The nose radii of cutting inserts are usually measured using a profile projector or toolmaker's microscope. Since only a sector of a circle is available for the measurement when using such instruments, the radius determined from these methods is inaccurate. A new approach to determine the nose radii accurately from 2-D images of the nose profile is presented in this work. The images were captured using a CCD camera with the aid of backlighting. The tool nose center in each digitized image is located based on the tool geometry and the curved nose profile is transformed into a linear profile using polar-radius transformation. The nose radius is then varied within ±10 pixels of the nominal radius and the average deviation from a straight line profile in the nose region in the polar-radius plot is evaluated. The radius corresponding to the minimum average deviation is identified as the most accurate radius value. The proposed method was verified using simulated inserts and tested on three types of turning inserts, namely triangular, rhombic and square inserts. The results obtained from this approach were compared with those measured using a profile projector and showed that the proposed method is a viable technique for accurate tool nose radius determination.
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Many factors influence the surface quality of a machined component. They can be divided broadly into four main groups: (a) Factors relating to cutting conditions, such as feed rate, cutting speed, depth-of-cut and presence or absence of cutting fluid, (b) factors relating to the cutting tool, such as tool wear, tool geometry, tool material and built-up-edge formation, (c) factors relating to cutting geometry, such as rake angle, feed angle, relief angle etc. and (d) factors relating to machine tool condition, such as stability, age, vibration etc. Due to the complex interaction of the multitude of factors, it is not easy to pin-point a single factor that has a major effect on the surface roughness of the finished workpiece.
Independent studies carried out in the past have shown that tool nose geometry, particularly nose radius, has a direct effect on the surface roughness of workpiece, wear of the cutting tool, residual stresses and chip formation. Chou and Song (2004) found that when nose radius increases the surface roughness decreases while the specific cutting energy increases. Xiao et al. (2003) reported that in conventional cutting of Inconel 600 a slight increase in nose radius increases the surface roughness significantly while in vibration cutting a decrease in surface roughness was noticed when nose radius increases up to 0.2 and thereafter the roughness increases gradually. Liang et al. (2007) used regression analysis to show that both feed-rate and tool radius affect the surface roughness of the finished workpiece significantly. Basheer et al. (2008) also found tool nose radius to be one of the factors that affected the surface roughness in machining of metal matrix composites. Several researchers have shown that the tool nose radius has a significant effect on the residual stress distribution on the machined surface when machining various types of materials (Liu et al. (2004), Nasr et al. (2007), Thiele and Melkote (2000), Hua et al. (2006)). The nose radius was also shown to affect the chip formation during cutting of silicon wafers (Liu et al. (2007)).
In finish turning and boring, where the depth of cut is less than the nose radius, the theoretical roughness value depends directly on the feed per revolution and the nose radius (Stephenson and Agapiou (2006)). Whitehouse (2002) showed that the average roughness varies inversely with the nose radius for a given feed rate. The importance of nose radii has prompted manufacturers to design tools of different radii for various machining conditions and surface finish requirement (Sandvik Coromat (2006)). Tools of small nose radii are generally used for smooth surface finish while larger radii are used for greater stability during machining, though too large a radius can lead to tool chatter and poor chip control.
For precision machining, it is essential that the tool nose radius has a close tolerance. Cutting tool inserts are generally manufactured with nose radius tolerances within ±10% as specified in the ISO3685 standard (ISO3685 International Standard (1993)). However, this small deviation may have a significant effect as far as precision high-speed machining is concerned. If the exact nose radius is known the tool path can be programmed precisely to obtain high dimensional accuracy in the finished product.
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Determining the actual nose radius of a cutting tool insert is almost impossible using conventional measuring equipments such as profile projectors or measuring microscopes. The reason is because only a sector of the tool nose is available for the radius measurement when using these instruments. Hopp (1994) showed that the radius of a circle is highly sensitive to errors in measurement when points extracted from a sector subtending less that 40° at the center are used in the radius and center point calculations. This paper introduces a novel approach to determine the nose radius accurately using polar-radius transformation of the 2-D nose profile where only a sector of the circle is available for the measurement. The proposed method uses the polar-radius transformation technique in nose wear measurement developed in our previous work (Mook et al. (2009)) .
To obtain a polar-radius plot of the tool nose profile, it is essential first to locate the center of curvature of the nose. The theory developed for finding the center of curvature was explained in detail in our previous publication (Mook et al. (2009)). It is included here since it forms the basis of the proposed method.
Figure 1 is a schematic diagram of the 2-D profile of a cutting tool insert in the nose region. The tool nose has a radius r. A(x1, y1) and B(x2, y2) are two points on the major cutting edge, while C(x3, y3) and D(x4, y4) are two points on the minor cutting edge. All the four points are located away from the curved nose region. The straight lines connecting A to B and C to D meet at M whose coordinates (xM, yM) are given by
where m1 and m2 are the gradients of the lines joining A-B and C-D, respectively. The angle b between the lines A-B and C-D represents the tool angle and is given by
The center of the tool nose O(xo, yo) can be located as follows. The perpendicular distance d from a point (h, k) to a straight line is given by the general expression,
Since the equations for the two lines A-B and C-D are given, respectively, by
the distances from the nose center O to the lines are given, respectively, by
Since d1 = d2 = r, by equating (5a) and (5b) we obtain
and yo can be found by substituting (6) into (5b) as
Thus, the nose center point O(xo, yo) of the cutting tool can be determined from equations (6) and (7). The radial distances from the nose center O to the boundary points on the nose profile are then determined as a function of angle q. Using this data a polar-radius plot of the nose profile can be obtained. The plot will reveal a straight line in the nose radius region and a symmetrical deviation from the straight line for points located away from the nose as shown in Figure 2. In Figure 2, qS and qE refer to the start point (P) and end point (Q) of the curved nose region. These are points where the straight and curved parts of the insert meet.
For a given tool profile if the exact nose radius is not known, the tool center can be determined by substituting a range of nose radii close to the exact radius. For each radius, the tool center can be determined from equations (6) and (7) and a polar plot can be obtained. The plot will be a straight line in the nose region only if the exact radius is used in the calculation of the nose center. An inexact nose radius will cause a deviation from the straight line plot. The nose radius value that produces the smallest deviation from the straight line will be the exact radius. This method of obtaining the exact radius is demonstrated using simulated cutting inserts in the next section. The reason for using simulated tools is because the exact radius is known a priori since the 'inserts' can be generated using CAD software.
3.1. Generation of simulated images of cutting tools
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To demonstrate the operation of the proposed method of determining the nose radius from a sector of a circle, several cutting tool inserts of different nose radii were generated using Autosketch v2.1 software. The simulated inserts were based on CCMT12 rhombic, TPGN16 triangular and SPGN12 square tools each having nominal nose radii of 0.4 mm, 0.8 mm and 1.2 mm (Figure 3) (Sandvik Coromat (2006)). The 'inserts' were converted into 8-bit grayscale uncompressed TIFF images using Matrox Inspector software. Figure 4 shows close-up views of the nose region for CCMT12 rhombic inserts. Two additional radii were introduced to increase the data available for verifying the method of determining the nose radius. By using simulated images the exact nose radius can be generated using the 'fillet' command available in the software. Since the image coordinates are in pixels a scaling factor was determined using the drawing units (in mm) and the corresponding units in pixels in the Matrox software for an entity of known dimensions. The scaling factors were 136 pixels/mm for triangular and square inserts and 162 pixels/mm for the rhombic inserts.
3.2 Algorithm for nose profile detection and polar-radius transformation
Figure 5 shows the various stages of the nose profile detection and polar-radius transformation algorithm for the simulated inserts. The first stage involves reading of the grayscale image of the insert. Since the image has one edge in the horizontal direction (Figure 3), it is rotated by about 5° counter-clockwise using bicubic interpolation to avoid a value of 0 for m2. This is done by using the imrotate command in Matlab in the second stage. In the third stage the image is binarized using global thresholding, thus converting it from 8-bit (0-255 grayscale range) to 1-bit (intensity values of 0 or 1). In the forth stage the image is scanned pixel-by-pixel to detect the two points A(x1, y1) and B(x2, y2) on the major cutting edge and another two points C(x3, y3) and D(x4, y4) on the minor cutting edge as in Figure 6(a). The intersection point M, nose angle b and nose center point O(xo,yo) are then determined using equations (1), (2), (6) and (7), respectively. The tool angles b for the various inserts are rhombic: 80°; triangular: 60°and square: 90°.
In the next stage the Canny edge operator is applied to detect the boundary (profile) of the cutting tool (Figure 6(b)). Using the tool center point, the region around the nose is divided into three quadrants as shown, i.e. Q1, Q2 and Q3,. The boundary pixels in each quadrant are determined by scanning within the quadrant. The distance dm of each edge pixel m from the tool center O and the angle qm measured counterclockwise from the center point are calculated using the following equations:
for quadrant Q1 (9a)
for quadrant Q2 (9b)
for quadrant Q3 (9c)
A plot of qm against the radial distance dm is the polar-radius plot of the tool nose profile as shown in Figures 6(c).
3.3 Determination of nose radius from polar radius plot
Figure 6(c) is the polar-radius plot for a simulated rhombic insert of radius 194 pixels, which corresponds to the exact nose radius of 1.20 mm. In the nose region, between 85° and 185°, the distance from the tool center to the edge has almost a constant value equal to the nose radius. The fluctuations observed in the plot are due to quantization of the digital image when it is converted from grayscale to binary. If the value of radius r used in the computation of the center of curvature of the nose (tool center) and radial distances is either too small or too large the polar radius plot will deviate from a straight line in the nose region as shown in Figure 7(a) for r=184 pixels (1.14 mm) and Figure 7(b) for r=204 pixels (1.26 mm), respectively. This deviation is due to the combined error in the radius value and tool center point calculation. Within the nose region, the average deviation d from the straight line profile is a minimum at the exact radius of 1.20 mm (194 pixels). Thus, by varying the radius r within a certain range of the nominal value and finding the radius that produces the smallest average deviation (dmin) from a straight line it is possible to determine the exact (or optimum) nose radius. The average deviation is determined from the deviation at each point on the profile within the nose region and the minimum average deviation dmin is determined as follows:
where k is the number of points along the nose profile.
3.4 Measurement of nose radii of real cutting tools
The measurement algorithm was applied to three different types of cutting inserts manufactured by Sandvik Coromant Ltd. (Sweden): Triangular (TPUN16-03-04), rhombic (CNGP12-04-04) and square (SPGN12-03-08). Two inserts of each type were used in the study. Images of the inserts were captured using a high-resolution CCD camera (1296-1024 pixels) (Model CV-A1 manufactured by JAI Co. Ltd., Japan) fitted with a 50-mm lens as in Figure 8. A 100-mm extension tube was used to increase the optical magnification. Distortions due to the use of long extension tube was verified and reported in our previous work (Shahabi and Ratnam (2009). The insert was mounted on a tool holder and illuminated by backlighting provided by a frequency-stabilized ring light. Sample images of the inserts are shown in Figure 9.
Since it is necessary to adjust the focusing ring each time a new insert is attached to the tool holder, the scaling factor was determined on each occasion using a 0.22 mm width pin gage (Part no. 313-101, Mitutoyo Corporation). The pin gage was placed at the same focal plane as the insert in both horizontal and vertical directions with reference to the image plane. The images of the pin gage were scanned along three sections and the pixel distances were noted. These were used to determine the horizontal and vertical scaling factors. Since the distances from the nose center to the edge profile are measured in the radial directions, an average scaling factor was determined.
To determine the optimum nose radius that gives the minimum average deviation from a straight line in the nose region the algorithm in Figure 5 was modified by including a loop that changes the nose radius from to pixels. For each value of nose radius the average deviation was determined and plotted against the radius. The radius corresponding to the minimum average deviation is determined automatically in the Matlab code.
3.5 Measurement of nose radii of using an optical profile projector
The radii of the inserts determined using the proposed system were compared with measurements made using an optical profile projector (Rax Vision: Model CPJ3015Z). The profile projector is a common instrument for 2-D measurement based on the projected image of an object. Ten points along the profile of the tool nose were selected for the measurement of radius for each insert. The measurement was repeated three times on each nose profile so that different points are selected for computing the radius.
4.0 RESULTS AND DISCUSSION
Figure 10(a) shows a plot of average deviation against nose radius for the simulated CCMT12 rhombic insert with an exact radius of 194 pixels (1.20 mm), while Figure 10(b) shows a similar plot for the TPUN16 triangular insert with an exact radius of 163 pixels (1.20 mm). For the rhombic insert the radius was varied from 184 pixels to 204 pixels. The optimum (exact) radius obtained by applying the algorithm in Figure 5 is 191 pixels where the deviation from straight line plot is minimum, i.e. an error of 1.57% compared to the actual value of 194 pixels. The optimum radii for all the other simulated tools were determined similarly and are shown in Table 1. For the 15 images tested the error in radii detected using the proposed algorithm varied from -4.9% to 3.7%. This error is mainly due to the quantization effect when the grayscale image was converted to binary. The small value of error shows that the nose radius can be determined accurately from the nose profile of the tool.
Table 2 compares the radii detected using the proposed algorithm with the nominal radii specified by the manufacturer. The scaling factor f used to convert the dimensions in pixels to millimeters for each insert is also given in the table. Figure 11 show a typical plot of average deviation from straight line against nose radius in the nose region for a triangular insert (Tool 1). For most of the inserts the nose radii determined using the proposed method were found to be 8% to 20% higher than the nominal radius. The average deviation is +11%, which is 1% larger than the 10% tolerance allowed in the nose radius according to ISO 3685 standard.
The nose radii measured using the profile projector is given in Table 3. Two observations can be made by comparing the data in Table 2 and Table 3: (i) The nose radii determined using the proposed method are about 9% to 22% higher than those measured using the profile projector and (iii) the nose radii measured using the profile projector are closer to the nominal radii with a average deviation of -3.2% compared to those measured using the proposed method. The data in Table 2 and Table 3 give the impression that the radii measured using the profile projector are more accurate compared to those determined using the proposed polar-radius transformation algorithm. This is because the radii measured by the profile projector are closer to the nominal radii compared to those determined by the algorithm. It is suspected that manufacturers measure the nose radii for quality checking using the nose profile, which essentially represents a sector of a circle of maximum angle of 90° for the square insert. The measured radii are then used to specify the nominal radii of the inserts. As verified by Hopp (1994), it is impossible to measure the radius of a circle accurately from a few points selected only from a sector of the circle. The points should ideally be located either 120° apart when three-point measurement is made or equally spaced around the circumference when more than three points are used. Thus, the measurement of the nose radius using a profile projector or any other device from a sector of a circle is inaccurate. The error increases as the angle subtended by the sector decreases. To confirm the error experimentally, guide holes on a precision stamped metal frame were measured using the profile projector using two methods:
Method 1: By selecting 10 points located around the circumference of the hole at approximately equal distances.
Method 2: By selecting 10 points within a 90° sector of the hole randomly located around the circumference.
Each measurement was attempted ten times and the results are shown in Table 4. As expected, the radius measurement using ten points around the circumference (Method 1) produced consistent values for the various attempts, with a standard deviation of 0.002 mm for hole 1 and 0.004 for hole 2. However, for hole 1 the radius measured from a 90° sector of the hole (Method 2) varied from 0.497 mm to 0.924 mm with a standard deviation of 0.129 mm. For hole 2, the radius varied from 0.568 mm to 0.837 mm with a standard deviation of 0.089 mm. These results experimentally verify that it is not possible to measure the radius accurately from only a sector of a circle.
Close observation of the polar radius plots for three types of inserts at the minimum average deviation, which corresponds to the most accurate radii, surprisingly show a significant deviation from a straight line in the nose region (Figures 12(a)-(c)), unlike for the simulated insert (Figure 6(a)). This deviation is suspected to be caused by manufacturing inaccuracy of the nose profile, which is not a true arc of a circle. Since the actual nose profile is not part of a perfect circle fitting a few points to find the radius based on the equation of a circle, as in the profile projector, will produce erroneous results.
A new method of determining the nose radii of cutting tool inserts is proposed in this paper. Compared to the conventional method of finding the nose radius from a sector of the circle represented by the tool nose, which is prone to measurement errors due to the limited angle, the proposed method transforms the curved nose profile into a linear profile using polar-radius transformation. The average deviation from a straight line in the nose region is evaluated for a range of input radii. The radius value that produces the minimum average deviation corresponds to the most precise measurement of the nose radius. Although this method is proven to be effective based on the simulated and real cutting tool inserts the accuracy of the measurement relies on the following factors:
The insert must be clean and free from dust particles when capturing the image. This will ensure that the gradients of the straight lines extending from the edges are determined accurately.
The insert must be aligned accurately so that the plane of the nose profile is perpendicular to the optical axis of the CCD camera. This will ensure that cosine errors due to misalignment are minimized.