1. Introduction

During the recent decades, the surface finish has become a significant variable in many engineering fields, especially in large optical engineering and laser applications. It is also a quality characteristic that can influence the performance and cost of mechanical parts. The roughness parameters are generally used to evaluate the surface finish. The surface roughness parameters are very important to ensure the satisfactory performance characteristics of the workpiece and reduce the concomitant tool wear. The ability to measure surface roughness is also very important for the control of manufacturing processes.

Many techniques for surface roughness measurement have been proposed [1–8]. These measurement techniques can be divided into two types, contact measuring method and non-contact measuring method [9–11]. The contact method has several disadvantages of off-line contact. It may be possible to change the characteristics of the rough surface during the mechanical contact between the stylus and the tested surface. In order to overcome these restrictions, the surface roughness can be measured rapidly and with improved accuracy by employing the non-contact methods. Beckmann and Spizzichino [12] proposed a theory on scattering of electromagnetic waves. Their study has been demonstrated that the RMS roughness was connected with the specular beam intensity. Peiponen and Tusboi [13] obtained a new parameter about surface roughness by calculating deviation of the roughness distribution. Domanski and Wolinski [14] proposed a new method for measuring roughness by utilizing optical fibers. Since then, many authors have studied this problem on surface roughness by employing optical fiber sensing technology based on laser scattering. In recent years, support vector machine (SVM) [15–18] was introduced into the measurement of surface roughness by some researchers. SVM is a popular and useful classification technique, offsetting theoretical defects of neural network. The work thought of SVM is parameters related to surface roughness as input, characteristic values obtained by the known surface roughness as the training set, and choosing appropriate SVM model and algorithm to predict the surface roughness value of the unknown samples.

In this study, a simple multi-wavelength fiber optic sensor [19] that combined the light scattering method with data acquisition system was presented. It is used to evaluate surface properties of the workpiece. To effectively predict surface roughness of the workpiece, a surface roughness prediction model based on SVM was proposed. The prediction model is employed, which is a useful technique for data classification and data regression prediction. The relative error for roughness measurement by using the software LIBSVM [20] is analyzed.

2. The prediction model

The surface roughness prediction model based on SVM was established. SVM is a useful technique for data classification. The concept of ε-insensitive loss function was introduced [21] and then SVM was extended to regression estimation of nonlinear system. The following will focus on the principle of the prediction model.

2.1 Support vector regression (SVR)

The regression problem of SVM is given sample data as input, predicting output according to input data. The sample data are divided into training sets and testing sets. Each instance in the training set includes a target value (i.e. the class labels) and several attributes (i.e. the features or observed variables). The goal of SVM is to produce a model based on the training data which predicts the target values of the test data given only the attributes of the test data.

Given a training set $\left(x_i,y_i\right)$$\left(i=1,\dots ,l\right)$, where $x_i\in R^n$ and $y_i\in Y=R$, the purpose of a support vector machine is an arbitrary given in N dimensional space to predict the corresponding output values. The function of the linear regression problem is limited to linear function, i.e. $y=g\left(x\right)=\left(\omega \cdot x\right)+b$.

SVM requires the solution of the following optimization problem:

The above regression problem is simplified into the dual problem by using the Lagrange function. The Lagrange function is defined as follows:

In order to minimize$\omega$, $b$ and${\xi }_i^{(\ast )}$, and maximize ${\alpha }^{(\ast )}$ and${\eta }^{(\ast )}$, The Lagrange function should satisfy

Then get

The nonlinear problem is generally converted into the linear problem according to the certain properties or the transformation. Here the nonlinear data is mapped into a higher dimensional space by the kernel function, and the linear problem can be solved in the higher dimensional space.

The transformation $X=\phi \left(x\right)$ introduced is as follows:

The Eq. (7) can be expressed as:

The variable ${\eta }^{(\ast )}$is eliminated with the constraint equation, Eq. (9) can be further simplified to

Assuming ${\overline{\alpha }}^{(\ast )}={\left(\overline{{\alpha }_1},{\overline{{\alpha }_1}}^{\ast },\dots ,\overline{{\alpha }_l},{\overline{{\alpha }_l}}^{\ast }\right)}^T$ is any solution of the Eq. (10), if there is a solution in the range of $\left(0,C\right)$, the solution can be obtained according to the following expression.

Letting

Assuming $\overline{{\alpha }_j}\in \left(0,C\right)$ or ${\overline{{\alpha }_k}}^{\ast }\in \left(0,C\right)$, then we have

The deceive function is defined as follows:

2.2 Kernel function

The training vectors $x_i$ are mapped into a higher dimensional space for SVM by the function $\phi$. SVM finds a linear separating hyper-plane with the maximal margin in this higher dimensional space. $C>0$is the penalty parameter of the error term. Furthermore, $K\left(x_i,x_j\right)\equiv \phi {\left(x_i\right)}^T\phi \left(x_j\right)$ is called the kernel function. The following four basic kernels are found in SVM books.

In general, the RBF kernel function is a reasonable first choice to establish the SVM. The RBF kernel function can nonlinearly map data into a higher dimensional space, so it can deal with the nonlinear relation between class labels and attributes. Moreover, the RBF model is relatively simpler than other kernel function. c and g are two important parameters in the RBF kernel function, and it can be found by using cross-validation.

2.3 LIBSVM method

The software LIBSVM [20] is a simple, easily useful and fast effective package to realize identification and regression of SVM. It not only provides the compiled files performed in the Windows system, also provides the source code, to achieve convenient improvement, modification and application in other operating systems.

The general procedure of using the software LIBSVM for data regression prediction as follows:

Based on above procedure, the surface roughness of workpieces can be obtained upon SVR by using the software LIBSVM. Then, the relative error of surface roughness is calculated to analyze the measurement accuracy on SVM method.

3. Results and discussion

To investigate the influence of surface roughness on light scattering characteristics and measure roughness by using the prediction model, the measured specimens with different roughness (R_a = 0.012μm, 0.025μm, 0.05μm, and 0.10μm), which are grinding samples produced by Precision Template Tool Factory in Harbin (meeting GB6060.2-85 standard), were tested by employing 650nm, 1310nm lasers, respectively. According to reference [19], the reflected intensity of the rough surface as data sets for testing (i.e. training sets and testing sets), is collected by multi-wavelength fiber optic sensor as shown Fig. 1.

 

Fig. 1 Experimental-setup for measuring surface roughness by multi-wavelength fiber sensor.

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3.1 Fiber optic sensors

In this study, the surface roughness and the major characteristics of the rough surface were investigated by optical fiber sensing technology. The fiber optic sensor is primarily designed, the end-face structure of the sensing probe of which consist of the transmitting fiber in the middle and six receiving fibers which are closely packed around the transmitting fiber according to equal-angle interval distribution as shown in Fig. 2(a).

 

Fig. 2 (a) Schematic diagram of the fiber bundle end face 2(b) Intensity of transmitting fiber under Gaussian distribution.

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According to the principle that the receiving fiber and the transmitting fiber are closely arranged, under the condition of not increasing the diameter of the fiber bundle, the maximum number of the receiving fiber is six. The structure cannot only ensure the receiving fiber receiving the reflected light within the scope of the corner of the space as large as possible, but is very beneficial to the structure of the fiber bundle, which is simple, reliable, small size. The intensity modulation function of the fiber bundle is considered to be the sum of the intensity modulation function of six single fiber pair. The transmitting fiber is single mode fiber, the light distribution of which is Gaussian distribution, so the emission intensity distribution is also Gaussian distribution as shown in Fig. 2(b). According to the theoretical mathematical background of the reference [19], the relationship between the reflected intensity and surface roughness and the working wavelength respectively can be mentioned detail. When the incident light irradiates perpendicularly the rough surface, ${\theta }_i={\theta }_s=0$(${\theta }_i$ and ${\theta }_s$are incidence angle and scattering angle, respectively.), the output intensity of fiber sensor for measuring surface roughness is [19]

Because of the unsystematic error (denoted as $\beta (\lambda )$), the precision measurement of the optical fiber sensing system is seriously affected. The unsystematic error can be effectively reduced by using multi-wavelength.

In our experiment, the multi-wavelength laser optical fiber sensing system for measuring surface characteristics was designed based on the single-wavelength fiber sensor (SWFS). The effects of the interference factors on double light source are $\beta ({\lambda }_1)$and $\beta ({\lambda }_2)$. Due to using the same optical fiber path, so we think this kind of influence is equal, i.e. $\beta ({\lambda }_1)=\beta ({\lambda }_2)$. The root mean square value σ can be calculated as below:

3.2 Experimental setup

The experimental setup for measuring roughness is shown in Fig. 1. The measurement system consists of diode laser, coupling device, fiber optic sensor, detector, data acquisition system. The diode lasers at 650 nm and 1310 nm were used as the source. All fiber jumpers work at 1550 nm, and the fiber probe of the fiber optic sensor adopts the special geometry design, and the output end is divided into two ways, which is used to transmit the laser with different wavelength. The optical fiber probe is fixed in the platform with the micrometer caliper, and the distance between the measured surface and the optical fiber probe can be changed by regulating the knob of the micrometer caliper. Two detectors can be used to separate the light source with different wavelength. The wavelength range of optical fiber coupling silicon detector (type: DET02AFC) with maximum power of 18mW and bandwidth of 1.2 GHz is 400~1100 nm. The wavelength range of optical fiber coupling InGaAs detector (type: DET01CFC) with maximum power of 70mW and bandwidth of 1.2 GHz is 800~1700 nm. The reflective intensity of the rough surface is obtained by data acquisition system (DAS) compiled by C language.

3.3 Measurement of the surface roughness

As shown in Fig. 3, the reflective intensity is the most sensitive to the change of the surface roughness in the peak area where surface roughness can be measured. So the range of the working distance chosen is 2.5 mm ~3.5 mm. The measured reflection intensity in the range of 2.5 mm ~3.5 mm is chosen as data set used by the prediction model. The processing of the model is as follow: Firstly, the collected data is divided into training and testing sets, converted to the format of an SVM package. Secondly, the collected data is processed under the computer DOS path. The function (the gridregression.py) is used for parameters optimization where v-fold cross-validation is used to obtain the optimum values of parameters c, g, and p. Then, the model is built through the optimal parameters and the statement SVM-train. Finally, the testing sets is analyzed and predicted by using the built model, which can be achieved by SVM-predict statement. The data obtained in the out.txt document is the desired result, which by comparing with the data of the testing sets, the measurement error can be obtained.

 

Fig. 3 The reflected intensity of grinding specimens ($R_a$ = 0.012μm, 0.025μm, 0.05μm, and 0.10μm) varying as the working distance under the different wavelengths. (a) ${\lambda }_1$ = 650 nm; (b) ${\lambda }_2$ = 1310 nm.

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Tables 1 and 2 show the surface roughness values obtained by using the model when the incident wavelengths are 650 nm and 1310 nm, respectively. The relative error of the surface roughness in each set is calculated. From Tables 1 and 2, it can be seen that the relative error is the largest at the roughness of 0.012μm. The mean squared error of regression prediction is 6.40444 × 10⁻⁷, and squared correlation coefficient is 0.999705 at 650 nm. The mean squared error of regression prediction is 6.72513 × 10⁻⁷, and squared correlation coefficient is 0.999838 at 1310 nm. The error range of surface roughness is 0.74% ~7.56% at 650 nm, and the error range of surface roughness is 1.03% ~5.92% at 1310 nm. The average relative error is about 2.669% at 650 nm, while it is about 2.431% at 1310 nm in Tables 1 and 2. As the incident wavelength increases, the average error decreases. The error of roughness measurement is less than 3% by using the model, which is acceptable. According to reference [19], the error range of surface roughness is 2.92% ~13.4% by the characteristic curves between surface roughness and the scattering intensity ratio which is obtained by multi-wavelength fiber sensor. It is obvious that the error of surface roughness based on the model is smaller than that by using the characteristic curves between surface roughness and the scattering intensity ratio.

Table 1. The surface roughness obtained by using the prediction model at 650 nm.

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Table 2. The surface roughness obtained by using the prediction model at 1310 nm.

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4. Conclusions

From the above experimental results and theoretical analysis on surface scattering characteristics of the grinding specimens with different surface roughness, it is known that multi-wavelength fiber sensor is suitable for surface roughness measurement of the workpiece, and it is more sensitive to the change of surface roughness in peak area. The error of roughness measurement is less than 3% by using the prediction model, which is acceptable. So, the surface roughness should be reduced as much as possible in high precision optical engineering. Useful help will be provided to study the precision measurement of the surface cleanliness level on particulate-contaminated mirrors.