Burr removal from measurement data of honeycomb core surface based on dimensionality reduction and regression analysis

This paper investigates Nomex honeycomb core material, which is fabricated from Nomex paper using adhesive bonding. Nomex paper is made up of aromatic polyamide (aramid) fibers immersed in resin, whose length and direction are variant. The fibers are of high strength, and are difficult to be thoroughly cut off during the machining process, which will produce different forms of burrs on the machined surface [20]. There mainly exist three geometrical shapes, as illustrated in figure 1.

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2.1.1. Lumpy burr.

2.1.2. Threadlike burr.

2.1.3. Dotted burr.

Dimensionality reduction is a process of reducing the dimension number of data [21]. The measured data of honeycomb cores are 3D, where burrs are very difficult to be removed, because of the complication of their distribution directions. Therefore, this paper does not directly deal with the measured data, but reduces its dimension number from three to two in order to reduce the difficulty of recognizing burrs from the measured data. In this study, dimensionality reduction is achieved by projecting the measured data into a plane to acquire the wanted 2D data [22]. Then, burrs will be recognized in the projection plane and then removed in the original measured data accordingly.

Firstly, vertical dimensionality reduction is employed on the measured data to remove burr I data, and then after burr I data are removed, horizontal dimensionality reduction will be employed to remove burr II data. In the process of vertical dimensionality reduction, the top view plane of the measured data is taken as the projection plane (plane I figure 2). As for horizontal dimensionality reduction, the projection plane is the one (plane II in figure 2) that is perpendicular to plane I and its projection line (l in figure 2) in plane I is the approximate line of the cell wall data. In the 2D space after each time of dimensionality reduction, the distribution line of the cell wall can be obtained, while burrs obviously deviate from the line. By this principle, burrs can be recognized in order to be removed from the original measured data.

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After dimensionality reduction, in the 2D space of the measured data, the distribution of the cell wall is considered to be close to a straight line or a curve with a certain width. Once the distribution line of the cell wall is determined, burr data can be recognized by analyzing the distance between itself to the distribution line, if the distance is larger than the set threshold, the data will be regarded as burr data.

However, the shape and position of the distribution line are indefinite, so its concrete mathematic function is also unknown. After observation on burrs, it is found that although some burrs are large in size, their positions are far from cell walls; Some are near to cell walls, but occupy small area. Therefore, the distribution line can be forecasted with regression analysis to fit the data using linear or non-linear function [23, 24]. The precision of regression analysis is affected by burr data, and the more burr data there are, the larger deviation there would be. As a result, regression analysis is repeatedly used to gradually approximate the true distribution line with the decrease of burr data, while the threshold for recognizing burr data is reduced gradually.

Burr I data are identified in the 2D space acquired through vertical dimensionality reduction, and then be removed from the measured data with regression analysis. Furthermore, the removal of burr I data is broken down into two procedures in order to achieve more accurate burr removal result, which are coarse and fine removal.

During vertical dimensionality reduction, the projection plane (plane I) is the xoy plane, and the result of every data point after dimensionality reduction is its x-coordinate and y-coordinate, as shown in figure 3. In the acquired 2D space, the actual distribution of the cell wall is a curve not a straight line. Non-linear regression model should be adopted to fit the curve, but it is greatly influenced by burr data. For this reason, in the coarse removal, linear regression model is taken for removing the majority of burr I data (l₁ in figure 3). Then in the fine removal, specific non-linear regression model (l₂ in figure 4), corresponding to the actual distribution of the cell wall, is adopted for removing the remainder burr I data. Once the distribution line is determined, burr data can be recognized by analyzing the distance between itself to the distribution line. If the distance is larger than the set threshold (such as d₁ in figure 3 and d₂ in figure 4), the data will be regarded as burr data.

• (1)  
  Coarse removal. Fit the dimension-reduced measured data with the linear regression model [25]:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(t)={{\alpha }_{1}}+{{\alpha }_{2}}t.\nonumber \end{align} \tag{ 1 }$$

  The variable t is considered to be an explanatory variable, while f(t) is considered to be a dependent variable. α₁ and α₂ are the parameters to be estimated. It is reasonable to take the axis who has the longer range as the explanatory variable for cell walls of different directions. Then parameters α₁ and α₂ can be determined automatically with the help of regression analysis tools in Matlab. The determination of other parameters used in the regression below is in the same way, and will not be stated again.In order to improve the accuracy of burr removal, regression analysis is used repeatedly (at least three times are recommended) with the thresholds gradually decreasing, in which only part of burr data is removed each time. Besides, in the coarse removal stage, the threshold of last regression analysis is a little larger than half of the width of the cell wall.
• (2)  
  Fine removal. For further fine removal, the actual distribution of the cell wall is taken into consideration. Besides, cell walls of different directions should be processed separately, due to their different distributions. The actual distribution of double cell wall data for all honeycomb cores is a straight line, while most of single cell walls are curves. In order to describe these different distributions, the available regression models includes.

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3.1.1. Linear regression model.

3.1.2. Logistic regression model.

For some types of honeycomb cores, the actual distribution of single cell wall data is close to 'S' shape, which can be expressed with logistic function [26]. The regression model is the logistic function:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(t)=\frac{L}{1+{{e}^{-k(t-{{t}_{0}})}}}.\nonumber \end{align} \tag{ 2 }$$

The variable t is considered to be an explanatory variable, while f(t) is considered to be a dependent variable. L, k and t₀ are the parameters to be estimated. The data need to be rotated or flipped to be placed with 'S' shape prior to using this regression.

3.1.3. Polynomial regression model.

Some single cell walls are neither straight line nor close to 'S' shape. The cell wall may not have a definite shape and is hard to be described uniformly. Such shape can be forecasted with a n-degree polynomial function [27].

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(t)={{\beta }_{0}}+{{\beta }_{1}}{{t}^{1}}+{{\beta }_{2}}{{t}^{2}}+\cdots +{{\beta }_{n}}{{t}^{n}}.\nonumber \end{align} \tag{ 3 }$$

The variable t is considered to be an explanatory variable, while f(t) is considered to be a dependent variable. β₀, β₁,..., β_n are the parameters to be estimated. Once the number of degree is determined, the polynomial function can be solved with Matlab software. Because of the irregularity of the cell wall shape, it is hard to give the concrete value of n, and 5–10 are recommended.

Similarly, in the fine removal, the remaining burr I data are removed for multiple times with gradually decreasing thresholds, where the threshold of last regression analysis is half of the width of the cell wall.

After burr I data are removed, to remove burr II data from the remaining measured data, horizontal dimensionality reduction is employed to obtain a 2D space, where burr II data are identified with regression analysis, and removed from the remaining measured data. Similar with the removal of burr I, the removal of burr II data also contains two procedures, which are coarse and fine removal.

During horizontal dimensionality reduction, to determine plane II, the approximate line of the cell wall in plane I is firstly determined by fitting the x-coordinate and y-coordinate of the measured data to a straight line, as shown in figure 5, where P₁(x₁, y₁) and P₂(x₂, y₂) are the intersection points of the line and the two axes, and α is its inclination angle. To achieve dimensionality reduction, a new coordinate system o'-x'y'z' is established with x'-axis along the approximate line. The coordinate components of any points in coordinate system o'-x'y'z' can be transferred and expressed in coordinate system o-xyz through coordinate transformation in equation (4):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[ \begin{array}{@{}cccccccccccccccccccc@{}} {{x}^{\prime}} \nonumber \\ {{y}^{\prime}} \nonumber \\ {{z}^{\prime}} \nonumber \end{array} \right]=T\left[ \begin{array}{@{}cccccccccccccccccccc@{}} x-{{x}_{1}} \nonumber \\ y-{{y}_{1}} \nonumber \\ z \nonumber \end{array} \right].\nonumber \end{align} \tag{ 4 }$$

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Where $\newcommand{\e}{{\rm e}} T=\left[ \begin{array}{@{}cccccccccccccccccccc@{}} \cos (\alpha) & \sin (\alpha) & 0 \\ -\sin (\alpha) & \cos (\alpha) & 0 \\ 0 & 0 & 1 \end{array} \right]$ is the transformation matrix.

In coordinate system o'-x'y'z', the result of every data point after dimensionality reduction is its x'-coordinate and z'-coordinate, as shown in figure 6, where plane II has been moved for a certain distance along y'-axis to make the schematic clearly. The distribution line (l₃ in figure 6) of the cell wall after dimensionality reduction reflects the height fluctuation of cell wall data. Then to identify and remove burr II data, the corresponding regression model is used for expressing the line. Taking x' as the explanatory variable, and z' as the dependent variable, the relationship between them can be expressed with a function (the concrete function is specified in its concrete removal stage):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{z}^{\prime}}=f\left( {{x}^{\prime}} \right).\nonumber \end{align} \tag{ 5 }$$

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From equations (4) and (5), the fitted value on the projection plane that every data point corresponds to is:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle z_{i}^{\prime}=f\left( x_{i}^{\prime} \right)=f\left( ({{x}_{i}}-{{x}_{1}})\cos (\alpha)+({{y}_{i}}-{{y}_{1}})\sin (\alpha) \right).\nonumber \end{align} \tag{ 6 }$$

The deviation between actual measured value and the fitted value is:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta h=\left| {{z}_{i}}-z_{i}^{\prime} \right|.\nonumber \end{align} \tag{ 7 }$$

Then remove the measured data whose ▵h is greater than the set threshold h.

The removal of burr II data also contains two procedures of coarse and fine removal.

• (1)  
  Coarse removal. In this stage, the selected regression model is a linear regression model.Here, part of burr II data are removed for multiple times with gradually decreasing thresholds, where the threshold of last regression analysis is a little larger than half of the height fluctuation range of cell wall data.
• (2)  
  Fine removal. For honeycomb cores, the measured result represents its micro profile, which is irregular and hard to be expressed with an exact shape. In this situation, polynomial regression analysis is suitable for describing the distribution thanks to its superior flexibility.

Similarly, the remaining burr II data are removed for multiple times with gradually decreasing thresholds, where the threshold of last regression analysis is half of the height fluctuation range of cell wall data.

In the different removal stage of burrs (coarse or fine removal) for removing different type of burrs (burr I or burr II), the suitable regression models are different. In summary, only linear regression is used in coarse removal, while the alternative regression models in the fine removal of burr I contain linear regression model, logistic regression model, and polynomial regression model, and only polynomial regression model is suitable for the fine removal of burr II. Because the cell walls of the same direction in a honeycomb core use the same regression model, only if all of them accord with identical shape, the corresponding regression model will be adopted. Otherwise, polynomial regression model is appropriate owing to its stronger adaptation.

As stated above, in every stage, burrs are removed repeatedly, corresponding to a set of thresholds, respectively. Each threshold can be expressed as a multiple of the final data band width w₀, where w₀ is the cell wall width when removing burr I, and is the height fluctuation range when removing burr II. Each set of thresholds can be determined by considering the following factors:

3.3.1. The number of repetitions.

3.3.2. The order of thresholds.

3.3.3. The last threshold.

3.3.4. The first threshold.

3.3.5. The threshold interval.

A linear laser displacement sensor (LJ-V7060, Keyence, Japan) was used for the measurement. Its specifications are summarized in table 1. During measuring, the incident laser was set along the cell wall, as shown in figure 7. The laser probe was mounted on a three-axis numerical control machine tool for scanning the surface of the specimen that was fixed to the workbench. The resolution of the measured data is 20 µm  ×  25 µm.

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The proposed method mainly depends on the regression models and the thresholds in different burr removal stages. These options are identical for the three specimens in this paper. According to different suitable regression models in the different removal stage for removing different type of burrs, the selected regression models are shown in table 2. After observation, the cell wall distribution fluctuate not very strongly, and quintic polynomial regression is close to the distribution. Such selection is also suitable for most measurement results except those whose cell walls correspond to specific regression models.

Table 3 presents the thresholds that are successively used in regression analysis of every burr removal stage, where d₀ and h₀ are the cell wall width and the height fluctuation range, respectively. The values of d₀ and h₀ are consistent for the same type of honeycomb cores. Considering the selection principle in section 3.3, both burr I data and burr II data are removed for five times of coarse removal and three times of fine removal, with thresholds of descending values and intervals. In coarse removal, the first thresholds are determined as 5d₀ and 5h₀, while the last ones are 0.75d₀ and 0.75h₀. In fine removal, the first thresholds are chosen as d₀ and h₀, while the last ones are 0.5d₀ and 0.5h₀. Then the thresholds in each burr removal stage can be determined as in table 3. Besides, the given sets of thresholds are effective on most of the measurement results in practical application.

To evaluate the effectiveness of the proposed burr removal method, experiments were carried out on a flat surface (figure 8), an inclined surface (figure 9) and a curved surface (figure 10), respectively. The honeycomb cores with flat surface and curved surface share the same cell size of 3.2 mm, while the cell size of the inclined honeycomb core is 4.8 mm. The three specimens are the ones without serious lattice deformation, severe burr level or cell wall laceration, and surface quality like these is usually visually inspected in production currently. The objective of this study is to evaluate the surface shape accuracy not the surface quality, so burr data are removed from the measurement result before calculating the surface shape accuracy.

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Regarding the tests on real measured data, accurate quantitative evaluations are not possible, since it is very difficult to judge whether a data point in the critical zone is a burr datum or a cell wall datum, owing to the roughness of cell walls. Alternatively, visual observations and descriptive statistics are employed for comparison before and after burr data are removed.

Cell wall detection is firstly employed on the measurement result, and then each detected cell wall is processed by the burr removal method successively. Accordingly, burrs on the whole measured honeycomb surface can be removed. To watch the burr removal results clearly, only a part of the measured surface is shown in the experiment results. Figures 8(c)–10(c) show the measurement results, then figures 8(d)–10(d) present the results of burr I data being removed, and burr II data are removed as shown in figures 8(e)–10(e).

For descriptive statistics, the root-mean-square deviation (RMSD) of a cell wall, referring to the dispersion level of the measure data, is specified as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm RMSD}=\sqrt{\frac{1}{n}\sum\limits_{t=1}^{n}{{{({{p}_{t}}-{{p}_{ft}})}^{2}}}}\nonumber \end{align} \tag{ 8 }$$

where, p_t (t  =  1, 2, ..., n) is data to be analyzed. p_ft, the corresponding predicted value of p_t, is forecasted with fitting based on the final remaining data. The smaller the RMSD is, the more stabilized the data are.

To demonstrate the performance of the proposed method, we calculate the data removal rate (DRR), and the root-mean-square deviation decline rate (RMSDDR), which are defined as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm DRR}=\frac{{\rm NOM{{\text -}}NFR}}{{\rm NOM}}\times 100\%\nonumber \end{align} \tag{ 9 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm RMSDDR}=\frac{{\rm RMSDOM{{\text -}}RMSDFR}}{{\rm RMSDOM}}\times 100\%\nonumber \end{align} \tag{ 10 }$$

where NOM  =  number of original measured data, NFR  =  number of final remaining data, RMSDOM  =  mean RMSD of the original measured data, and RMSDFR  =  mean RMSD of the final remaining data. The statistical results of the three specimens are shown in table 4.

Figure 8 shows the burr removal results of the flat specimen. In the original measured data (figure 8(c)), there exist lots of burr data, including threadlike burrs, lumpy burrs and dotted burrs, which correspond to the surface of honeycomb cores (figure 8(b)). After before and after contrast, burr data are effectively removed in the final remaining data, while there are no obvious accidentally deletions of cell wall data. Circle A gives an example of a typical burr I, while circle B is a burr II, and both are successfully removed. After burr data are removed (figure 8(e)), the range of data decreases from 0.97 mm to 0.28 mm, and obvious visible burrs no longer exist. As shown in table 4, 24.83% of the original measured data are removed, while the RMSD is reduced by 42.84%.

The results of the inclined specimen are shown in figure 9. In the aerospace application, inclined surfaces are widely used, thus, the applicability on inclined surfaces for the proposed method is of great practical significance. From figures 9(b) and (c), it can be seen some burrs are near from cell walls, which are considered to be difficult to be removed. As illustrated in circle C, such burr has been successfully removed with the proposed method. The removal of burr data decreases the data range from 8.09 mm to 2.75 mm, corresponding to the height difference of the inclined surface. The percentage of removed data reaches 21.20%, decreasing the RMSD by 65.82%.

We further demonstrated the effectiveness of our burr removal method on the curved surface (figure 10). Despite the more complicated shape, burrs contained in the curved specimen are effectively removed. The curved shape is able to be seen clearly in the final remaining data (figure 10(e)). The removed burr data, accounting for 28.06% of the original measured data, reduce the RMSD by 57.79%.

By comparison between figures 8(c)–10(c) and 8(e)–10(e), it can be found that obvious burr data have been successfully removed. The final remaining data (figures 8(e)–10(e)) are in the form of hexagonal grid, corresponding to the measured surfaces in figures 8(b)–10(b). Besides, the cell wall data are smooth, continuous, and complete, which proves that the majority of the cell wall data are not accidentally removed. Moreover, it can be seen from the results that RMSDFR is much larger than DRR, exhibiting the great improvement on the measured data with the proposed method. Therefore, our method is capable of removing burr data and keeping the cell wall data.