$ drive motor, $T_{e_i}$ is the electromagnetic torque of the $ith$ drive motor, $G_i=J_{m_i}+J_{d_i}/({\eta }_i\ast R_{g_i}^2)$, $J_{m_i}$ is the rotation inertia of the $ith$ drive motor, $J_{d_i}$ is the rotation inertia of the $ith$ wheelset, and ${\eta }_i$, $R_{{\mathrm{g}}_i}$ are the transmission efficiency and gearbox ratio between the $ith$ wheelset and the drive motor.

By selecting the wheelset velocity and the load torque as the estimation variables, the constructed sliding mode-based load observer is expressed as follows:

$$\left[\begin{array}{c}{\displaystyle \frac{d}{dt}}{\widehat{w}}_i\hfill \\ {\displaystyle \frac{d}{dt}}{\widehat{T}}_{L_i}\hfill \end{array}\right]=\left[\begin{array}{cc}0& {\displaystyle \frac{-1}{G_i}}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\widehat{w}}_i\hfill \\ {\widehat{T}}_{L_i}\hfill \end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{G_i}}\hfill \\ 0\hfill \end{array}\right]T_{e_i}+\left[\begin{array}{c}1\\ {\delta }_i\end{array}\right]\partial \left(e_{w_i}\right)$$

(5)

where ${\widehat{w}}_i$ is the estimated velocity of the $ith$ wheelset, ${\widehat{T}}_{L_i}$ is the estimated load torque of the $ith$ drive motor, $\partial \left(e_{w_i}\right)$ is the compensation function of the $ith$ SMLO, ${\delta }_i$ is the gain coefficient of the $ith$ SMLO, and $e_{w_i}$ is the estimation error of the $ith$ wheelset velocity, $e_{w_i}=w_i-{\widehat{w}}_i$.

Combining Equations (4) and (5), the differential function of the SMLO estimation error is expressed as follows:

$${\dot{e}}_{w_i}={\displaystyle \frac{-1}{G_i}}{e}_{T_i}-\partial \left(e_{w_i}\right)$$

(6)

where $e_{T_i}$ is the estimation error of the load torque of the $ith$ drive motor.

The integral sliding mode surface and the exponential reaching law are adopted to construct the compensation function of the SMLO.

$$S_{w_i}=e_{w_i}+c_{w_i}\int e_{w_i}dt$$

(7)

$${\dot{S}}_{w_i}=-{\kappa }_{1\_w_i}\mathrm{sgn}\left(S_{w_i}\right)-{\kappa }_{2\_w_i}{S}_{w_i}$$

(8)

where $S_{w_i}$ is the integral sliding mode surface of the $ith$ SMLO with respect to the velocity error, ${\dot{S}}_{w_i}$ is the exponential reaching law of the $ith$ SMLO with respect to the velocity error, $c_{w_i}$ is the gain coefficient of the $ith$ sliding mode surface, and ${\kappa }_{1\_w_i}$, ${\kappa }_{2\_w_i}$ are the reaching coefficients of the $ith$ reaching law.

Based on Equations (6)–(8), the compensation function of SMLO is expressed as follows:

$$\partial \left(e_{w_i}\right)=c_{w_i}{e}_{w_i}+{\kappa }_{1\_w_i}\mathrm{sgn}\left(S_{w_i}\right)+{\kappa }_{2\_w_i}{S}_{w_i}$$

(9)

By selecting appropriate sliding surfaces and reaching-law gain coefficients, the load torque of the $ith$ wheelset can be obtained quickly and accurately, $c_{w_i}=150$, ${\kappa }_{1\_w_i}=260$, ${\kappa }_{2\_w_i}=200$, ${\delta }_i=-900$. The estimates obtained from the SMLO serve as an important reference for reconstructing the inputs of the line gradient observer, which is crucial for ensuring the proposed scheme’s robustness across different railway lines.

(2) Sliding mode-based line gradient observer (SMLGO)

The train operational dynamics model on the ramp railway line is expressed as follows:

$${\displaystyle \frac{d{v}_t}{dt}}=-\left(\mathrm{a}+b{v}_t+c{v}_t^2\right)+{\displaystyle \frac{1}{M}}\sum _{i=1}^{n}F{s}_i-g\alpha$$

(10)

where $a,b,c$ are the resistance coefficients of the train during operation, and $\alpha$ is the line gradient.

The friction force in Equation (10) is reconstructed by incorporating the estimates obtained from the SMLO, and the SMLGO is subsequently developed based on sliding mode variable structure theory. The expression of SMLGO is written as follows:

$${\displaystyle \frac{d{\widehat{v}}_t}{dt}}=-\left(\mathrm{a}+b{\widehat{v}}_t+c{\widehat{v}}_t^2\right)+{\displaystyle \frac{1}{M}}\sum _{i=1}^n\left({\displaystyle \frac{{\eta }_i·R_{g_i}}{r_{\mathrm{i}}}}{\widehat{T}}_{L_i}\right)-g·{\lambda }_{\alpha }·F\left(\overline{\alpha }\right)$$

(11)

where ${\widehat{v}}_t$ is the estimated velocity of the train, $r_{\mathrm{i}}$ is the rotational radius of the $ith$ wheelset, ${\lambda }_{\alpha }$ is the gain coefficient related to the line gradient, $\overline{\alpha }$ is the estimation error of the line gradient, and $F\left(\overline{\alpha }\right)$ is the sliding mode function related to the line gradient. By selecting appropriate gain coefficients, ${\lambda }_{\alpha }=0.8$, the line gradient can be obtained.

$$F\left(\overline{\alpha }\right)=\left\{\begin{array}{c}+1,if\overline{\alpha }\ge 0\\ -1,if\overline{\alpha }<0\end{array}\right.$$

(12)

$$\widehat{\alpha }={\lambda }_{\alpha }·F\left(\overline{\alpha }\right)$$

(13)

(3) Friction characteristic calculation

The friction characteristics of the wheelset are not only related to the condition of the wheelset–rail interface but also closely associated with the axle load of the wheelset. On the flat railway line (FRL), the axle load remains constant. However, when the train enters the ramp railway line (RRL), the wheelset axle load shifts, resulting in certain differences in the friction characteristics among the wheelsets of each carriage. It should be emphasized that such variations caused by transitions between different railway types can lead to fluctuations in wheelset friction, which may in turn result in false identification of wheelset anomalies.

The axle-load model of each wheelset in each carriage under the ramp railway line is expressed as follows:

$$\left\{\begin{array}{c}{N}_{AL\_1}^{RRL}=Wgcos\alpha +{\displaystyle \frac{Wg}{L}}(Z-z)sin\alpha -{\displaystyle \sum _{i=1}^4}F{s}_i\left({\displaystyle \frac{Z-z}{2L}}+{\displaystyle \frac{Z}{2L_b}}\right)\\ N_{AL\_2}^{RRL}=Wgcos\alpha +{\displaystyle \frac{Wg}{L}}(Z-z)sin\alpha -{\displaystyle \sum _{i=1}^4}F{s}_i\left({\displaystyle \frac{Z-z}{2L}}-{\displaystyle \frac{Z}{2L_b}}\right)\\ N_{AL\_3}^{RRL}=Wgcos\alpha -{\displaystyle \frac{Wg}{L}}(Z-z)sin\alpha +{\displaystyle \sum _{i=1}^4}F{s}_i\left({\displaystyle \frac{Z-z}{2L}}-{\displaystyle \frac{Z}{2L_b}}\right)\\ N_{AL\_4}^{RRL}=Wgcos\alpha -{\displaystyle \frac{Wg}{L}}(Z-z)sin\alpha +{\displaystyle \sum _{i=1}^4}F{s}_i\left({\displaystyle \frac{Z-z}{2L}}+{\displaystyle \frac{Z}{2L_b}}\right)\end{array}\right.$$

(14)

where $N_{AL\_1}^{RRL},N_{AL\_2}^{RRL},N_{AL\_3}^{RRL},N_{AL\_4}^{RRL}$ are the axle loads of each wheelset in each carriage under ramp railway line, Z is the height of the coupler above the rail surface, z is the height of the bogie, L is half of the distance between the centers of the two bogies, and $L_b$ is half of the wheelbase of the wheelset.

By reconstructing the axle-load model based on the estimations from SMLO and SMLGO, the friction characteristics of the wheelset on the ramp railway line are determined as follows:

$$\left\{\begin{array}{c}{\widehat{N}}_{AL\_1}^{RRL}=Wgcos\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)+{\displaystyle \frac{Wg}{L}}(Z-z)sin\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)-{\displaystyle \sum _{i=1}^4}{\displaystyle \frac{{\eta }_i·R_{g_i}}{r_{\mathrm{i}}}}{\widehat{T}}_{L_i}\left({\displaystyle \frac{Z-z}{2L}}+{\displaystyle \frac{Z}{2L_b}}\right)\\ {\widehat{N}}_{AL\_2}^{RRL}=Wgcos\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)+{\displaystyle \frac{Wg}{L}}(Z-z)sin\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)-{\displaystyle \sum _{i=1}^4}{\displaystyle \frac{{\eta }_i·R_{g_i}}{r_{\mathrm{i}}}}{\widehat{T}}_{L_i}\left({\displaystyle \frac{Z-z}{2L}}-{\displaystyle \frac{Z}{2L_b}}\right)\\ {\widehat{N}}_{AL\_3}^{RRL}=Wgcos\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)-{\displaystyle \frac{Wg}{L}}(Z-z)sin\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)+{\displaystyle \sum _{i=1}^4}{\displaystyle \frac{{\eta }_i·R_{g_i}}{r_{\mathrm{i}}}}{\widehat{T}}_{L_i}\left({\displaystyle \frac{Z-z}{2L}}-{\displaystyle \frac{Z}{2L_b}}\right)\\ {\widehat{N}}_{AL\_4}^{RRL}=Wgcos\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)-{\displaystyle \frac{Wg}{L}}(Z-z)sin\left({\lambda }_{\alpha }·F\left(\overline{\alpha }\right)\right)+{\displaystyle \sum _{i=1}^4}{\displaystyle \frac{{\eta }_i·R_{g_i}}{r_{\mathrm{i}}}}{\widehat{T}}_{L_i}\left({\displaystyle \frac{Z-z}{2L}}+{\displaystyle \frac{Z}{2L_b}}\right)\end{array}\right.$$

(15)

where ${\widehat{N}}_{AL\_1}^{RRL},{\widehat{N}}_{AL\_2}^{RRL},{\widehat{N}}_{AL\_3}^{RRL},{\widehat{N}}_{AL\_4}^{RRL}$ are the reconstructed axle load of each wheelset in each carriage under the ramp railway line.

Integrating the load torque of the wheelset, the calculation model of the friction characteristics of the train under different railway lines are determined as follows:

$${\mu }_i^{\ast }=\left\{\begin{array}{c}{\displaystyle \frac{{\eta }_i·R_{{\mathrm{g}}_i}}{W·g·r_i}}·{\widehat{T}}_{L_i}, if train on FRL\\ {\displaystyle \frac{{\eta }_i·R_{{\mathrm{g}}_i}}{N_{AL\_i}^{RRL}·r_i}}·{\widehat{T}}_{L_i}, if train on RRL\end{array}\right.$$

(16)

where ${\mu }_i^{\ast }$ is the friction coefficient used to characterize the friction characteristics of the $ith$ wheelset under different railway lines.

3.2. FCA-Based Multiple Wheelsets Anomaly Identification

The identification of abnormal wheelset friction in trains is divided into two stages: anomaly detection and anomaly localization. In the first stage, anomaly detection is performed at the carriage level, which effectively reduces the impact of friction fluctuations when the train transitions from FRL to RRL. In the second stage, once the carriage-level abnormal condition is detected, a localization algorithm is applied to accurately pinpoint the faulty wheelsets.

(1) Friction dynamic evaluation-based anomaly detection

The friction anomaly detection function for the train is expressed as follows:

$$D_{A\_Slip}\left(k\right)=\left|\sum {\mu }_i^{\ast }\left(k\right)-{\displaystyle \frac{1}{m}}\sum _{\sigma =k-m}^{k-1}\sum {\mu }_i^{\ast }\left(\sigma \right)\right|$$

(17)

where $D_{A\_Slip}\left(k\right)$ is the detection feature at the $kth$ instant, ${\mu }_i^{\ast }\left(k\right)$ is the friction coefficient of the $ith$ wheelset at the $kth$ instant, ${\mu }_i^{\ast }\left(\sigma \right)$ is the friction coefficient of the $ith$ wheelset at the $\sigma th$ instant, and m is the number of sampled friction coefficients.

$$R_{A\_Slip}\left(k\right)=\left\{\begin{array}{c}1,D_{A\_Slip}\left(k\right)>\Delta \mu \\ 0,D_{A\_Slip}\left(k\right)\le \Delta \mu \end{array}\right.$$

(18)

where $R_{A\_Slip}\left(k\right)$ is the detection result at the $kth$ instant, and $\Delta \mu$ is the detection threshold.

If $R_{A\_Slip}\left(k\right)=0$, the train is operating normally; if $R_{A\_Slip}\left(k\right)=1$, the train has the friction fault and proceeds to wheelset anomaly localization.

(2) Coordinate distance distribution-based anomaly localization

The abnormal localization of the wheelset is achieved through the coordinate analysis of the four wheelsets of each carriage. The coordinate-reconstructed expressions of the friction coefficient for each wheelset are as follows:

$$\left\{\begin{array}{c}{\sigma }_1={\displaystyle \frac{1}{\epsilon +{\widehat{\mu }}_1}}\\ {\sigma }_2={\displaystyle \frac{1}{\epsilon +{\widehat{\mu }}_2}}\\ {\sigma }_3={\displaystyle \frac{1}{\epsilon +{\widehat{\mu }}_3}}\\ {\sigma }_4={\displaystyle \frac{1}{\epsilon +{\widehat{\mu }}_4}}\end{array}\right.$$

(19)

where ${\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4$ are the reconstructed friction characteristics of each wheelset, and $\epsilon$ is the constant.

By assigning ${\sigma }_1$ and ${\sigma }_2$ to the X-axis and ${\sigma }_3$ and ${\sigma }_4$ to the Y-axis, the coordinate distribution of the four wheelsets for each carriage can be described as follows:

$$\left\{\begin{array}{c}\left(x_1,y_1\right)=\left({\sigma }_1,{\sigma }_3\right)\\ \left(x_1,y_2\right)=\left({\sigma }_1,{\sigma }_4\right)\\ \left(x_2,y_1\right)=\left({\sigma }_2,{\sigma }_3\right)\\ \left(x_2,y_2\right)=\left({\sigma }_2,{\sigma }_4\right)\end{array}\right.$$

(20)

Let ${\sigma }_i^n$ denote the wheelset coordinates under normal operating conditions and ${\sigma }_i^f$ denote the wheelset coordinates under abnormal fault conditions.

The coordinate distribution of the wheelset after the occurrence of an abnormal fault is shown in Figure 3. Each coordinate plot contains four coordinate points constructed from projected coordinates. Figure 3a–d show the coordinate distributions of wheelset-1, wheelset-2, wheelset-3, and wheelset-4, respectively, after the occurrence of the abnormal condition. The coordinate point corresponding to the faulty wheelset is always the farthest from the origin of the coordinate system, which serves as an important basis for locating the abnormal wheelset based on the coordinate distribution.

Based on Figure 3, we can observe that the coordinates of abnormal wheelsets are always located at the periphery, corresponding to the yellow region, whereas the coordinates of normally operating wheelsets are always near the origin, corresponding to the green region. Therefore, the abnormal coordinate function based on the Euclidean distance is proposed.

$$\left\{\begin{array}{c}{J}_1\left[k\right]=\sqrt{{\sum }_{j=k-n}^k\left(x_1{\left[j\right]}^2+y_1{\left[j\right]}^2\right)}+\sqrt{{\sum }_{j=k-n}^k\left(x_1{\left[j\right]}^2+y_2{\left[j\right]}^2\right)}\\ J_2\left[k\right]=\sqrt{{\sum }_{j=k-n}^k\left(x_2{\left[j\right]}^2+y_1{\left[j\right]}^2\right)}+\sqrt{{\sum }_{j=k-n}^k\left(x_2{\left[j\right]}^2+y_2{\left[j\right]}^2\right)}\\ J_3\left[k\right]=\sqrt{{\sum }_{j=k-n}^k\left(x_1{\left[j\right]}^2+y_1{\left[j\right]}^2\right)}+\sqrt{{\sum }_{j=k-n}^k\left(x_2{\left[j\right]}^2+y_1{\left[j\right]}^2\right)}\\ J_4\left[k\right]=\sqrt{{\sum }_{j=k-n}^k\left(x_1{\left[j\right]}^2+y_2{\left[j\right]}^2\right)}+\sqrt{{\sum }_{j=k-n}^k\left(x_2{\left[j\right]}^2+y_2{\left[j\right]}^2\right)}\end{array}\right.$$

(21)

where $J_1\left[k\right],J_2\left[k\right],J_3\left[k\right],J_4\left[k\right]$ are the anomaly fault localization functions of wheelset-1, wheelset-2, wheelset-3 and wheelset-4.

Based on Figure 3 and Equation (21), when a certain wheelset experiences an abnormal friction fault, its positioning function attains the maximum value compared with the positioning functions of normal wheelsets. Therefore, a positioning function for the wheelset with abnormal friction is established as follows:

$$J_{max}\left[k\right]=\left\{J_1\left[k\right],J_2\left[k\right],J_3\left[k\right],J_4\left[k\right]\right\}$$

(22)

If $J_{max}\left[k\right]=J_1\left[k\right]$, the identification algorithm determines that the first wheelset is in an abnormal friction state. If $J_{max}\left[k\right]=J_2\left[k\right]$, the identification algorithm determines that the second wheelset is in an abnormal friction state. If $J_{max}\left[k\right]=J_3\left[k\right]$, the identification algorithm determines that the third wheelset is in an abnormal friction state, If $J_{max}\left[k\right]=J_4\left[k\right]$, the identification algorithm determines that the fourth wheelset is in an abnormal friction state. The schematic diagram of the proposed friction anomaly identification scheme is shown in Figure 4.

4. Experimental

4.1. Experimental Setup

The proposed scheme in this paper is verified by the HIL-based platform, as shown in Figure 5, which consists of the power supply, dSpace, control board, drive board, and the host computer. In this platform, the power supply equipment provides energy to the control board and the driver board. The dSpace acts as a simulator running the model, thereby receiving control signals. The proposed method in this paper is implemented on the control board, with a running frequency of 5 kHz, while the physical models of the train and wheelset are simulated on the dSpace (LABBOX 19-SLOT). During the testing process, the friction module in dSpace is switched via ControlDesk 7.6 to enable model transitions under different operating conditions, thereby reproducing abnormal friction scenarios of varying severity. In addition, during the validation process, the wheelset speed and train speed in the dSpace model are primarily collected and transmitted to the control board for corresponding friction estimation, feature extraction, and evaluation.

In this section, three classical cases are designed. Case 1: The train transitions from FRL to RRL under normal operating conditions. Case 2: The early abnormal fault occurs in the first wheelset. Case 3: The serious abnormal fault occurs in the third wheelset. These cases are used to verify the accuracy of the proposed scheme in identifying the friction anomaly conditions under different railway lines.

4.2. Experimental Results

4.2.1. Case1

In this case, the train transitions from FRL to RRL at the fifth second while maintaining normal operating conditions. Figure 6 shows the dynamic variation trend of the friction coefficients for the four wheelsets of each carriage. On the FRL, the axle load of each wheelset is constant, and the friction coefficient of each wheelset follows the same trend. When the train enters the RRL, the axle load of each wheelset shifts, causing different trends in the friction coefficients of each wheelset. However, this is a normal physical mechanism, and the wheelsets are still in a normal operating state. Traditional methods overlook the changes in friction characteristics caused by such railway line transitions, which can easily lead to false detection of these changes as a friction anomaly.

Figure 7 shows the trend of characteristic changes in the wheelset’s operational state. From Figure 7, it is clear that when the train transitions from FRL to RRL, the detection features of the traditional method exhibit significant fluctuations, triggering the detection threshold. This indicates that the traditional method is influenced by changes in wheelset axle load under this condition, leading to false detection of wheelset anomalies. In contrast, the proposed method maintains stable detection features under the same condition and does not trigger the detection threshold. Therefore, the proposed method can adapt to different railway lines during train operations and demonstrates stronger robustness in friction anomaly detection performance.

Based on the above analysis, we can conclude that, compared to traditional methods, the proposed anomaly detection does not produce false positives when the train transitions from FRL to RRL. We also conducted extensive experimental validation, and in this case, traditional methods had an almost 100% false positive rate. This is because most methods overlook the friction fluctuations caused by the switch between different railways. In contrast, the proposed method had an almost 0% false positive rate, effectively improving the detection accuracy of train friction anomalies.

4.2.2. Case2

Under this condition, wheelset-1 of the train experiences an early friction anomaly. Figure 8 shows the friction coefficient variation trend for each wheelset. Before the third second, the friction coefficients of all wheelsets maintain a consistent and steady change. However, after the onset of the anomaly, the friction coefficient of wheelset-1 suddenly drops and then slowly recovers to a balanced friction state similar to the other wheelsets. In reality, at this point, wheelset-1 is in an early abnormal state. If the train increases its driving force, it is highly likely that wheelset-1 will rapidly enter a severe abnormal state. Therefore, accurately locating the faulty wheelset is crucial for ensuring the safe operation of the train.

Figure 9 shows the feature variation trend of wheelset anomaly localization. When the wheelsets are in normal operating conditions, the localization features of each wheelset remain stable. After the anomaly fault occurs in wheelset-1, the fault features used for anomaly localization show significant fluctuations. The feature variation trend for wheelset-1 is the most noticeable. Based on the anomaly identification mechanism of the proposed scheme, the faulty wheelset-1 can be accurately localized. This also demonstrates the superior performance of the proposed scheme in the localization of faulty wheelsets.

Based on the above analysis, it can be concluded that the proposed method is capable of accurately localizing abnormal wheelsets in this case. In addition, extensive experimental validations were conducted, and a statistical analysis of the localization accuracy under early-stage friction anomalies in the train was performed. The results show that even under such early abnormal operating conditions, the proposed approach still achieves a localization accuracy of 96%. This also demonstrates that the proposed method can effectively identify the relevant wheelsets at the early stage of friction anomalies in trains.

4.2.3. Case3

In this case, wheelset-3 of the train experienced a serious friction anomaly. Figure 10 shows the friction coefficient variation trend for each wheelset. Compared to the early friction anomaly, although the friction coefficient of wheelset-3 showed a slight recovery after the serious friction anomaly fault, it still did not return to a stable state. Furthermore, the friction coefficient of wheelset-3 eventually entered a sharp decline. If such a serious anomaly is not localized in time, it can easily lead to significant incidents such as abnormal vibrations or derailment.

Figure 11 shows the feature variation trend of wheelset anomaly localization under serious anomaly conditions. Compared to early friction anomalies, the localization feature of the wheelset anomalies exhibit more significant fluctuations under these serious anomaly conditions. Especially at the third second, after wheelset-3 experiences a serious anomaly, the anomaly feature representing wheelset-3 shows a large amplitude change. Similarly, based on the recognition mechanism of the proposed scheme, the anomaly fault of wheelset-3 can be precisely localized.

Based on the above analysis, it can be concluded that the proposed method is capable of accurately locating severely abnormal wheelsets in this case. In addition, extensive experimental validations are conducted, and a statistical analysis of the localization accuracy under conditions of serious friction anomalies in the train is performed. The results indicate that, under such severely abnormal conditions, the proposed approach achieves a localization accuracy of 98%. This further demonstrates that the proposed method maintains strong identification performance across varying levels of abnormal operating conditions.

4.3. Discussion

The experimental validation of the three cases demonstrates the superiority of the proposed scheme under different operating conditions. In Case 1, where the train transitions from FRL to RRL, the proposed scheme is unaffected by the friction fluctuations caused by the railway line change. In Case 2 and Case 3, the proposed scheme can accurately localize the wheelsets experiencing early anomaly faults and serious anomaly faults. All cases confirm that the proposed scheme can adapt to different operating lines, offering greater robustness compared to traditional methods, while also precisely localizing the faulty wheelsets.

5. Conclusions

This paper presents a sliding mode coordinate positioning-based friction anomaly monitoring scheme for multiple wheelsets in train traction drive systems. The proposed framework integrates friction characteristic observation and coordinate-based anomaly localization to achieve reliable monitoring of wheelset–rail friction conditions. First, a MSMF-based friction characteristic observer is designed to estimate key variables related to the friction state. By combining the SMLO with SMLGO, the proposed observer can effectively reconstruct wheelset friction characteristics under different railway operating conditions, including FRL and RRL. Second, an FCA-based anomaly identification method is developed. The friction characteristics of each wheelset are mapped into a coordinate domain, and the Euclidean distance distribution among the reconstructed coordinates is used to locate the abnormal wheelset. Finally, the effectiveness of the proposed scheme is verified. Compared with traditional identification approaches, the proposed scheme demonstrates improved robustness and adaptability under varying railway operating conditions.

Overall, the proposed scheme provides a practical solution for friction monitoring in multi-wheelset traction systems. Future work will focus on extending the approach to more complex operating scenarios and integrating it with traction control strategies to further enhance train operational safety and reliability. Additionally, we will develop optimized control strategies for abnormal states based on this monitoring method, aiming to effectively suppress wheelset anomalies and further enhance the safety and reliability of train operations. Furthermore, investigating how to validate the robustness of the proposed method under disturbances such as system noise and parameter uncertainties in real-world railway systems will be an important direction for future research.