J. Cent. South Univ. (2018) 25: 230-240

DOI: https://doi.org/10.1007/s11771-018-3732-9

A prediction method of rail grinding profile using non-uniform rational B-spline curves and Kriging model

YANG Yue(杨岳)¹, QIU Wen-sheng(丘文生)^{1, 2}, ZENG Wei(曾威)¹,XIE Huan(解欢)³, XIE Su-chao(谢素超)¹

1. School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China;

2. Guangzhou Large Scale Track Maintenance Machine Running and Maintenance Division,Guangzhou Railway Co., Guangzhou 511400, China;

3. School of Mechanical Engineering, Xijing University, Xi’an 710123, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Non-uniform rational B-spline (NURBS) curves are combined with the Kriging model to present a prediction method of the rail grinding profile for a grinding train. As a worn rail profile is a free-form curve, the parameterized model of a rail profile is constructed by using the cubic NURBS curve. Taking the removed area of the rail profile cross-section by grinding as the calculation index of the grinding amount, the grinding amount calculation model of a grinding wheel is established based on the area integral formula of the cubic NURBS curve. To predict the grinding amount of a grinding wheel in different modes, a Kriging model of the grinding amount is constructed, taking the travel speed of a grinding train, the grinding angle and grinding pressure of a grinding wheel as the variables, and considering the grinding amount of a grinding wheel as the response. On this basis, the prediction method of the rail grinding profile is presented based on the order-forming mechanisms. The effectiveness of this method is verified based on a practical application.

Key words: rail grinding; NURBS; Kriging; worn rail; target profile

Cite this article as: YANG Yue, QIU Wen-sheng, ZENG Wei, XIE Huan, XIE Su-chao. A prediction method of rail grinding profile using NURBS curves and Kriging model [J]. Journal of Central South University, 2018, 25(1): 230–240. DOI: https://doi.org/10.1007/s11771-018-3732-9.

1 Introduction

Rail grinding can prolong the service life of rails and improve the riding stability of a rail vehicle. Nowadays, the research on rail grinding is mainly focused on these three aspects: firstly, summing up and obtaining the influence of grinding on the rail based on long-term grinding experience in practice [1], which could provide reasonable guidance for the appropriate selection of a grinding mode [2]; secondly, research on rail grinding regarding improvement of the rail service property, such as repair rail damage [3, 4], profile optimization of rail grinding [5–8], and improvement of the wheel/rail contact performance of rail vehicles [9]; and thirdly, research on the rail grinding mechanism, such as thermodynamics during the rail grinding [10, 11], factors that influence grinding quality [12], the influence of grinding wheel properties on the grinding result [13], and the rail forming grinding mechanism [14–16]. At present, the research on rail grinding is mainly focused on the analysis of grinding mechanism and the factors that influence grinding quality based on the literature above. Furthermore, there are few studies on the practical grinding process at grinding sites, which means that the above studies cannot provide direct support for grinding operations at grinding sites and the improvement of rail grinding quality is limited to some extent.

In the practical rail grinding process, a corresponding target profile of rail grinding needs to be designed based on the damage and wear status of the on-site rail. Then a reasonable grinding mode that includes appropriate grinding parameters of a grinding train is selected based on experience to ensure that the obtained final grinding profile is consistent with the pre-designed target profile. However, selection of the grinding mode method based on experience cannot fully guarantee that the obtained final grinding profile will be consistent with the target profile. Therefore, it is necessary to predict the final grinding profile obtained with a certain grinding mode before practical grinding construction, which could provide a basis for the selection and optimization of the rail grinding mode.

To predict the final grinding profile of a worn rail in any grinding mode, the parameterized model of the rail cross-sectional curve is constructed by using a cubic non-uniform rational B-spline (NURBS) curve. Taking the area of the rail profile cross-section removed by grinding as the evaluated index of the grinding amount, the grinding-amount calculation model of a single grinding wheel is also established based on the algorithm for computing the integral values for cubic NURBS boundaries [17]. Considering the main factors influencing the grinding amount, the grinding-amount Kriging model of a grinding wheel is constructed by using experimental design and practical grinding construction. Finally, the prediction method of the rail grinding profile based on NURBS-Kriging is presented according to the rail grinding order-forming mechanics [15]. This could provide a basis for the selection and optimization of the rail grinding mode of a grinding train in practice.

2 Rail grinding mechanisms

In order to repair a worn rail profile by using the end surface of a grinding wheel, a grinding train has to place a group of high-speed spinning grinding wheels above the rail at different angles and to exert axial pressure to grind the profile by using the end surface of the grinding stone. The rail grinding mechanism is shown in Figure 1.

The grinding mode of a grinding train is composed of different values of travel speed v of a grinding train, grinding angle θ and grinding press p. During the grinding construction process, a differential grinding mode is selected to control the final grinding profile obtained based on the actual damage and wear status of the on-site rail [12].

3 Parameterized model of rail profile based on NURBS curve

To calculate the grinding amount S in differential grinding mode, the predicted profile needs to be adjusted and modified. This means that a parameterized model of the rail profile should be constructed. As the profile of the worn rail is a free curve with apparent irregularity, this work uses the NURBS curve to fit the worn rail profile curve. A NURBS curve is a unified mathematical form that is used to describe standard analytical curves and free curves [18]. Therefore, it could satisfy the parameterized modeling requirement of the worn rail profile curve. Usually, the cubic NURBS curve will be able to meet the requirements of engineering applications in practical settings [19]. Hence, the cubic NURBS curve is selected to establish the parameterized model of the worn rail profile curve.

Figure 1 Rail grinding mechanisms

3.1 NURBS curve

A NURBS curve of degree p [20] can be defined as

              (1)

where {P_i} are the reference points, {w_i} are the weight factors, and {N_i,p(u)} are the B-spline basis functions of degree p that are defined in the aperiodic and non-uniform knot vector U (u₀, u₁, …, u_m). The {N_i,p(u)} are defined as

 (2)

where U is calculated by

          (3)

3.2 Geometric profile of a worn rail

A worn rail of an operating route is selected as the object to construct the parameterized curve; the polar coordinates are used, setting the centre of the railhead curve as the origin, and the concrete polar coordinate is shown in Figure 2. Then, the rail profilometer is used to measure the coordinate values of the rail profile, and 33 points are obtained considering the adjustment accuracy. The concrete values are shown in Table 1.

3.3 Parameterization of worn rail profile based on cubic NURBS curve

The 33 points in Table 1 are selected as the data points to reconstruct the reference points {P₁, …, P₃₅} as presented in Table 2 using the NURBS interpolation method [21]. The knot vector U shown in Eq. (4) is calculated using the chord length parameterization method, and all of the weight factors {ω_i} in each reference point are preset to 1. Then, the parameterized model of the rail profile is established as shown in Figure 3.

Figure 2 Polar coordinates of worn rail profile

               (4)

4 Calculation model of grinding amount for a grinding wheel

To forecast the final obtained grinding profile after the grinding construction of n-grinding wheels for a grinding train in a grinding mode, the grinding amount of every single grinding wheel needs to be calculated. The simplified conditions of the calculation model are as follows:

Table 1 Coordinate values of worn rail profile

Table 2 Coordinate values of reference points

Figure 3 Parameterized curve of worn rail profile based on cubic NURBS

1) All of the grinding amounts are equal whatever the practical rail profiles are if the grinding mode is the same;

2) There is no deformation and wear of the grinding wheel during the grinding process;

3) The contact pressure between grinding wheel and rail is constant and is not affected by the vibration of the grinding train.

To calculate the actual grinding amount of a grinding wheel, the removed area of the rail cross-section S ground by a single grinding wheel is regarded as the calculation index. The coordinate values of the rail profile before and after grinding are measured by a rail profilometer, and the parameterized curve of a worn rail before and after grinding is constructed by using the established rail profile parameterized model based on the obtained coordinate values. Then the problem of calculating the grinding amount is transformed into the problem of solving the area S enclosed by the grinding wheel’s undersurface line and the parameterized curve. The details of the calculation principle are illustrated in Figure 4.

Figure 4 Calculation principle of grinding amount

To calculate the grinding area S of a grinding wheel, the cubic parameterized curve l_n is transformed into a cubic rational Bézier curve, as shown in Eq. (5), based on the area integral formula of the cubic NURBS curve [17].

                (5)

where {B_i,3(t)} is a Bernstein polynomial.

The knot vector of the Bézier curve is:

                             (6)

where the new knot is:

         (7)

The new weight factor and new reference point are presented as

(8)

where i=0, 1, …, n if s=0 and i=0, 1, …, n+s if s=1, 2, …, l. Also, can be calculated by

       (9)

Then the integral parameter S_i is:

      (10)

where e=–ω_i0+3ω_i1–3ω_i2+ω_i3; p, m, n, R₁, R₂, R₃, Q₁, Q₂, Q₃, Z₁, Z₂, and Z₃ are undetermined coefficients that can be calculated by Eq. (19) in Ref. [17].

Therefore, the area enclosed by the NURBS curve segment N₁AFN₄, curve segment N₁N₂, N₃N₄, and the x-axis is The grinding amountwhere is the trapezoidal area enclosed by points N₁, N₂, N₃, and N₄.

5 Grinding-amount Kriging model of a single grinding wheel

5.1 Basic form of a Kriging model

In order to increase the calculation efficiency of the prediction model for grinding amount S, this work establishes a surrogate model of the grinding amount. A Kriging model is a kind of unbiased estimation model that gives the minimum variance while considering the spatial correlation of variables sufficiently. As it can analyze the tendency of sample points, a Kriging model can be used to fit the nonlinear characteristics of the grinding amount and grinding parameters. A typical Kriging model is constructed of two parts a polynomial and a random distribution [22]:

y(X)=F(, X)+Z(X)                      (11)

where X is the variable of the Kriging model and X) is the regression model found by the known function group that depends on X and can be calculated as

(12)

whereis the regression coefficient, f(x_i) is a predetermined base function, and n is the number of sample points of the training sample.

Z(X) is the error of a stochastic process and has a mean value of 0. Besides, the covariance between any two points is:

           (13)

where σ² is the variance of the random process, R is an n×n order symmetric positive definite diagonal matrix, and R(x_i, x_j) is the spatial correlation function between any two points x_i and x_j of the sample and is usually described by the Gaussian correlation function, as provided in Eq. (14) in engineering applications.

            (14)

where h is the number of variables, and are the k-th components of x_i and x_j in the training sample respectively, θ_k is the correlation coefficient and can be calculated using maximum likelihood estimation (MLE, E_ml) method shown in Eq. (15), and is the difference between and

E_ml=            (15)

Then the response value and variance estimation of any sample point x are calculated by

                (16)

               (17)

where r(x)^T is the relevance vectors of length n, f is a one-dimensional column vector of length h, and Then the response value of the grinding amount S can be calculated using Eq. (16).

5.2 Variables of Kriging model

The main grinding parameters that affect the grinding amount are the travel speed of the grinding train v, the grinding angle θ, and the grinding pressure p based on the grinding principles of a grinding train. Therefore, to construct the grinding amount Kriging model of a grinding wheel, the travel speed v, grinding angle θ, and grinding press p are selected as the variables and the grinding amount S is selected as the response value.

As shown in Figure 5, taking the right side of a rail line as the object, the angle θ between the grinding wheel undersurface line and horizontal direction is the grinding angle, and the value range of the grinding angle θ is [0°, 180°].

Figure 5 Variables of grinding-amount Kriging model

Assuming that the grinding angle is a positive angle when θ∈[0°, 90°] and a negative angle when θ∈[90°, 180°], then the grinding angle θ∈{[0, 90°]∪(–0°, –90°]}. The value range of the grinding angle is –15°–70° based on the “Management Measures of High Speed Railway Rail Grinding” [23] released by Chinese National Railway Corporation. Hence, the value range of the grinding angle θ∈[–15°, 70°]. As the grinding pressure p is adjusted by controlling the current magnitude of the micro-electro-hydraulic system of the grinding wheel, the unit of current magnitude, the ampere (A), is regarded as a grinding parameter. To determine the value range of the grinding pressure p and grinding speed v, the GMC-96x grinding train is taken as the object, and the values of p∈[3A, 15A] and v∈[25 km/h, 30 km/h], which are frequently used in practice, are set as the value ranges. Finally, the variables and their value ranges of the Kriging model are shown in Table 3.

Table 3 Design variables of Kriging model and their value ranges

5.3 Construction of grinding-amount Kriging model

The determinate variables of the Kriging model are changed to construct different types of grinding modes and imported to the GMC-96x grinding train to grind a section of rails, and the profiles before and after grinding are measured using a profilometer. Then the grinding-amount Kriging model is constructed as shown in Figure 6 after all of the grinding amounts S are calculated using the model constructed in Section 4.

5.4 Rail grinding experiment

In order to guarantee the prediction accuracy of a Kriging model, the training sample size must be 3k, where k=(n+1)(n+2)/2 and n is the number of design variables [24]. As the number of design variables in the grinding amount Kriging model is 3, the training sample size of the Kriging model is 30.

Thirty groups of design variable combinations are obtained by using the Latin hypercube experimental design method to form 30 types of grinding modes. They are imported to the GMC-96x grinding train to grind a section of rail, and the coordinate values of profiles before and after grinding are measured using a profilometer. And the rail profile curve is obtained using NURBS interpolation method. Then the corresponding grinding amounts S are obtained using the grinding-amount calculation method in Section 4. The concrete values of the grinding amounts and the corresponding grinding parameters are illustrated in Table 4.

5.5 Grinding-amount Kriging model

The correlation matrix R and unit column vector f are calculated based on the training sample of the Kriging model, as shown in Table 4. The correlation coefficient of the Kriging modelis calculated as follows:

            (18)

Thenis calculated using R, y, f, andas

   (19)

The estimated value of unbiased variance of the Kriging model is calculated using Eq. (16). Then, the correlation parameters of θ_k are calculated using a maximum likelihood estimation, as shown in Eq. (15). The concrete values of the correlation parameters are provided in Eq. (20).

θ_k=[0.142, 0.086, 0.0001]                  (20)

The relevance vectors r(x)^Tare calculated after the correlation parameters θ_k are determined, and the response value can be calculated using Eq. (16). Then the Kriging model of the grinding amount S is constructed as presented in Figure 7.

Figure 6 Construction of grinding-amount Kriging model

Table 4 Training samples of Kriging model

Figure 7 Grinding-amount Kriging model of a grinding wheel

5.6 Error test of Kriging model

In order to test the fitting accuracy of the constructed Kriging model, the root mean square error (RMSE, E_rms) method [25] is used to estimate the fitting accuracy of the Kriging model.

The mean square error of any response value of the Kriging model is calculated using Eq. (21) based on Eq. (16).

      (21)

where μ=f ^TR^–1r–fθ_k.

To guarantee the fitting accuracy of the Kriging model, the RMSE must be minimal and greater than 0. In the error test, 25 groups of design variables in Table 4 are selected randomly and plugged into the Kriging model to calculate the predicted values of the grinding amount S. On this basis, the predicted values of the grinding amount S are compared with the corresponding values obtained by the actual grinding experiment shown in Table 4. The RMSE value of the Kriging model is 0.25123, indicating that the constructed Kriging model has high prediction accuracy and can be used to fit the impact of grinding parameters on the grinding amount S.

6 Grinding profile prediction of a worn rail based on NURBS-Kriging

6.1 Prediction mechanisms of rail grinding profile based on NURBS-Kriging

A grinding mode can be broken up into n grinding wheels to grind a worn rail based on order-forming mechanisms. In Figure 8, there are three grinding wheels in a grinding mode. The formed profile is P₀P₁Q₁Q_e after grinding by the a-th grinding wheel; the formed profile is P₀ P₁P₂Q₂Q_e after grinding by the b-th grinding wheel; and the final profile obtained is P₀ P₁P₂P_nQ_nQ_e after grinding by the 3rd grinding wheel.

Figure 8 Order-forming mechanisms of rail grinding

To calculate the final profile after rapid grinding, the grinding profile prediction method is designed, as shown in Figure 9. Combined with the grinding-amount Kriging model constructed in Section 5, the order-forming mechanisms of rail grinding are shown in Figure 8.

In Figure 9, the curve segment between the two intersections needs to be deleted after the intersection operation of every grinding wheel’s undersurface line and rail profile, and a new rail profile is obtained. This calculation process is repeated until the intersection-computation of all the grinding wheels’ undersurface lines and rail profiles has been done. The final curve obtained is the predicted grinding profile.

According to the above rail grinding profile prediction method, in order to obtain the final grinding profile in different grinding modes, the intersections P_n and Q_n of every grinding wheel’s undersurface line and rail profile need to be calculated first. Therefore, the intersection algorithm must be designed.

6.2 Intersection algorithm of a grinding wheel’s undersurface line and rail profile curve

To obtain the two intersections P_n and Q_n of a grinding wheel’s undersurface line and the parameterized rail profile curve, the intersection algorithm is designed, as shown in Figure 10, based on the area integral formula of the cubic NURBS curve in Section 4.

The grinding parameters v_i, θ_i and p_i are selected randomly from the design space to construct a grinding mode, and the grinding amount predicted value S_i is calculated by using the grinding-amount Kriging model. Two points P_i and Q_i are found in the rail profile curve to ensure that the slope of the line which is the undersurface line of the grinding wheel, is equivalent to the tangent value tanθ_i of the grinding angle θ_i. The area enclosed by the line and the curve segment between points P_i and Q_i is calculated by the presented area-calculation method of the cubic NURBS curve. Points P_i and Q_i are the two intersection points P_n and Q_n that need to be found if is equivalent to S_i. Otherwise, the points P_i and Q_i are changed until the two intersection points P_n and Q_n are found.

7 Application of rail grinding profile prediction method

Taking the worn rail profile defined by the coordinate values in Table 1 as the object, the grinding mode of the GMC-96x grinding train, as shown in Table 5, is used to grind the worn rail profile, the predicted rail profile is obtained using the established rail profile prediction method, and the concrete result is shown in Figure 11. It is consistent with the rail profile measured by a profilometer after grinding in actual construction.

Figure 9 Rail grinding profile prediction method

Figure 10 Intersection algorithms of a grinding wheel’s undersurface line and rail profile curve

Table 5 Grinding mode of worn rail for GMC-96x train

8 Conclusions

In order to predict the final rail grinding profile obtained before rail grinding construction and to provide a basis for the reasonable selection of a rail grinding mode, a method of prediction of the rail grinding profile using the NURBS curve and Kriging model is presented. The concrete conclusions are as follows:

1) Considering the worn rail profile curve as a free-form curve, the cubic NURBS curve is used to construct the parameterized model of the worn rail profile curve, which is convenient for the precise calculation of the grinding amount.

2) The removed cross-sectional area of rail profile is regarded as the grinding-amount calculation index, and the grinding-amount calculation model is established based on the area integral formula of the cubic NURBS curve. It guarantees the precision of the calculation of the grinding amount.

Figure 11 Prediction result in grinding mode as shown in Table 5

3) To improve the calculation efficiency of the grinding amount, the grinding-amount Kriging model is constructed, in which the variables are the grinding speed v, grinding angle θ, and grinding pressure p and the response is the grinding amount S. It provides a high calculation efficiency and precise surrogate model for prediction of the rail grinding profile.

4) To predict the rail grinding profile of the worn rail after grinding, the rail grinding profile prediction method based on NURBS-Kriging is presented according to the order-forming mechanisms. A practical application is used to verify the effectiveness of this method.

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(Edited by FANG Jing-hua)

中文导读

一种基于非均匀有理B样条曲线和Kriging模型的钢轨打磨廓形预测方法

摘要：铁路钢轨打磨可以改善车辆运行质量，延长钢轨使用寿命。为了保证钢轨打磨后可以获得预定的打磨目标廓形，需要在打磨之前对钢轨打磨廓形进行预测，为打磨施工中打磨模式的优选提供依据。本文设计了一种基于NURBS-Kriging模型的钢轨打磨廓形预测方法，实现了不同打磨模式下钢轨打磨廓形的预测。具体包括：考虑磨损钢轨断面廓形为自由曲线，应用三次NURBS 曲线构建了钢轨廓形的参数化模型；基于三次NURBS曲线面积精算公式，设计钢轨打磨量的计算模型；为预测单个砂轮的打磨量，将钢轨断面打磨面积作为打磨量衡量指标，以打磨角度、打磨压力和打磨速度三个打磨参数为变量，以打磨面积为响应量，构建单砂轮打磨量的Kriging模型；根据钢轨打磨顺序成形原理，提出基于NURBS-Kriging模型的钢轨打磨廓形预测方法，并通过工程应用验证了该方法的有效性。

关键词：钢轨打磨；非均匀有理B样条(NURBS)曲线；Kriging；磨损钢轨；目标廓形

Foundation item: Project(51405516) supported by the National Natural Science Foundation of China; Project(2015JJ2168) supported by the Natural Science Foundation of Hunan Province, China

Received date: 2016-03-30; Accepted date: 2016-07-20

Corresponding author: ZENG Wei, Lecturer, PhD; Tel: +86–18807418236; E-mail: zengwei1987@yeah.net