Mathematical Problems in Engineering

Volume 2019, Article ID 6913215, 10 pages

https://doi.org/10.1155/2019/6913215

## The Calculation of Roughness Uncertainty by Fitting B-Spline Filter Assessment Middle Lines

^{1}Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300350, China^{2}School of Mathematics, Tianjin University, Tianjin 300350, China

Correspondence should be addressed to Xiaochen Sun; moc.361@227cxs

Received 19 September 2018; Accepted 8 January 2019; Published 11 February 2019

Academic Editor: Leonid Shaikhet

Copyright © 2019 Gaiyun He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The roughness and uncertainty are important parameters of surface morphology. The least square middle line method is often used to estimate the roughness and its uncertainty. However, the roughness and its uncertainty obtained by the least square middle line method are inaccurate. This paper proposes a method to calculate exactly the roughness and its uncertainty by piecewise fitting the smooth B-spline filter assessment middle lines. A B-spline smoothing filter is selected to determine the assessment middle line of roughness. The B-spline filter can not only give the accurate roughness, but also obtain the smooth assessment middle line. The model of roughness uncertainty is proposed by piecewise fitting B-spline filter middle lines as the quadratic curves. The S-shaped test part is used to verify the model of roughness uncertainty.

#### 1. Introduction

Surface topography affects measurement accuracy, wear resistance, fatigue strength, and stability of fit, etc. It is an important factor to evaluate surface quality. The surface morphology is composed of the shape profile, waviness profile, and roughness profile. The roughness and its uncertainty are studied in this paper. The International Organization for Standardization (ISO) issued the corresponding surface roughness standard ISO 468-1982 in 1982 [1]. This standard stipulates some evaluation parameters of surface roughness, such as , and it also gives the selection criteria of evaluation length and sampling length. The roughness is often extracted from the surface morphology. More and more measuring instruments are used to get the surface morphology, such as the contact surface profiler [2], laser interferometric profiler [3], and noncontact digital optical profiler [4]. The contact surface profiler is the most commonly used instrument. It is used to measure the surface profile. Because it is connected with the computer, the surface profile information can display on the computer [2]. When the surface morphology is obtained, the method of extracting the roughness needs to be considered. However, the existing commercial software cannot accurately extract the roughness and calculate the uncertainty of roughness at the same time. The Gaussian filter which is included in the existing commercial software is the most common method to extract roughness at present. In ISO 11562, the Gaussian filter was adopted as a standard to calculate the roughness [5]. The development of the Gaussian filter is relatively mature. In order to decrease the amplitude deviation of the Gaussian filter, the cascaded first-order Butterworth filters and the cascaded moving average filters were used to extract the roughness. The combination of the two filters can decrease the amplitude deviation greatly [6].

However, the Gaussian filter has the edge effect and large fluctuation, which is not suitable for fitting the roughness assessment middle line [7]. Due to the large fluctuation of the middle line obtained by the general filter, the B-spline smoothing filter is proposed to obtain the middle line for the roughness evaluation. In ISO/TS 16610-22, the B-spline filter middle line was also adopted as a standard to evaluate the roughness [8]. The middle line obtained by the B-spline smoothing filter is relatively smooth, which is convenient for fitting the middle line subsequently. Generalized B-spline signal processing was proposed to process the band limited signals. The principle of the B-spline filter was given in detail [9]. Considering the smoothness of the B-spline filter middle line, it is convenient for us to fit the middle line.

When we know the expression of the fitted middle line, the uncertainty of roughness needs to be calculated. For the uncertainty of roughness, it usually refers to the indirect measurement of uncertainty, which is the A-type uncertainty [10]. Generally, the Guide to the Expression of Uncertainty in Measurement (GUM) method is used to calculate the uncertainty of indirect measurement. The GUM method is the standard to calculate the uncertainty [11].

Arencibia adopted the GUM method in combination with the coordinate measuring machine to calculate the uncertainty of roundness or cylindricity errors. Considering the correlation of variables, the uncertainty of the measurand was calculated by the GUM method [12]. The use of calibrated parts in ISO/TC 15530-3 is also a method to calculate the uncertainty. In 2017, paper [13] compared the GUM method and the ISO/TC 15530-3 method of evaluating flatness error uncertainty about an optical flat side. The ISO/TC 15530-3 method needs the calibrated parts and long-term experiments and the result of the GUM method differs greatly with the ISO/TC 15530-3 method, which makes the ISO/TC 15530-3 method inappropriate to verify the GUM method. In addition to the GUM method and the ISO/TC 15530-3 method, the Monte Carlo method (MCM) is also used to calculate the uncertainty [14]. Combining the Monte Carlo method with the theory of error ellipse, the Monte Carlo method is proposed to estimate the measurement uncertainty of circular characteristics [15]. The MCM and Bayesian estimation were combined to calculate the uncertainty, and the Bayesian principle was used to analyze the prior distribution information related to the MCM [16]. The Monte Carlo method relies on the choice of the number of experiments, while the adaptive Monte Carlo method (AMCM) can increase the number of experiments until the results are stable. Fang et al. evaluated the uncertainty in the linear model and nonlinear model by the AMCM [17]. However, both the MCM and AMCM have strict requirements on the distributions of measure parameters. When the distributions of measure parameters are uncertain, the uncertainty obtained by the MCM and AMCM will be inaccurate. However, the MCM and AMCM can still be used to verify the GUM method. Wen et al. chose the GUM method to calculate the uncertainty of cylinder error and used the adaptive Monte Carlo method to verify the results of the GUM method [18]. Cao et al. adopted two verification ways of the MCM to verify the GUM method: the interval comparison and the probability comparison, both of which are appropriate verification ways [19]. First, they calculated the uncertainty of roughness in the GUM method. Then the result of the GUM method was verified in the AMCM by the interval comparison.

From what have been discussed above, because the middle line obtained by the Gaussian filter is uneven, it is not convenient to fit the curves. This paper adopts the B-spline smoothing filter, which can obtain the smooth middle line. In this situation, the effect of fitting the B-spline middle line is better. All current articles fit the entire middle line, which makes the residual sum of squares of curves fitting rather big. This paper piecewise fits the B-spline smoothing filter middle lines to reduce the residual sum of squares of curves fitting. Furthermore, the middle line is often fitted as the straight line, which also increases the residual sum of squares. This paper adopts the quadratic curve to fit the middle line so that the fitted curves more clearly reflect the trend of the middle lines. The model of roughness is established based on the two quadratic fitting curves of the middle line. When we obtain the roughness model, the formulas of the uncertainty in GUM can be obtained considering the correlation between variables in the model. Then we can get the value of the uncertainty by calculating the formulas of the uncertainty. Finally, the AMCM verifies the proposed GUM method. The main works of this paper are given as follows:(1)We adopt the B-spline smoothing filter to obtain the smooth middle line.(2)We propose a piecewise fitting method of the B-spline smoothing filter middle line.(3)Substituting the straight line, we use the quadratic curve to fit the middle line.(4)We apply our method to the S-shaped test part and employ the AMCM to verify the proposed GUM method.

The structure of this paper is as follows: In Section 2, the roughness middle line is extracted by B-spline filtering, the roughness middle line is fitted, and the mathematical model of roughness is established. In Section 3, the formula of the GUM method is obtained. In Section 4, an experiment on S-shaped test part roughness is presented, and the results by the GUM method and the AMCM are analyzed and discussed. Section 5 draws a conclusion that the GUM method to calculate the uncertainty by fitting B-spline filter assessment middle lines is valid.

#### 2. The Roughness Model under the B-Spline Filter

##### 2.1. The Principle of the B-Spline Filter

The roughness middle line obtained by the B-spline filter is smooth and the B-spline filter can remove the influence of noise. Therefore, comparing with the Gaussian filter, it is more convenient to extract the roughness middle line. Then the middle lines are piecewise fitted. The principle of the B-spline filter [20] is the combination of the IIR digital filter and the B-spline function, namely, the series of IIR digital filter and weighted average filter, as shown in Figure 1. The amplitude deviation between the roughness middle lines extracted by the Gaussian filter and the two-stage cascade smooth B-spline filter is less than 1%, so the smooth B-spline filter with the two-stage cascade is selected to calculate the middle lines of roughness [21]. The transfer function formula of is shown as follows [17]:where the subscript 2 of represents the two-stage cascade and is actually the IIR digital filter function obtained based on . is obtained by the z-transform of the smooth B-spline coefficient . is the indirect B-spline transformation. The calculations of these variables are shown in Figure 1.