Uncertainty Analysis and Evaluation of Gauge Measurement in Track Geometry Inspection Systems

Abstract

To ensure the credibility of measurement data from the Track Geometry Detection System (TGDS) and to achieve its dynamic and accurate evaluation, this paper analyzes and assesses the sources of uncertainty in the measurement of track geometric irregularities by the track inspection system based on a calibration test bench in the laboratory. To address the issue that the track inspection system is prone to sporadic outliers under electromagnetic interference and vibration, while conventional statistical methods are sensitive to outliers and tend to overestimate the repeatability uncertainty, this paper introduces a robust statistical method based on median absolute deviation (MAD) to evaluate the uncertainty introduced by repeatability. This robust approach effectively suppresses the influence of outliers by using the median instead of the mean and absolute deviations instead of squared deviations, thereby yielding a more realistic and reliable estimate of repeatability. Taking track gauge measurement as an example for uncertainty evaluation, experimental results show that the expanded uncertainty U = 0.64 mm, which satisfies one-third of the tolerance requirement for track gauge measurement, verifying the feasibility of the proposed method. The quantitative results of uncertainty sources in this paper can be used as Type B input for uncertainty evaluation in field practical measurements, providing a reliable metrological basis for the uncertainty evaluation of track inspection systems. Meanwhile, the dynamic evaluation of track inspection systems is realized, filling the gap in their dynamic and reliable evaluation under complex interferences.

1. Introduction

With the continuous increase in railway mileage in China, by the end of 2025, the national railway operating mileage had reached 165,000 km, of which high-speed railways accounted for over 50,000 km, establishing the world’s largest and most advanced high-speed railway network [1]. To evaluate track quality and ensure operational safety, railway tracks require regular inspection [2]. The Track Quality Index (TQI) is a technical indicator that comprehensively evaluates the state of track geometric irregularities using mathematical statistical methods [3]. This index quantifies the smoothness of the track structure by calculating the sum of the standard deviations of seven geometric parameters—including longitudinal level (left and right), alignment, gauge, cross-level, and twist—over a specified track section. A smaller TQI value indicates better track quality. Therefore, accurately measuring track geometric irregularities plays a crucial role in maintaining railway quality and ensuring the safety of both life and property [4].

Track inspection technology [5] is a key technology for ensuring track safety in railway transportation engineering. It is primarily implemented through two methods: static inspection (using manual or light measurement trolleys to measure parameters such as gauge and cross-level) and dynamic inspection (using track inspection cars under train loads). The evolution of dynamic inspection technology has gone through three generations. Currently, dynamic inspection technology has become the mainstream approach. Therefore, ensuring the accuracy and reliability of geometric irregularity measurements obtained from track inspection systems has become a critical challenge.

The core significance of measurement uncertainty evaluation in track inspection systems lies in quantifying the degree of confidence in inspection results [6], thereby providing a scientific basis for safe track operation and maintenance as well as system optimization. It ensures the reliability of inspection results by clarifying the error range of measurements, avoiding misjudgment of track defects caused by ambiguous results, and reducing operational and maintenance risks. Furthermore, it identifies the primary sources of uncertainty, offering targeted directions for equipment calibration and algorithm optimization. Additionally, it meets standards and compliance requirements, aligning with the metrological requirements for measurement equipment in the rail transit industry, ensuring that inspection data possess legal validity and mutual recognition to support processes such as acceptance and assessment.

In the study of uncertainty sources in track inspection system measurements and vehicle–track coupling, Han Zhi et al. [7] proposed a dynamic calibration method for track inspection systems based on actual railway lines, introduced a bisection method to reduce systematic measurement errors, and performed an analysis and uncertainty evaluation. Wang Yan [8] proposed a method based on an integrated calibration test bench for track inspection systems, enabling full-system operating condition simulation and parameter traceability calibration. Xu et al. [9] developed a three-dimensional train–track interaction model that couples the train and track into a complete system matrix. Wang et al. [10] investigated a wayside hunting detection system for monitoring the absolute displacement of wheelsets. Fu et al. [11], based on the dynamic response analysis of vehicle–track coupling, designed the track structure and discussed the vibration responses of the vehicle and track under different operating conditions by calculating the coupling system of an HTS maglev train.

Currently, the evaluation of measurement uncertainty for track inspection systems faces several challenges. First, traditional methods based on actual railway lines are not only time-consuming and labor-intensive but also struggle to account for the influence of various operating conditions (such as vibration amplitude and spatial attitude) on the inspection quality of the track inspection system. Second, the operational environment of the track inspection system involves a dynamically coupled scenario between the track and the vehicle [12], whereas traditional methods are based on static evaluation, making it difficult to accurately assess the measurement uncertainty of the track inspection system. Therefore, this paper analyzes and dynamically evaluates the sources of uncertainty in the measurement process of the track inspection system based on a calibration test bench in the laboratory.

The experimental method involves mounting the object under test (i.e., the track inspection system) on a calibration test bench to measure a simulated track. The geometric irregularity values measured by the track inspection system are compared with the reference values, and the indication errors for each geometric irregularity are output. By varying the test bench parameters and initial conditions, the potential sources of uncertainty affecting the measurements of the track inspection system are analyzed. Finally, this paper takes the uncertainty evaluation of the gauge measurement process as an example, conducting an uncertainty evaluation of the gauge indication error using the Guide to the Expression of Uncertainty in Measurement (GUM). The track inspection system is subject to electromagnetic interference and vibration in actual measurements [13,14,15]. Therefore, the robust statistical method [16] is used to evaluate repeatability uncertainty, better reflecting real operating conditions and avoiding outlier interference in traditional methods.

The analysis and quantification of uncertainty sources in the measurement process of the track inspection system can serve as Type B input sources for subsequent evaluations, simplifying the complexity of the actual evaluation process while enhancing the reliability of the evaluation results.

2. Measurement Principle

2.1. Measurement Principle of the Track Geometry Inspection System

Gauge refers to the minimum inner distance between the left and right rails at the same cross-section of the track, measured at a point 16 mm below the top surface of the rail head. The nominal value of standard gauge is 1435 mm, as shown in Figure 1.

Figure 1 shows a schematic diagram of the track inspection system. The system employs laser camera-based non-contact measurement technology [17] to detect track geometric irregularities. The laser and camera form an integrated unit, with the line laser arranged perpendicular to the longitudinal centerline of the rail. The camera captures cross-sectional images of the rail illuminated by the laser at a specific angle. After processing by the laser camera assembly, the image data are transmitted to a real-time processing computer, which ultimately outputs the measurement results of track geometric irregularities.

2.2. Calibration Method of the Track Geometry Inspection System

A schematic diagram of the track geometry parameter system calibration test bench constructed in the laboratory is shown in Figure 2. This calibration test bench is based on a multi-dimensional vibration test platform and mainly comprises the following components:

• Vehicle body vibration simulator and bogie vibration simulator: used to simulate the position and attitude variations in the train during actual operation.
• Track vibration simulator: used to simulate changes in track superelevation.
• Geometric irregularity vibration simulator: used to simulate variations in track geometric irregularities.
• Measurement calibration device: used to measure changes in track irregularities, providing reference values for evaluating the indication errors of track geometry parameters.

The specific functions and structures of each sub-test bench are as follows:

• Vehicle Body Attitude Simulation Sub-Test Bench: A six-degree-of-freedom vibration test bench is adopted, which independently controls the actuators to achieve three-dimensional translation and three-dimensional rotation, fully replicating the dynamic attitude of the vehicle body.
• Frame Attitude Simulation Sub-Test Bench: This six-DOF vibration test bench simulates the bogie’s lateral/vertical displacements and three-dimensional rotation. As the support platform for the vehicle body simulator, it reproduces the frame’s dynamic attitude and the vehicle-frame coupling. Four triaxial accelerometers are symmetrically mounted at the corners, with the IMU co-located for high-precision multi-directional vibration and angular rotation sensing, providing fundamental system data.
• Track Simulation Sub-Test Bench: An eight-actuator structure is adopted. Multi-degree-of-freedom electric actuators precisely control the track inclination and curvature, simulating the dynamic variation in the overall track alignment and the six-degree-of-freedom spatial changes along the longitudinal direction, thereby meeting typical operating conditions.
• Track Geometric Irregularity Simulation Sub-Test Bench: Composed of two sets of horizontal linear modules and vertical vibration modules, it simulates the lateral and vertical motions of the track, respectively, and can generate track geometric irregularity parameters with different wavelengths and amplitudes.

This test bench can simulate variations in train attitude, bogie attitude, and track conditions under actual operating conditions, thereby achieving dynamic coupling between the vehicle and the track. Different parameters are set to replicate variations under diverse operating conditions. Geometric irregularity reference values from the calibration test bench are compared with outputs of the track inspection system. The geometric irregularity indication error is therefore derived to realize calibration.

During the experiment, the track inspection system acquires the track irregularity parameters at sampling points in real time and uploads them to a computer. The computer processes the output values from the track inspection system and the reference values from the test bench at the sampling points in real time, outputting the indication errors. The measurement model for the gauge indication error based on this measurement process can be calculated using Equation (1):

$${\mathrm{\Delta }}_{\mathrm{c}}=Z_{\mathrm{cg}}-Z_{\mathrm{g}}$$

(1)

where Z_cg is the gauge irregularity output by the track inspection system, and Z_g is the gauge reference value provided by the calibration test bench.

2.3. Quantification Principle for Type B Sources

The core of Type B uncertainty quantification lies in converting known error limits or interval information into standard uncertainty by assuming a probability distribution, using non-statistical methods. It does not rely on repeated measurements but instead utilizes prior information such as instrument manuals, calibration certificates, and empirical knowledge to transform interval quantities—such as maximum permissible error and expanded uncertainty—into standard uncertainty that can be combined with the Type A component.

3. Optimized Evaluation Method for Repeatability Experiments

3.1. Uncertainty Evaluation Using the GUM Method

The GUM method is an internationally accepted core approach for measurement uncertainty evaluation. It systematically assesses the dispersion of measurement results through a four-step procedure: model establishment, component quantification, uncertainty combination, and uncertainty expansion, ensuring that the measurement results meet international standards. The specific evaluation process is shown in Figure 3.

Measurement repeatability is an important indicator for evaluating the precision of a measurement system [18]. According to the Guide to the Expression of Uncertainty in Measurement, the conventional approach for evaluating repeatability typically uses the Bessel formula to calculate the experimental standard deviation as a Type A evaluation, based on empirical assumptions. However, the Bessel formula is highly sensitive to outliers; a single outlier can significantly exaggerate the standard deviation, thereby affecting the accurate assessment of the track inspection system’s precision.

In contrast, robust statistical methods do not require the identification or removal of outliers and can minimize the influence of anomalous values on statistical analysis results [19].

3.2. Repeatability Uncertainty Evaluation Based on Robust Statistics

3.2.1. Classical Bessel Method

For n independent repeated measurements of the same fixed rail, a gauge sequence x₁, x₂, …, x_n is obtained. The arithmetic mean is given by:

$$\overline{x}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i}$$

(2)

The standard uncertainty introduced is given by the following equation:

$$s(x)=\sqrt{{\displaystyle \frac{1}{n-1}}{\displaystyle {\sum }_{i=1}^n{(x_i-\overline{x})}^2}}$$

(3)

The standard uncertainty introduced by repeatability, denoted as u_A, is given by s(x).

3.2.2. Robust Statistical Method Based on MAD

The core of robust statistics lies in using the median and the median absolute deviation (MAD) to replace the mean and standard deviation. The calculation steps are as follows:

• Calculate the median of the measurement series:

$$\tilde{x}=\mathrm{median}(x_i)$$

(4)

2.

Calculate the median of the absolute residuals, MAD:

$$\mathrm{MAD}=\mathrm{median}(\left|x_i-\tilde{x}\right|)$$

(5)

3.

Normality test and robust standard deviation estimation:

The classic formula for converting MAD into an estimate of standard deviation is:

$$s_{\mathrm{robust}}=1.4826\times \mathrm{MAD}$$

(6)

where the factor 1.4286 is based on the assumption that the data follow a normal distribution. Therefore, before applying this coefficient, a normality test must be performed on the measurement series.

In this study, the Shapiro–Wilk test is adopted. The test was conducted on the 20-gauge indication error data points in

Table 1. Gauge indication error data (mm).

No.	Value	No.	Value	No.	Value	No.	Value
1	0.18	6	0.20	11	0.16	16	0.17
2	0.19	7	0.20	12	0.16	17	0.18
3	0.17	8	0.18	13	0.14	18	0.18
4	0.16	9	0.17	14	0.16	19	0.10
5	0.19	10	0.16	15	0.15	20	0.08

, yielding a statistic of $W=0.82$ with a corresponding $p<0.05$. At a significance level of $\alpha =0.05$, the null hypothesis of normality is rejected, indicating that the data are not normally distributed [20].

Therefore, this study adopts the following robust strategy: directly using the MAD itself as the scale estimator without multiplying by the normality correction factor, i.e.,

$$s_{\mathrm{robust}}=\mathrm{MAD}$$

(7)

This estimator is insensitive to outliers and faithfully reflects the actual dispersion of the data.

3.3. Experimental Data and Calculation

On the test bench of the track inspection system, continuous repeated measurements were performed on a fixed rail section, and the gauge indication error data were obtained as shown in Table 1. The number of measurements was n = 20.

Based on the calculations performed using Equations (2)–(6), the data were obtained. A comparison of the calculation results from the two methods is shown in

Table 2. Comparison of Calculation Results (mm).

Method	Location Estimator	Scale Estimator	Uncertainty Component
Classical Bessel	0.164	0.030	0.030
Robust Statistical Method (MAD)	0.170	0.015	0.015

.

3.4. Bootstrap-Based Statistical Significance Comparison

The classical Bessel method and the robust statistical method (MAD) yield different repeatability uncertainty estimates in Table 2. However, the difference between point estimates alone is insufficient to demonstrate that the robust method is statistically significantly superior to the Bessel method. Therefore, this paper employs the Bootstrap resampling method to compare the stability of the two estimators.

Bootstrap is a nonparametric statistical method that does not require any assumptions about the data distribution, making it particularly suitable for small-sample data with outliers. The specific steps are as follows:

• Randomly draw 20 samples with replacement from the original 20 gauge indication error data to form a Bootstrap sample.
• For this Bootstrap sample, calculate both the Bessel standard deviation and the MAD.
• Repeat the above process $B=\mathrm{10,000}$ times to obtain the Bootstrap distributions of the two statistics.
• Based on the Bootstrap distributions, calculate the standard error and the 95% confidence interval for each estimator.

The calculation results are summarized in

Table 3. Bootstrap analysis results of the Bessel standard deviation and MAD (mm).

Statistic	Bootstrap Mean	Standard Error SE	95% Confidence Interval
Bessel standard deviation	0.0301	0.0068	[0.022, 0.047]
MAD	0.0152	0.0021	[0.011, 0.019]

.

Bootstrap analysis shows that the standard error of MAD is much smaller than that of the Bessel standard deviation, and the 95% confidence interval width of MAD is considerably narrower than that of the Bessel method. Moreover, the two confidence intervals do not overlap at all, indicating that MAD provides higher stability and precision in estimating repeatability uncertainty, and that the difference between the two methods is statistically significant.

3.5. Results Analysis

As shown in Figure 4, the majority of the data lie within the range of 0.14 mm to 0.20 mm, with a median of approximately 0.17 mm. The box plot exhibits two distinct outliers (0.10 mm and 0.08 mm), which are significantly lower than the main data range and may be attributed to occasional factors during the measurement process, such as vibration or laser spot interference.

As shown in Table 2, the repeatability uncertainty obtained by the classical Bessel method is 0.030 mm, while that obtained by the robust statistical method is 0.015 mm. This significant difference indicates that the two low-value outliers in the original measurement data have a substantial influence on the Bessel formula. Due to the squaring of the deviations between the outliers and the mean, the classical standard deviation is markedly inflated. In contrast, the robust statistical method, which employs MAD scale estimation based on the median to calculate absolute deviations, effectively suppresses the interference of outliers. The resulting value better reflects the true repeatability of the track inspection system under normal operating conditions.

4. Uncertainty Modeling

4.1. Uncertainty Model Analysis

The track inspection system has a complex structure involving various sensor-integrated components. Unlike simple measurement devices that can be disassembled for internal analysis, the track inspection system comprises intricate components such as sensors, algorithms, and data processing modules, making it difficult to deconstruct and analyze the error sources at each stage. Therefore, a black-box model can be adopted to directly evaluate the uncertainty of the overall measurement results. In this study, the track inspection system is treated as a black-box model for the uncertainty evaluation of its gauge measurement results. A black-box model is a simplified representation of a complex system; it does not require disassembly of internal components, algorithms, or physical mechanisms but instead constructs a mathematical model or statistical relationship by observing the correspondence between inputs and outputs [21].

Based on the actual operating environment, running speed, track conditions, and the operational methods of the instrument (such as sampling frequency), the main sources of uncertainty can be identified. As illustrated in Figure 5, during the measurement process of track geometric irregularity parameters, the measurement uncertainty of the track inspection system includes the following components and is transmitted to the final measurement results during the measurement process.

(1)

Uncertainty introduced by the measurement inaccuracy of the track inspection system u₁

(2)

Uncertainty introduced by the repeatability of the indication error measurement u₂

(3)

Uncertainty introduced by variations in the amplitude of the inspection vehicle body u₃

(4)

Uncertainty introduced by different sampling frequencies of the track inspection system u₄

(5)

Uncertainty introduced by different spatial attitudes of the track inspection system u₅

(6)

Uncertainty introduced by different sampling speeds u₆

(7)

Uncertainty introduced by different directions of the track inspection system u₇

(8)

Uncertainty introduced by temperature variations u₈

4.2. Uncertainty Introduced by Measurement Inaccuracy of the Track Inspection System

Due to factors such as sensor accuracy, the track inspection system itself introduces a certain deviation between the measurement results and the actual values, thereby contributing to uncertainty in the measurement process. Using an optimized evaluation method, the uncertainty introduced by the measurement inaccuracy of the track inspection system can be calculated from the parameters obtained during system calibration, and is expressed as shown in:

$$u_1=\sqrt{{\displaystyle (\frac{\mathrm{\Delta }E}{\sqrt{3}}{)}^2}+u_E^2}$$

(8)

where ΔE and u_E are the indication error and the uncertainty, respectively, obtained during calibration of the track inspection system.

4.3. Uncertainty Introduced by Repeatability of Indication Error Measurement

During the measurement of geometric irregularities by the track inspection system, there exist random errors that cannot be completely eliminated (for instance, the track inspection system consists of various sensors and an inertial measurement unit, which inevitably introduce random noise), leading to inconsistencies in measurement data under identical experimental conditions. The measurement uncertainty introduced by repeatability experiments can be calculated using the robust statistical method.

$$u_2=s_{\mathrm{robust}}$$

(9)

where $s_{\mathrm{robust}}$ is the estimated robust standard deviation.

4.4. Uncertainty Introduced by Vehicle Body Vibration Amplitude

When the track inspection system operates under different working conditions (such as varying track conditions and track quality), vehicle body vibration occurs [22]. A larger vibration amplitude leads to reduced stability of the sensor measurement reference, resulting in different measurement outcomes. Therefore, different vibration amplitudes of the vehicle body introduce uncertainty, which can be obtained using the range method:

$$u_3={\displaystyle \frac{R_{\max }}{C_{\mathrm{n}}}}$$

(10)

where R_max is the range among the mean gauge indication errors under different vibration amplitudes during the experiment, and C_n is the corresponding range coefficient.

4.5. Uncertainty Introduced by Different Sampling Frequencies of the Track Inspection System

When the track inspection system operates at different sampling frequencies, phenomena such as signal distortion or signal loss may occur, ultimately leading to deviations between the measurement results and the true values that cannot be quantified, thereby introducing uncertainty. This uncertainty component can be obtained using the range method:

$$u_4={\displaystyle \frac{R_{\max 1}}{C_{\mathrm{n}}}}$$

(11)

where R_max1 is the range among the mean gauge indication errors under different sampling frequencies during the experiment.

4.6. Uncertainty Introduced by Measurement Under Different Spatial Attitudes of the Track Inspection System

The measurement reference and algorithmic model of the track inspection system are established based on an ideal track attitude. However, due to factors such as centrifugal force in track design and ballast settlement during track laying, the track may exhibit different spatial attitudes [23,24]. These variations can lead to measurement reference mismatch, deviation of sensor measurement conditions from the ideal state, and increased model approximation errors, ultimately manifesting as dispersion in the measurement results. The uncertainty introduced by this process can be obtained using the range method:

$$u_5={\displaystyle \frac{R_{\max 2}}{C_{\mathrm{n}}}}$$

(12)

where R_max2 is the range among the mean gauge indication errors under different spatial attitudes of the track inspection system during the experiment.

4.7. Uncertainty Introduced by Different Sampling Speeds

During train operation, the running speed cannot be exactly the same. The test bench simulated measurements of track geometric irregularity parameters under different train speeds, and it was found that high speeds lead to larger sensor sampling intervals and signal attenuation, making it difficult to accurately capture transient rail variations. Therefore, different sampling speeds need to be considered as a source of uncertainty. The uncertainty introduced by different sampling speeds is given by:

$$u_6={\displaystyle \frac{R_{\max 3}}{C_{\mathrm{n}}}}$$

(13)

where R_max3 is the range among the mean gauge indication errors measured by the track inspection system under different sampling speeds during the experiment.

4.8. Uncertainty Introduced by Different Directions of the Track Inspection System

When the track inspection system measures the same track in forward and reverse directions, discrepancies may arise, introducing uncertainty. This discrepancy originates from sensor asymmetry, direction-dependent wheel-rail contact, differences in running vibration, phase deviation in signal processing, unidirectional track defects, and environmental asymmetry. It constitutes a systematic uncertainty component related to the measurement direction and must be separately accounted for in the evaluation. The uncertainty introduced by forward and reverse direction measurements is given by:

$$u_7={\displaystyle \frac{R_{\max 4}}{C_{\mathrm{n}}}}$$

(14)

where R_max4 is the maximum indication error at the same sampling point under different measurement directions.

4.9. Uncertainty Introduced by Temperature

Temperature variations alter the physical dimensions of the rail, causing deviations between the measured values and the actual state of the track. According to the thermal expansion equation, the dimensional change in the rail under different temperatures can be quantified as follows:

$$\mathrm{\Delta }L=l\cdot a\cdot \mathrm{\Delta }t$$

(15)

where ΔL is the change in width at a point 16 mm below the top surface of a single rail; a is the linear expansion coefficient of the rail; and Δt is the temperature difference relative to the standard reference temperature of 20 °C for dimensional measurement, l is the rail width at a point 16 mm below the top surface of the rail head. Since gauge measurement is defined as the distance between the two rails, dimensional changes in both rails contribute to variations in the gauge. Evaluated according to the Type B method, the uncertainty introduced by temperature is given by:

$$u_8=\sqrt{2}\cdot {\displaystyle \frac{\mathrm{\Delta }L}{\sqrt{3}}}$$

(16)

5. Case Study: Measurement Uncertainty of Gauge Parameters

Taking the uncertainty evaluation of the gauge measurement process as an example, the track inspection system is mounted on the test bench to measure a simulated track. The test bench inputs the collected train operation data from actual lines (e.g., bogie frame data transmitted to the bogie simulator) to generate corresponding dynamic coupling between the vehicle body simulator and the bogie simulator. The collected train vibration amplitude is set to a reference amplitude, and experimental conditions such as amplitude multiplier, sampling speed, sampling frequency, and track spatial attitude can be varied according to the experimental requirements.

During the experiment, the positioning accuracy of the test bench, as given by its calibration certificate, is 0.01 mm. The uncertainty introduced by this factor is negligible compared to other components. Therefore, the uncertainty component arising from the calibration test bench is not considered in the following sections.

During the uncertainty evaluation process, the range coefficients can be obtained from the national metrological technical specification JJF 1059.1-2012, Evaluation and Expression of Uncertainty in Measurement. The range coefficients used in this paper are presented in

Table 4. Table of Range Coefficients.

n	2	5	6
Cn	1.13	2.33	2.53

[25].

5.1. Uncertainty Introduced by Measurement Inaccuracy of the Track Inspection System u₁

In this study, an optimized evaluation method is adopted for the uncertainty introduced by the measurement inaccuracy of the track inspection system. The uncertainty is calculated based on the indication error ΔE obtained from actual calibration. The uncertainty of the calibration process is u_E = 0.11 mm, and the indication error is ΔE = 0.325 mm (derived from the calibration test data of the track inspection system). According to Equation (8), the uncertainty component u₁ is obtained as:

$$u_1=\sqrt{{\displaystyle (\frac{0.325}{\sqrt{3}}{)}^2}+{0.11}^2}=0.217\mathrm{mm}$$

(17)

5.2. Uncertainty Introduced by Repeatability u₂

The track inspection system was mounted on the test bench for experimentation. Under constant experimental conditions and with the track simulation subsystem kept stationary (i.e., the gauge of the measured object remained unchanged), 20 measurements were performed, with sampling conducted at the same sampling point. The uncertainty introduced by repeatability was calculated using the robust statistical method. As shown in Table 2, the uncertainty introduced by repeatability is u₂ = 0.015 mm.

5.3. Uncertainty Introduced by Vehicle Body Vibration Amplitude u₃

The vibration amplitude of the train under normal operating conditions, as collected from actual measurements, was set as the reference amplitude (1×). Under otherwise constant experimental conditions, the vibration amplitude was varied, and measurements were conducted on the same standard gauge at amplitude multipliers of 1, 1.5, 2, 2.5, and 3. The mean gauge indication error for each measurement was calculated, and the results are presented in

Table 5. Mean Gauge Indication Error Under Different Vibration Amplitudes.

Amplitude multiplier	2.5	5	7.5	10	15
Mean Indication Error (mm)	0.0898	0.0543	0.0516	0.0446	−0.0017

.

According to Table 4, the maximum gauge difference caused by vibration amplitude is R_max = 0.0915 mm. Therefore, the uncertainty introduced by vehicle body vibration amplitude is obtained from Equation (10):

$$u_3=0.0393\mathrm{mm}$$

(18)

5.4. Uncertainty Introduced by Different Sampling Frequencies of the Track Inspection System u₄

With the vibration amplitude and sampling speed of the test bench kept constant, the track inspection system was used to measure the simulated track at different sampling frequencies. The mean gauge indication errors obtained under each condition are shown in

Table 6. Mean gauge indication error under different sampling frequencies.

Sampling frequency (Hz)	0.5	1	1.5	2	2.5
Mean Indication Error (mm)	0.081	0.0187	0.056	0.0905	0.0762

:

Therefore, the range of gauge indication errors caused by different sampling frequencies is 0.0728 mm. The uncertainty introduced by different sampling frequencies of the track inspection system is obtained from Equation (11):

$$u_4=0.0312\mathrm{mm}$$

(19)

5.5. Uncertainty Introduced by Measurement Under Different Spatial Attitudes of the Track Inspection System u₅

Variations in the spatial attitude of the track can be quantified through the track simulation subsystem as angular variations in the RX, RY, and RZ directions. A deviation of 0.5° in each direction (covering both normal and extreme conditions) was applied. With other experimental conditions kept constant, the influence of different angular variations in the three directions on the measurement results of the track inspection system was investigated on the test bench. The data are shown in

Table 7. Mean gauge indication error under different angular attitudes.

Different Angular Attitudes	RX0.5°	RX-0.5	RY0.5	RY-0.5	RZ0.5	RZ-0.5
Mean Indication Error (mm)	0.0837	−0.129	−0.1566	0.218	0.0849	−0.1864

.

Therefore, the range of gauge indication errors shown in Table 6 is 0.4044 mm. The uncertainty introduced by different angular attitudes is obtained from Equation (12):

$$u_5=0.1736\mathrm{mm}$$

(20)

5.6. Uncertainty Introduced by Different Sampling Speeds u₆

With the track geometric irregularity simulation subsystem kept unchanged (i.e., the gauge remained constant), the track inspection system was tested at different speeds (25 km/h, 50 km/h, 120 km/h, 200 km/h, 300 km/h, and 350 km/h) to obtain gauge indication error results. Six groups of measurement data were collected at the different speeds, each group containing 1000 consecutive gauge indication error values. Since the reference gauge remained unchanged, the mean value of each group was calculated, as shown in

Table 8. Mean Gauge Indication Error Under Different Sampling Speeds.

Speed (km/h)	25	50	120	200	300	350
Mean Indication Error (mm)	0.018	0.013	0.008	0.005	0.007	0.008

.

The range among the mean gauge indication errors in Table 7 is R_max3 = 0.013 mm. Therefore, the uncertainty introduced by different sampling speeds u₆ is obtained from Equation (13):

$$u_6=0.005\mathrm{mm}$$

(21)

5.7. Uncertainty Introduced by Different Measurement Directions u₇

With all other experimental conditions kept constant, the track inspection system was used to measure the same standard gauge section in the forward and reverse directions, respectively. Two sets of gauge indication error data were obtained; the two sets of experimental data, along with the data segment at the sampling point where the maximum range occurs, are presented in

Table 9. Indication Error of Track Gauge in Different Directions (mm).

Forward	Reverse	Range Value
0.13	0.03	0.1
0.16	0.03	0.13
0.18	0.03	0.15
0.20	0.04	0.16
0.23	0.05	0.18
0.24	0.05	0.19
0.23	0.05	0.18
0.22	0.06	0.16

.

The two sets of data curves are shown in Figure 6.

Figure 7 shows the curve of the range of gauge indication errors under different measurement directions. As illustrated in the figure, the maximum range is R_max4 = 0.190 mm. Therefore, the uncertainty introduced by different measurement directions can be obtained from Equation (14):

$$u_7=0.1681\mathrm{mm}$$

(22)

5.8. Uncertainty Introduced by Temperature Variation u₈

The operating temperature range of the track inspection system is estimated based on an extreme temperature difference of ±20 °C. The width of the rail at a point 16 mm below the top surface is 70 mm, and a is the linear expansion coefficient of the rail, taken as 12 × 10⁻⁶/°C. The physical dimensional change caused by temperature is obtained from Equation (15):

$$\mathrm{\Delta }L=0.0168\mathrm{mm}$$

(23)

Therefore, the measurement uncertainty of the gauge introduced by extreme temperature variation is obtained from Equation (16):

$$u_8=0.0137\mathrm{mm}$$

(24)

5.9. Calculation of Combined Standard Uncertainty and Expanded Uncertainty

Based on the analysis, since the above uncertainty components are independent of each other, the measurement uncertainty in the process of gauge measurement by the track inspection system is given by:

$$u_c=\sqrt{u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2}$$

(25)

Since the uncertainty components are independent of each other, the measurement uncertainty in the process of gauge measurement by the track inspection system is calculated using Equation (25), yielding a combined standard uncertainty of u_c = 0.33 mm.

To verify the normality assumption and reasonably determine the coverage factor, we supplemented the analysis with the Monte Carlo method (MCM). The distribution was tested using the Monte Carlo method. The results of the MCM are shown in Figure 8. According to the MCM calculation procedure, the combined standard uncertainty u_c1 and the expanded uncertainty U₉₅ were obtained as u_c1 = 0.3291 mm and U₉₅ = 0.6348 mm. Then the coverage factor k can be calculated as:

$$k={\displaystyle \frac{U_{95}}{u_{c1}}}=1.93$$

(26)

Taking a coverage factor of k = 1.93, the expanded uncertainty is U = 0.64 mm.

Table 10. Summary of uncertainty components and their contribution analysis.

Uncertainty Component	Standard Uncertainty (mm)	Proportion
u1	0.217	43.4%
u2	0.015	0.21%
u3	0.0393	1.42%
u4	0.0312	0.9%
u5	0.1736	27.79%
u6	0.005	0.02%
u7	0.1681	26.1%
u8	0.0137	0.17%

presents the summary of uncertainty components and their contribution analysis. It can be seen from the table that the uncertainties introduced by the measurement inaccuracy of the track inspection system and by different spatial attitudes account for relatively high proportions. Subsequently, the evaluation scheme can be improved based on these main sources of uncertainty.

6. Conclusions

This paper presents an analysis and evaluation method for uncertainty sources in the measurement process of a track inspection system based on a calibration test bench. For the first time, the dynamic coupling between the train and the track is incorporated into the evaluation method, ensuring the accuracy and repeatability of track geometry parameter inspection data. The robust statistical method is integrated into the repeatability evaluation of the track inspection system, effectively suppressing the interference of outliers and yielding measurement results that better reflect the true repeatability under actual measurement conditions.

Taking gauge measurement as an example, an uncertainty evaluation experiment was conducted for the measurement process of the track inspection system. The experimental results show that the expanded uncertainty is U = 0.64 mm, satisfying the requirement that the uncertainty be less than one-third of the maximum permissible error for gauge inspection of track inspection vehicles, thereby validating the feasibility of the proposed method. Furthermore, the analysis and evaluation results of the uncertainty sources presented in this paper can serve as Type B uncertainty inputs for subsequent field evaluations, providing a reliable metrological basis for the uncertainty evaluation of track inspection systems and enabling a faster and more accurate assessment.

The uncertainty evaluation method and the quantified Type B input sources proposed in this paper have clear engineering application value. In routine inspections of high-speed railways, this method can be embedded into the data processing system of comprehensive inspection trains to enable dynamic accuracy assessment. In multi-source data fusion, it can serve as a basis for weight allocation. Furthermore, the obtained expanded uncertainty of 0.66 mm provides a quantified boundary for controlling the confidence interval in gauge exceedance judgments, thereby supporting precise decision-making in track maintenance and repair. Future work may focus on further improving the dynamic response accuracy of the test bench to reduce the gap between the simulated and actual track environments.