A 3D terrain reconstruction method for virtual grinding wheel based on matrix convolution operation

Abstract

The present invention proposes a method for reconstructing the 3D topography of a virtual grinding wheel based on matrix convolution operation, which reconstructs the grinding wheel topography by convolution of a digital filter matrix and a random matrix. The actual grinding wheel topography is collected by a 3D optical profilometer and statistically analyzed. Statistical parameters such as mean value, standard deviation, skewness and kurtosis are used to quantitatively analyze the distribution law of the grinding wheel topography height data. The protruding abrasive particles on the grinding wheel surface are extracted, and the shape of the abrasive particles is statistically analyzed. The abrasive particles after statistics are used as a filter function and convolution calculation is performed with a random matrix to realize the reconstruction of the random grinding wheel topography. The actual collected grinding wheel topography is used as a reference for reconstructing the virtual grinding wheel topography. For the grinding wheel topography at different wear stages, the Johnson transformation is used to transform and calculate the grinding wheel topography digital matrix. The present invention solves the problem of poor consistency between the reconstruction of virtual grinding wheel topography and the actual grinding wheel topography, and improves the accuracy of the reconstruction of the grinding wheel topography.

Description

Technical Field

[0001] The present invention proposes a virtual grinding wheel topography reconstruction method, particularly relates to the randomness of the abrasive shape on the grinding wheel surface and the change of the abrasive shape in different wear stages, and belongs to the field of grinding processing technology. Background Art

[0002] Grinding wheels are composed of abrasive grains and a binder. The protrusions of abrasive grains on the wheel surface exhibit a certain degree of randomness, including their position, protrusion height, and shape. To deepen our understanding of grinding behavior, virtual grinding wheels are often used in grinding simulations to explore the interaction between abrasive grains and material and the material removal mechanism during grinding. Grinding wheel topography plays a crucial role in analyzing the grinding process and forms the basis for grinding process simulations [Wei L, Zhaohui D, Yuanyuan S, et al. Parametric evaluation and three-dimensional modeling for surface topography of grinding wheel [J]. International Journal of Mechanical Sciences, 2019, 155: 334-342]. Unfortunately, measuring the entire grinding wheel topography is a time-consuming and labor-intensive task. However, the complete grinding wheel topography is required for the simulation calculation of the grinding process. Therefore, most studies have focused on the reconstruction of virtual grinding wheel topography [D.AD, AW, RBA survey of recent grinding wheel topography models [J]. International Journal of Machine Tools and Manufacture, 2005, 46 (3): 343-352] and used it for the simulation calculation of the grinding process. Suto [Suto and T.Sala, Simulation of grinding process based on wheel surface characteristics, Journal of the Japan Society of Precision Engineering, 1979, 45 (535), 775-780] established an empirical model of the grinding process by measuring the number of active cutting edges and the wear area, but did not consider the size of the abrasive particles. Chen Xun [Xun C, WB R.Analysis and simulation of the grinding process. Part I: Generation of the grinding wheelsurface[J]. International Journal of Machine Tools and Manufacture, 1996, 36(8): 871-882] constructed a virtual grinding wheel topography based on random numbers, assumed the shape of the abrasive grains to be spherical, and used the reconstructed virtual grinding wheel for simulation calculations of grinding wheel dressing and grinding processes.In their study of the surface creation mechanism of monocrystalline silicon, Hao NL, Tian BY, Li DZ, et al. Analytical modeling of ground surface topography in monocrystalline silicon grinding considering the ductile-regime effect [J]. Archives of Civil and Mechanical Engineering, 2017, 17(4): 880-893, Li et al. assumed that the abrasive particles were truncated into a cone shape. Summarizing the above research, when constructing virtual grinding wheel topography, the abrasive particles were assumed to have a regular shape. By statistically analyzing the size, position, and protrusion height of the abrasive particles on the actual grinding wheel surface, the reconstruction of the virtual grinding wheel topography was completed.

[0003] During the grinding process, the topography of the grinding wheel at different grinding stages is not static, but a dynamic process. Regarding the study of grinding wheel topography at different grinding stages, Chen Xun [Xun C, WB R, BM, et al. Analysis and simulation of the grinding process. Part IV: Effects of wheel wear [J]. International Journal of Machine Tools and Manufacture, 1998, 38 (1): 41-49] found that the wear of the grinding wheel affects the quality and efficiency of grinding, and proposed that the grinding wheel has different topographic characteristics after dressing and wear. Akshit Choudhary [Akshit C, NR B. Influence of 3D topography on tribological behavior of grinding wheel [J]. Procedia Manufacturing, 2020, 48: 533-540] used a laser confocal microscope to collect the topography of the grinding wheel at different wear stages and found that as the grinding progressed, the density of the abrasive tip gradually decreased, and more abrasive particles came into contact with the workpiece by friction, which led to a gradual increase in the grinding force. Hwang[Hwang TW, Evans CJ, Malkin S. High Speed ​​Grinding of Silicon Nitride with Electroplastic Diamond Wheels, Part 2: Wheel Topography and Grinding Mechanisms[J]. Journal of Manufacturing Science and Engineering, 2000, 122(1): 42-50]'s research shows that compared with the normal distribution, the probability distribution curve of the abrasive protrusion height shows obvious skewness.Yao Peng [Peng Y, Yadong G, Takeshi M, et al. Investigation of wheel wear mechanisms during grinding optical glasses through statistical analysis of wheel topography [J]. Int. J. of Abrasive Technology, 2012, 5(1): 33-47] measured the wheel topography of a diamond grinding wheel at different stages of grinding optical glass and found that as grinding progressed, the height of the abrasive protrusion gradually changed from a normal distribution to a skewed distribution. Statistical parameters such as mean, standard deviation, skewness, and kurtosis were used to describe the distribution characteristics of the abrasive protrusion height. Hamid Jamshidi [Jamshidi H, Gurtan M, Budak E. Identification of active number of grits and its effects on mechanics and dynamics of abrasive processes [J]. Journal of Materials Processing Tech., 2018, 273: 116-239] used extreme functions to describe the probability distribution of the abrasive protrusion height at different grinding stages.

[0004] Compared with the virtual grinding wheel reconstructed by assuming the abrasive grains to be of standard shape, the virtual grinding wheel topography reconstruction method proposed in the present invention can well represent the randomness of the abrasive grain size, position, protrusion height and shape, and realizes the change of the abrasive grain shape through Johnson transformation, which greatly improves the consistency between the virtual grinding wheel topography and the actual grinding wheel topography. Summary of the Invention

[0005] Purpose of the Invention: This invention addresses the current problem of insufficient representation of abrasive grain randomness in virtual grinding wheel topography reconstruction. This invention proposes a 3D virtual grinding wheel topography reconstruction method based on matrix convolution operations. This method can fully represent the randomness of abrasive grain size, position, protrusion height, and shape on the grinding wheel surface. This invention also provides a method for calculating the change in grinding wheel topography during different wear stages. By performing a Johnson transform on the grinding wheel topography digital matrix, the change in abrasive grain size, protrusion height, and shape is achieved while maintaining the position of the abrasive grains.

[0006] Technical solution: In order to achieve the purpose of the above invention, the present invention adopts the following technical solution:

[0007] A virtual grinding wheel 3D topography reconstruction method based on matrix convolution operation, the steps are as follows:

[0008] Step 1: Use a 3D profilometer to collect the grinding wheel topography and determine the grinding wheel topography characteristics.

[0009] Step 2: Use mean, standard deviation, skewness and kurtosis to quantitatively analyze and evaluate the distribution pattern of grinding wheel landform height data.

[0010] Step 3: extract the abrasive grains protruding from the surface of the grinding wheel, perform statistics on the shapes of the extracted abrasive grains, and obtain statistical results of the abrasive grain shapes.

[0011] Step 4: The statistically obtained abrasive shape matrix is ​​used as the convolution kernel to perform convolution calculation with the random matrix to obtain the grinding wheel topography with random characteristics, which can well describe the randomness of the abrasive size, position, protrusion height and shape.

[0012] Step 5. For the grinding wheel topography at different wear stages, the Johnson transformation is used to convert and calculate the grinding wheel topography digital matrix. Under the condition that the abrasive position remains unchanged, the abrasive size, protrusion height and shape are converted and calculated, which provides a new method for reconstructing the grinding wheel topography at different wear stages.

[0013] Step 6: Conduct grinding tests and surface topography simulation calculations to verify the feasibility of the above-mentioned grinding wheel topography reconstruction method.

[0014] The beneficial effects of the present invention are:

[0015] (1) The present invention proposes a method for quantitative analysis and evaluation of grinding wheel landforms, using statistical parameters (mean, standard deviation, skewness and kurtosis) to analyze and evaluate grinding wheel landforms.

[0016] (2) In order to address the problem that the abrasive grains on the grinding wheel surface are currently assumed to be of regular shape, which reduces the accuracy of the reconstruction of the virtual grinding wheel topography, the present invention uses a convolution method to reconstruct the grinding wheel topography, which can restore the randomness of the abrasive grain shape on the grinding wheel surface. A random grinding wheel topography reconstruction method is proposed, which can well describe the randomness of the size, position, protrusion height and shape of the abrasive grains on the grinding wheel surface, and improve the consistency between the virtual grinding wheel topography and the actual grinding wheel topography.

[0017] (3) The present invention proposes a method for reconstructing the grinding wheel topography at different wear stages. Without changing the position of the abrasive grains, the size, protrusion height and shape of the abrasive grains are changed, providing a new method for reconstructing the grinding wheel topography at different wear stages.

[0018] In summary, the present invention aims at the problem of low accuracy in reconstruction of grinding wheel landforms, and reconstructs grinding wheel landforms by convolution of digital filter matrix and random matrix. Actual grinding wheel landforms are collected by Sensofar (3D optical profilometer) and statistical analysis is performed, and statistical parameters such as mean value, standard deviation, skewness and kurtosis are used to quantitatively analyze the distribution law of grinding wheel landform height data. The abrasive grains protruding from the grinding wheel surface are extracted, and statistical analysis of the abrasive grain shape is performed. The abrasive grains after statistics are used as filter functions and convolution calculation is performed with the random matrix to realize the reconstruction of random grinding wheel landforms. The grinding wheel landforms actually collected are used as a reference for reconstruction of virtual grinding wheel landforms. For grinding wheel landforms in different wear stages, Johnson conversion is used to convert the grinding wheel landform digital matrix into calculation. The present invention solves the problem of poor consistency between virtual grinding wheel landform reconstruction and actual grinding wheel landforms, and the feasibility of the method is verified by comparing grinding tests with simulation calculation results. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 Schematic diagram of the diamond grinding wheel dressing process in Example 1;

[0020] Figure 2 is the actual grinding wheel topography image acquired by the 3D profilometer in Example 1;

[0021] Figure 3 is the statistical result of the actual grinding wheel landform height data in Example 1;

[0022] Figure 4 is the statistical calculation result of the actual grinding wheel landform height data in Example 1;

[0023] Figure 5 It is a schematic diagram of the contact state between the abrasive and the workpiece;

[0024] Figure 6 It is the result of abrasive grain extraction from actual grinding wheel topography;

[0025] Figure 7 is the actual wear particle statistical shape;

[0026] Figure 8 It is a schematic diagram of the digital filter matrix convolution calculation process;

[0027] Figure 9 It is to reconstruct the virtual grinding wheel topography;

[0028] Figure 10 It is the abrasive grain extraction result of virtual grinding wheel topography;

[0029] Figure 11 It is a schematic diagram of Johnson transformation of grinding wheel landform;

[0030] Figure 12It is a schematic diagram of the experimental process of grinding GH4169;

[0031] Figure 13 These are the results of GH4169 grinding tests and simulation calculations; (a) the grinding test results; (b) the simulation calculation results.

[0032] In the figure: 1. Three-jaw chuck; 2. Silicon carbide rod; 3. Diamond grinding wheel; 4. Grinding wheel spindle; 5. Abrasive; 6. Binder; 7. Pits; 8. Part where the abrasive contacts the material; 9. Normal projection of the part where the abrasive contacts the material; 10. Tangential projection of the part where the abrasive contacts the material; 11. GH4169 material; 12. Microcrystalline corundum grinding wheel. DETAILED DESCRIPTION

[0033] The invention is further described in detail below with reference to the accompanying drawings and specific examples:

[0034] Example 1

[0035] A virtual grinding wheel landform reconstruction method comprises the following steps:

[0036] Step 1: Use 3D profilometer to collect grinding wheel topography and determine the grinding wheel topography characteristics. The grinding wheel dressing process and the actual grinding wheel topography collected are as follows: Figure 1 and Figure 2 The collected grinding wheel topography is used as the basis for reconstructing the virtual grinding wheel topography. The machine tool used in the diamond grinding wheel dressing test 3 is a STUDER S131 internal and external cylindrical grinder. Figure 1 In the figure, the diamond grinding wheel 3 is fixed on the grinding wheel spindle 4, and the silicon carbide rod 2 is fixed by the three-jaw chuck 1. The diamond grinding wheel and the silicon carbide rod rotate in opposite directions. The grinding wheel dressing parameters are shown in Table 1:

[0037] Table 1 Grinding wheel dressing parameters

[0038]

[0039] The 3D topography of the grinding wheel surface was characterized using sensorfar (3D optical profilometer), with a sampling area size of 500 × 500 μm. 2 , the spatial sampling interval is 0.5 μm, and the grinding wheel topography data consists of a 1000 × 1000 matrix, such as Figure 2 As shown, the surface of the grinding wheel is composed of protruding abrasive grains 5, bonding agent 6 and pits 7 formed by the falling of abrasive grains.

[0040] Step 2: Use mean, standard deviation, skewness and kurtosis to quantitatively analyze and evaluate the distribution of grinding wheel landform height data. The specific implementation steps are as follows:

[0041] The grinding wheel landform height data has a certain degree of randomness. MATALB software is used to perform statistical analysis on the collected grinding wheel landform height data. The statistical results are as follows: Figure 3 Compared with the normal distribution, the grinding wheel landform data shows a certain deviation. In order to quantitatively analyze the grinding wheel landform height data, the statistical parameters mean, variance, skewness and kurtosis are used to describe the distribution law of the grinding wheel landform height data. The statistical calculation results of the grinding wheel landform are shown as follows: Figure 4 shown.

[0042] The calculation formula of the average value ζ1 is shown in (1):

[0043]

[0044] Where h is the height of the grinding wheel landform, and n is the total statistical data.

[0045] The standard deviation ζ2 reflects the degree of aggregation of the data set, and the calculation formula is shown in (2):

[0046]

[0047] Skewness ζ3 is the standard third-order moment of the sample data. Skewness ζ3 is a measure of the direction and degree of skewness of the statistical data distribution and is a numerical characteristic of the degree of asymmetry of the statistical data distribution. The larger the absolute value of skewness ζ3, the greater the degree of skewness of the data set. The calculation formula is shown in (3):

[0048]

[0049] Kurtosis ζ4 is the standard fourth-order moment of sample data. Kurtosis ζ4 is a statistic that describes the steepness of the distribution of all sample data in the population. The kurtosis of the normal distribution is 3. The calculation formula of kurtosis ζ4 is shown in (4):

[0050]

[0051] Step 3: The part where the abrasive contacts the workpiece material is irregular, such as Figure 5 As shown. The abrasive grains 5 protruding from the grinding wheel surface are extracted, as shown Figure 6 As shown, statistical analysis is performed. When the statistical result is large enough, the shape of the abrasive particles has a certain degree of certainty. The extracted abrasive particle shapes are statistically analyzed to obtain the abrasive particle shape matrix, as shown in Figure 7 shown.

[0052] Step 4: The abrasive shape matrix obtained by statistics is used as the convolution kernel to perform convolution calculation with the random matrix to obtain the random grinding wheel topography. The details are as follows:

[0053] (1) The reconstruction of the virtual grinding wheel is based on the statistical law of the grinding wheel topography height data to establish the virtual grinding wheel random topography. The convolution calculation can represent the characteristics of two functions, namely the determinism of the abrasive shape after statistics and the randomness of a single abrasive particle. The convolution process of the digital matrix is ​​as follows: Figure 8 As shown, the calculation formula is shown in formula (5):

[0054]

[0055] Wherein the matrix b represents the convolution kernel, is a digital filter matrix, the digital filter matrix of the present invention is the abrasive shape matrix obtained by the above statistics, η is a random digital matrix, and h is a grinding wheel topography digital matrix.

[0056] (2) The virtual grinding wheel topography obtained by convolution calculation is as follows Figure 9 As shown in the figure, the abrasive particles are extracted from the virtual grinding wheel topography to obtain the abrasive particle size, position, protrusion height and shape. Figure 10 shown.

[0057] Step 5: For the grinding wheel landforms at different wear stages, Johnson transformation is used to convert and calculate the grinding wheel landform digital matrix to obtain the grinding wheel landforms at different wear stages.

[0058] (1) As the grinding wheel gradually wears, the grinding wheel topography changes continuously at different grinding stages. The grinding wheel topography at different grinding stages obtained by Johnson transformation is as follows: Figure 11 As shown in Figure 6, Johnson transformation changes the abrasive particle size, protrusion height and shape when the abrasive particle position is below the bottom edge. The general formula of Johnson transformation is shown in formula (6):

[0059]

[0060] Where h is a normal random sequence, h' is a random sequence with specified skewness and kurtosis, γ and δ are shape parameters, ε is the center offset, and λ is the scale factor.

[0061] (2) For different non-normal random sequences, Johnson has different transformation forms, including the bounded system S B , unbounded system S U and the lognormal system S L There are three types. As shown in Table 2:

[0062] Table 2 Three types of Johnson transformation

[0063]

[0064] Step 6: Conduct GH4169 grinding test. The experimental device is as follows: Figure 12As shown, the internal and external cylindrical grinding test was conducted using a STUDER S131 internal and external cylindrical grinding machine. A microcrystalline corundum grinding wheel 12 with a 20 mm diameter, 20 mm width, and 150-mesh grit was mounted on the grinding wheel spindle 4. The GH4169 material 11 was secured using a three-jaw chuck 1. After installation, a circular runout tester was used to measure the circular runout of the workpiece surface to ensure it was within 3 μm. Simulation calculations of the GH4169 surface topography were also performed to verify the feasibility of the aforementioned grinding wheel topography reconstruction method. The experimental parameters are shown in Table 3:

[0065] Table 3 Grinding test parameters

[0066]

[0067] The surface morphologies obtained by experiment and simulation are shown in Figure 2. Figure 13 As shown, Figure (a) is the result of the grinding experiment; Figure (b) is the result of the simulation calculation. It can be seen that the surface roughness Sa obtained by the experiment and the simulation are 0.713μm and 0.749μm respectively, the maximum difference in surface height Sz are 5.609μm and 5.353μm respectively, and the errors of Sa and Sz on the workpiece surface are 5.04% and 4.56% respectively. The difference in the surface morphology of the workpiece obtained by the experiment and the simulation calculation is slight, which is caused by the randomness of the abrasive grains on the grinding wheel surface. Based on the grinding test and simulation results of GH4169, the feasibility of the virtual grinding wheel topography reconstruction method proposed in this invention is verified.

[0068] This embodiment does not impose any formal restrictions on the shape, material, structure, etc. of the present invention. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are within the scope of protection of the technical solution of the present invention.

Claims

1. A virtual grinding wheel 3D topography reconstruction method based on matrix convolution operation, characterized in that: Here are the steps: Step 1: Use a 3D profilometer to collect the grinding wheel topography and determine the grinding wheel topography characteristics; Step 2: Use mean, standard deviation, skewness and kurtosis to quantitatively analyze and evaluate the distribution pattern of grinding wheel landform height data; Step 3: extracting the abrasive grains protruding from the surface of the grinding wheel, and statistically analyzing the shapes of the extracted abrasive grains to obtain statistical results of the abrasive grain shapes; Step 4: The abrasive shape matrix obtained by statistics is used as a convolution kernel to perform convolution calculation with the random matrix to reconstruct a grinding wheel landform with random characteristics, which is used to express the randomness of the abrasive size, position, protrusion height and shape. The process of reconstructing and establishing the virtual grinding wheel random landform is as follows: (1) Convolution calculation represents the characteristics of two functions, namely, the determinism of the wear particle shape after statistics and the randomness of a single wear particle. The calculation formula of the digital matrix is ​​shown in formula (5): The matrix b represents the convolution kernel, is the digital filter matrix, and the digital filter matrix is ​​the average wear particle shape matrix h obtained by the previous statistics. mean , η is a random number matrix, h is a grinding wheel landform number matrix; (2) The virtual grinding wheel topography obtained by convolution calculation is used to extract abrasive particles, and the abrasive particle size, position, protrusion height and shape are obtained for the simulation calculation of grinding; Step 5. For the grinding wheel topography at different wear stages, the Johnson transformation is used to convert and calculate the grinding wheel topography digital matrix. Under the condition that the abrasive position remains unchanged, the abrasive grain size, protrusion height and shape are converted and calculated to adapt to the reconstruction of the grinding wheel topography at different wear stages.

2. The method for reconstructing 3D topography of a virtual grinding wheel based on matrix convolution operation according to claim 1, characterized in that: In step 2, MATALB software is used to perform statistical analysis on the collected grinding wheel landform height data; The calculation formula of the average value σ1 is shown in (1): Among them, h is the height of the grinding wheel landform, and n is the sum of statistical data; The standard deviation σ2 reflects the degree of aggregation of the data set, and the calculation formula is shown in (2): Skewness σ3 is the standard third-order moment of sample data. Skewness σ3 is a measure of the direction and degree of skewness of the statistical data distribution and is a numerical characteristic of the degree of asymmetry of the statistical data distribution. The larger the absolute value of skewness σ3, the greater the degree of skewness of the data set. The calculation formula is shown in (3): Kurtosis σ4 is the standard fourth-order moment of sample data. Kurtosis σ4 is a statistic that describes the steepness of the distribution of all sample data in the population. The kurtosis of the normal distribution is 3. The calculation formula of kurtosis σ4 is shown in (4):

3. The method for reconstructing 3D topography of a virtual grinding wheel based on matrix convolution operation according to claim 1, characterized in that: The specific process of step five is as follows: (1) As the grinding wheel gradually wears, the grinding wheel topography changes continuously at different wear stages. The grinding wheel topography at different wear stages is obtained by Johnson transformation. Johnson transformation changes the size, protrusion height and shape of the abrasive grains without changing the position of the abrasive grains. The general formula of Johnson transformation is shown in formula (6): Where h is a normal random sequence, h' is a random sequence with specified skewness and kurtosis, γ and δ are shape parameters, ε is the center offset, and λ is the scale factor; (2) For different non-normal random sequences, Johnson has different transformation forms, including the bounded system S B , unbounded system S U and the lognormal system S L Three types:

4. The method for reconstructing 3D topography of a virtual grinding wheel based on matrix convolution operation according to claim 1, characterized in that: In step three, the abrasive particle shapes are statistically analyzed to obtain an abrasive particle shape matrix, and the process is as follows: (1) Query the diameter of the abrasive grains according to the grinding wheel model and calculate the size of the pixel area occupied by the abrasive grains. The calculation formula for the length of the pixel area occupied by the abrasive grains is shown in (7): Among them, n x 、n y Represents the length in the horizontal and vertical directions respectively, d g Indicates the abrasive particle diameter, l b Represents the distance between adjacent pixels; (2) Obtain the position and height of the abrasive particle apex. The abrasive particle apex is the highest point of the abrasive particle, that is, the maximum value of the pixel area occupied by a single abrasive particle. The calculation formula is shown in (8): in, Indicates that the point is the abrasive particle vertex. To distinguish the abrasive particle vertex from other positions, I and J are used to represent the fixed position of the abrasive particle. In order to eliminate the influence of the irregular shape of the binder on the shape of the abrasive, the apex height of the abrasive should be greater than the height of the binder. The calculation formula is shown in (9): The calculation formula of wear particle position area is as follows: (3) Obtaining the shape of the abrasive particles. After the above steps, the vertex position (I, J) of the abrasive particles and the corresponding height value have been obtained. The calculation formula for extracting the abrasive particle shape is shown in (10): (4) The extracted abrasive particles are superimposed to obtain the average shape of the abrasive particles. The extracted single abrasive particle shape matrices are superimposed and averaged to obtain the average abrasive particle shape matrix h mean , h mean The convolution kernel matrix b is used as the grinding wheel topography reconstruction calculation.

5. The method for reconstructing 3D topography of a virtual grinding wheel based on matrix convolution operation according to claim 1, characterized in that: In steps 3 and 4, the abrasive grains in the reconstructed virtual grinding wheel topography are extracted, and the position, height, and shape of the abrasive grains on the virtual grinding wheel surface are recorded and used for grinding simulation calculations to verify the feasibility of the method for reconstructing the grinding wheel topography.

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