Error Measurements for Flank Milling - University of Waterloo
Error Measurements for Flank Milling Chenggang Li Stephen Mann Sanjeev Bedi University of Waterloo, Waterloo, Ontario, Canada. N2L 3G1. Phone:(519)888-4567 Fax(519)888-6197
Abstract This paper presents and compares methods of error metrics used to measure the error for the flank milling of a machined ruled surface. The aim of this work is to propose a new scheme for error approximation that is easy to implement and gives better error assessment when using optimization techniques to determine tool position. We propose using as the error calculation an approximation to the distance from the desired surface along the direction of the surface normal. Key words: 5-axis machining, error metric, optimization
1
Introduction
Flank milling is used to machine turbine blades, impellers and other engineering objects. Many techniques have been developed over the last ten years for flank milling. The availability of 5-axis Computer Numerically Controlled (CNC) machines has brought flank milling to the forefront and many researchers have focused their attention to developing better methods of tool positioning for flank milling applications. The main goal in these methods is to determine the trajectory of a cylindrical (flat end mill) or a conical tool so that the surface swept by the tool as it moves closely matches the desired surface. In flank milling methods, the tool is first positioned based on surface properties of one or two points on the surface and then the tool position is optimized by lifting and/or twisting the tool to reduce an error metric. So far most researchers have attempted to machine ruled surfaces with flank milling. One reason for using a ruled surface is the simplicity of defining the surface. Another reason is that until recently the shape produced by a tool moving and 1 2 3
[email protected], Graduate student, Mechanical Engineering [email protected], Associate Professor, Computer Science [email protected], Professor, Mechanical Engineering
Preprint submitted to Elsevier Preprint
12 April 2004
rotating at the same time was not known, but observations showed that if the rotation is small the machined surface was close to a ruled surface. Although methods of generating surfaces swept by moving tools are now well known, no work in the literature focuses on machining of doubly curved surfaces. In this work, as well, the focus is limited to machining ruled surface, although much of our error metric analysis should be applicable to more general classes of surfaces. The ultimate goal is for the machined surface to match the desired surface as closely as possible. This suggests using a closest point error metric for the optimization methods. However, there are two difficulties with using a closest point error metric for optimizing the tool path: first, the metric is expensive to compute, as the simulated machined surface has a complex representation. Indeed, a closed form for the machined surface is rarely used, and approximations are computed and used instead. A second problem with using a closest point error metric with an optimization method is that as we optimize the tool path, we change the machined surface. Thus, not only does the distance to the closest point change, but the direction to the closest point also changes, making the optimization task more difficult. In this paper, we look at error metrics used when optimizing tool paths for the flank milling of ruled surfaces. Many error metrics have been developed by different researchers. Use of different error metrics leads to different solutions. This paper studies the error metrics proposed by other researchers and proposes a new error metric that we feel is most appropriate when using optimization methods for tool positioning.
2
Background
A ruled surface is generated between two guiding lines (T (u) and B(u)) by connecting corresponding points on the two lines with a straight line called the generating line of the ruled surface (Figure 1(a)). The machined surface is produced by the 5-axis movement of a cutting tool. It can be generated mathematically by identifying grazing curves at numerous points on the tool path and connecting the connective grazing curves into a wire mesh representing the swept surface (Figure 1(b)). The grazing curve is a function of the tool, cutting surface and the velocity of the tool axis. As a tool sweeps through the stock, the tool cuts the stock at numerous points on its surface. The points that lie towards the front of the tool are machined away as the tool leaves its current position. Only those points for which the moving direction is perpendicular to the normal of the surface of the cutting tool are left behind on the stock as a curve on the machined surface; this is the grazing curve. The grazing points are related to the motion of the tool and are easy to identify. These are the points where the velocity of the tool axis is perpendicular to the surface normal. So, the equation of those points that leave a mark on 2
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Fig. 1. (a) The desired ruled surface. (b) The machined surface, made by connecting grazing curves. (c) Ruled Surface and Grazing Surface superimposed.
the stock is P (u, v) = S(u, v) +
dS(u,v) du dS(u,v) du
× T (u, v)
R,
× T (u, v)
(1)
where S(u, v) is any point on tool axis, T (u, v) is the tool axis vector and R is the cutter radius corresponding to S(u, v). Different tool positions will generate different grazing curves, which can be joined together into a surface [1]. Consider the situation in Figure 2. In this figure, T (u) and B(u) are the guiding curves, (3) is a ruled line lying on the ruled surface, and P1 is a point on the ruled line. The surface machined by the moving tool is a collection of grazing curves (one of the grazing curves is superimposed on the tool in Figure 3). The match between the desired shape and the machined surface can be measured by the shortest distance between a point on the desired surface and some point C on the machined surface. The error at point P1 is the distance between the machined surface and the point P1 . The question addressed in this paper is in what direction should we measure this distance? While some times one will want to consider the distance between parametrically equivalent points, and at other times one will want the geometric shortest distance, in this paper we argue that for using optimization methods to find the tool orientation, the distance for the error metric should be along the ruled surface normal direction at P1 . In Figure 2, the surface normal passing through point P1 is (1) and the intersection point between the machined surface and normal (1) is the point C. The length of P1 C is our proposed error of the machined surface at point P1 . What ever form of closest point method is used to compute C, the point C has a complex mathematical relationship with the ruled line and the point P1 . This makes it difficult to calculate the exact error in a closed form manner. Generally, there are two reasons for this difficulty. First, the equation of the machined surface is not known, so the closed form calculation of shortest distance is not possible and an iterative numerical technique is required 3
Fig. 2. (1) surface normal at point P1 . (2) tool axis. (3) ruled line. (4) curve on surface
to calculate it. Second, the machined surface depends on the trajectory of the tool and any change in the trajectory effects the error calculation. Thus, using the error calculation to re-position the tool changes the error in the neighbourhood. For these reasons, focusing on the exact error at large computational cost may be futile and a simple error approximation may be better. Researchers who use error assessment to improve tool positioning have used a variety of different approximations. It is a goal of this paper to put the different error approximations into perspective and propose a new scheme for error approximation that is easy to implement and gives a better error assessment. This new method will be compared with established techniques. We begin with a review of previous techniques. Rehsteiner et al. [2] suggested a technique to position the cutting tool. In their method, the tool axis is first positioned parallel to the rule line and tangent to the middle point of the rule, then the tool is rotated about the surface normal (at middle point of rule) until it touches the boundary curves. The interferences at boundary curves are checked to try to get error-free surfaces. Undercutting will occur with this method. Stute et al. [3] developed another method to position the tool. In their work, the tool axis is positioned at two points. One point is on the surface normal through the top point of the rule line and the other point is on the surface normal through the bottom point of the same rule line. Both of these points are offset by the value of milling cutter radius. The maximum overcut is produced in the middle of the ruled surface. Liu [4] presented a double point offset (DPO) method to generate the cutter location. The idea behind this method is to offset two points at parametric values of 0.25 and 0.75 on the 4
rule line along the surface normal direction. The offset distance is the tool radius. The two offset points are then used to define the tool axis orientation. Bohez et al. [5] developed a different algorithm to locate the tool. They require that the tool to be tangent to a point P on the rule line while the tool axis is parallel to the same rule line. The point P is selected so that the angles between the surface normal at P and the normals on the two guiding curves are equal. The resulting overcut can be further diminished by moving the tool axis away from the surface along the surface normal at point P until the tool is tangent to the two guiding curves. The maximum deviation will be around the middle area of the surface. This error can be further reduced by using multiple machining passes. In Tsay and Her’s method [6], the cutting tool is initially tangent to the boundary curves at the same parametric value u, and then by varying the angle between the tool axis and the rule line as well as the distance between the tool axis and ruling, the tool position is optimized. Statistical analysis is used to minimize the error measured in the plane perpendicular to the rule line. Monies et al. (using a conical tool) and Redonnet et al. (using a cylindrical tool) [7–10] suggest that the tool can be positioned to be tangent to the ruled surface at three points by slightly changing the angle between the tool axis and the rule line, which can further reduce the surface error. Seven transcendental equations are developed in their paper to define each tool position. Bedi et al. [1] developed a strategy to roll a cylindrical cutting tool along two guiding rails. The tool remains tangent to the guiding rails and the contact points have the same parametric value. Each tool position is obtained by solving two transcendental equations simultaneously and the maximum overcut will lie at the middle curve of the surface. The error at each point of the surface can be predicted. Menzel et al. [11] modified this strategy to develop a triple tangent flank milling method. In this method, the tool is initialized by the Bedi et al. strategy, and then the contact points are moved along the rule line and finally the tool is twisted to generate an optimized tool position. The optimized tool position reduces the deviation. This technique can also be extended to a conical tool as well. All of these methods try to reduce the error between the machined surface and the desired ruled surface even though different error measurements are used. These techniques are described in the next section.
3
Previous Methods for Error Measurement
Before presenting our error metrics, we begin with a review of other methods for approximate error measurement. These previous methods broadly fall into two categories: the Radial Methods and the Parametric Methods. In addition, some work has been done using a closest point metric. 5
Fig. 3. Radial Error Calculation. (1) tool axis. (2) ruled line. (3) grazing curve. (4) curve on surface
3.1 The Radial Method
In the radial method, the tool is assumed to be stationary. The error is estimated by calculating the shortest distance from P1 (see Figure 3), which lies on the ruled line of interest, to the surface of the tool. In our discussion, we will consider both cylindrical and conical tools (most of the previous work has focused on cylindrical tools). For a cylindrical tool, we could estimate the error as |P10 − P | = R − |P − P1 |, where P10 is the point on the cylinder closest to P1 , P is the point on the tool axis closest to P1 , and R is the radius of the cylinder. Note that for a cylinder, P , P1 , and P10 are collinear. This is the method used as the error metric in several papers. However, except in some situations, this would be an under estimation of the error, since P10 will in general not lie on the machined surface (i.e., P10 is not on a grazing curve and either has or will be machined away by the moving tool). A more accurate radial error calculation proceeds by taking the grazing point A corresponding to P , finding D, the closest point on the rule line to A, and estimating the error as |DA|. In the case of a cylindrical tool, DA is normal to the cylinder surface. However, for a conical tool, DA is not perpendicular to the tool surface and the error must be calculated by multiplying |DA| with the cosine of the cone angle. Figure 4(a) shows an example of the error along a ruled line calculated using the Radial Method. This method of error measurement was used in Liu’s strategy, Bohez et al.’s strategy and Stute et al.’s strategy. However, in these strategies, the cutting tool is a cylinder and the error is obtained by subtracting the shortest distance (point on ruled line to cylinder tool axis) from the tool radius. In Bedi et al.’s strategy, the error is modified by a constant related to tool parameter because they used a conical cutter. 6
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Fig. 4. (a) Error Calculated Using Radial Method at u = 0.25; (b) Error Calculated Using Parametric Method at u = 0.25; (c) Error Calculated Using Motion Method at u = 0.25
Fig. 5. Parametric Error Calculation. (1) tool axis. (2) ruled line. (3) grazing curve. (4) curve on surface
3.2 The Parametric Method
In the Parametric Method, the error is computed between points on the ruled surface and points on the grazing curves that have the same parameter values. Thus, the error at a point P (ui , vi ) on the ruled surface is given by ε = P (ui , vj ) − G(ui , vj ),
where G(ui , vi ) is the corresponding grazing point. Figure 5 graphically shows the Parametric Method for calculation of error. Point P1 is a point on ruled line and point A is the corresponding grazing point. The length of P1 A is the approximate error ε. Figure 4(b) shows an example of the error along a ruled line calculated using the Parametric Method. 7
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Fig. 6. Schematic Illustration of Error Measurement Methods: (a) Radial (b) Parametric (c) First Order. Error between point A on the ruled surface and the machined surface is given by the dark line. Parametrically, A corresponds to point C on the tool/machined-surface.
3.3
Closest Point
Monies et al. also suggested a strategy similar to the Parametric Method. However, after using the Parametric Method as a starting point, the authors then search for the smallest error ε by looking over a small surface surrounding the point. This method, exhaustively searching for a proper grazing point to get the smallest error ε, is a computationally challenging job.
4
Methods for Error Measurement
The Radial Method for error measurement has the advantage of being simple to compute. However, the approximation seems poor, since as illustrated in Figure 3 and Figure 6(a), this method is not really calculating the error between the point on the ruled surface and the machined surface. The Parametric Method (Figure 6(b)) appears to answer this problem, and it does give an exact measurement of the error between the parameterized ruled surface and the parameterized machined surface. While the Parametric Method will clearly be useful for many purposes, it is a parametric error measure rather than a geometric error measure. In particular, given the sweeping motion of the tool, the Parametric Method will fail to give an accurate measurement of the shortest distance between ruled surface and the machined surface. The method of Monies et al. [9] addresses the question of computing a geometric error between the two surfaces. In many situations, computing the shortest distance between the ruled surface and the machined surface is the appropriate metric. However, there are two concerns with the closest point method: first, it is computationally expensive, and second, when the distance is used in optimization techniques, the direction of error at each point is not constant. The lack of a constant direction of error means that both the direction vector and the distance along the error direction should be considered in the optimization, making the problem more difficult. As an alternative, we propose using the distance from the point S(u, v) along the normal n(u, v) to the ruled surface. While similar to the Closest Point method of Monies et al., with 8
this method the direction of error is constant. However, computing the distance from the ruled surface to the machined surface along the surface normal has similar computational issues as the Monies et al. method. In particular, since the exact mathematical model of the machined surface is unknown and is often approximated with a piecewise linear surface, we would have to search a group of polygons to find the closest point. As a compromise between the accuracy of the error measurement and speed, we propose the Tangent Plane method, which finds the closest point between S(u, v) along the normal direction n(u, v) to a first order approximation to the machined surface (Figure 6(c)). We also suggest a second method, the Motion Method, which finds the closest point between a rule line and a moving grazing point, in essence making a linear approximation to both the ruled surface and to the machine surface, rather than computing the distance between a fixed point on the ruled surface to a planar approximation to the machined surface.
4.1 The Tangent Plane Method
In the tangent plane method, we construct the tangent plane to the grazing surface at G(ui , vi ) and find the intersection of this plane with the line from P (ui , vi ) in the direction of the normal to the ruled surface, n(ui , vi ). The tangent plane to the grazing surface at a grazing point is easily computed, since we know the direction of motion of the grazing point and we can compute the tangent to the grazing curve. The error plot for this method on our running example is discussed further at the end of Section 4.2.
4.2 The Motion Method
The Motion Method is our second method for approximating the error between a ruled surface and the machined surface. Starting with a point P1 on the ruled line, we compute the corresponding grazing point A. We then compute the shortest distance between the ruled line containing P1 and the line through the grazing point A in the direction motion of this grazing point. For this method, at a point A along the grazing curve, we would like to construct a line M in the direction of motion of the grazing point. If there is no rotation, then the direction of motion is given by dS(u, v)/du, which is readily seen from Equation 1 to be the cross product of the the tool axis with the tool radial line passing through A. However, the true direction of motion is unknown for 5-axis motion, since the 5-axis motion of a point A on the grazing curve differs from dS(u, v)/du by a vector that lies in the tangent plane of the tool at A. If the rotation motion is small compared to the translational motion, then this direction will give an acceptable approximation to the direction of motion of the grazing point. Regardless, 9
Fig. 7. Motion method error calculation. (1) tool axis T. (2) ruled line. (3) grazing curve. (4) curve on surface.
using dS(u, v)/du as the direction of M yields a line that is tangent to the surface swept by the moving tool. The error at this point on the grazing curve is estimated as the shortest distance between the ruled line of the surface and the line M . Figure 7 shows this method graphically. Parametrically, point A on the grazing curve corresponds to point P1 on the ruled surface. From point A, the shortest distance to the tool axis T is calculated as is r = |AB|, where B is the closest point on the tool axis. Then, the line M is constructed to pass through A in the direction N = T × (B − A). The shortest distance between line M and ruled line (2) is our approximation to the error. Note that this approximation computes an error from the line M to a point P2 on the ruled line, where P2 6= P1 ; Note also that Figure 7 is exaggerated – in practice, points P1 and P2 nearly coincide. Figure 4(c) shows error along the ruled line u = 0.25 calculated using the Motion Method. The error for the Tangent Plane method is similar, with the two methods producing results within 1% of one another. In our work, the two methods produced similar enough results that we only give the results for the Motion Method.
5
Discussion and Experimental Results
Several methods for error measurement are described in this paper. In this section, we wish to address several issues, including (1) Are the Tangent Plane Method and the Motion Methods good approximations of the Closest Point Method? (2) How do the various methods compare, both as methods for assessing error in machined surfaces and in being used for optimizing tool positions? To address these questions, we ran a variety of simulations and measured data off one machined part. 10
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Table 1 Control points for guiding trails [mm]
Fig. 8. Machining Part
In this paper, we give the results for machining a particular ruled surface that is comprised of two B´ezier curves where the control points for guiding rails are given in Table 1. Further, we machined this part and measured the closest point error for the ruled line u = 0.9. We machined this part using both a non-optimization technique and an optimization technique [12], using the motion method to optimize the tool position. Figure 8 shows the test part using the optimization technique. Both the Motion Method and the Tangent Plane Method are approximations of the Closest Point Method. To see how good the approximation is, we compared the Motion Method to the Closet Point Method. Figure 9 shows the error plots of these two methods for one ruled line of our example ruled surface; further, we compare these errors to the error in our machined surface. The graph shows that all three error plots are similar. (Figure 10(b) shows a comparison of the Motion Method to the Radial Method and to the Parametric Method on the same data.) Figure 10 compares the error using different methods to measured data. The tool used was a conical tool and it was positioned with Bedi et al.’s strategy and its extension. The cone parameters are R1 = 19.252mm, R2 = 3.175mm, h1 = 3mm, h0 = 60mm. Measuring experimental error introduces additional difficulties into the error metric question. Errors such as the part fixture error, machine tolerances, etc., must be considered and filtered in the experimental results. These issues aside, a correspondence between points on the machined surface and points on the mathematical surface must be established. Radial error metrics no longer make sense (since the tool is no longer present), and to use a Parametric error metric would require appropriately parameterizing the machined surface which is not 11
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Fig. 9. Comparison of Motion Method and Closest Point (u = 0.9). u=0.9 u=0.25
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Fig. 10. (a) Errors Comparing (u=0.25); (b) Error Comparing (u=0.9)(c) Error Comparing (u=0.9) (after optimization); (d) Error Comparing (u=0.9) (after optimization)
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Fig. 11. Optimization using different metrics.
feasible. Instead, we used a Closest Point error metric, which is typically the form of error metric used for experimental data. (A Tangent Plane error metric could also be used, since the ruled surface is known, but the motivating factors for using the Tangent Plane error metric for tool position optimization are not relevant when measuring experimental error.) Figure 10(a) and Figure 10(b) displays the results before optimization and Figure 10(b) also contains the experimental results. These graphs show that the resulting error from the Motion Method is close to the experimental result. The Radial Method gets the similar results, but the resulting error has some deviation. The Parametric Method can also reflect the error variation, but huge deviations may exist in its results. Figure 10(c) and Figure 10(d) show the same tool position as Figure 10(b) after optimization. In Figure 10(c), we see that the Parametric Method yields a large error estimate compared to the other methods. Figure 10(d) is a view of the error plots at higher resolution. Again, we see that the experimental result is almost the same as the error given by the Motion Method. As a final test, we simulated using the three methods (Radial, Parametric, and Motion) to optimize toolpaths, and compared the resulting errors. In all three cases, although different error metrics were used for the optimization, the final error measurements were computed using the Motion Method. Figure 11 shows the results of these tests. In all three cases, we see that the surface obtained using the Parametric Method has the highest error. We also see that the Radial Method gives results similar to the Motion Method; that is a result of, in our example surface, the Radial and Motion Methods having similar error results for the sample tool positions (i.e., see Figures 10(a)(b)). After optimization, note that the Radial Method fails to give an accurate estimation to the measured error as shown in Figure 10(d).
6
Conclusion
In this paper, three different error measurement methods are studied and compared. Two new strategies, the Tangent Plane Method and the Motion Method, were proposed, both of which 13
closely reflect the surface error. Both new methods essentially make linear approximations in two dimensions; the difference is in whether both dimensions are on the machined surface (as in the Tangent Plane Method) or if only one dimension is on the machined surface with the other being on the ruled surface. Both methods are good approximations to the Closest Point Method, and use of either one should produce similar results. The calculation procedure of both new methods is simple and robust. The Radial Method sometimes gets similar results to the Motion Method, but the calculated error has some deviations especially when the tool is twisting. The Parametric Method can reflect the error changing in general conditions, but has large deviations if the tool is twisted during the optimization. While our work in this paper suggests that the Tangent Plane or Motion Method is most appropriate for optimizing tool paths, note that there are situations in which it is likely not the best method. In particular, in the example in this paper, the desired surface is smooth. If we were to machine a surface with a ridge or corner (a C 1 discontinuity), then use of the Motion Method would potentially smooth the ridge, and thus some other method is likely needed to accurately machine the surface in such regions.
References
[1] S. Bedi, S. Mann, and C. Menzel. Flank milling with flat end cutters. Computer-Aided Design, 35:293–300, 2003. [2] F. Rehsteiner and H.J. Renker. Collision-free five-axis milling of twisted ruled surfaces. CIRP ANNALS, 42(1):457–461, 1993. [3] G. Stute, A. Storr, and W. Sielaff. Nc programming of ruled surface for five axis machining. Ann. CIRP, 28(1):267–271, 1979. [4] X. Liu. Five-axis NC cylindrical milling of sculptured surfaces. Computer-Aided Design, 27(12):887–894, 1995. [5] E.L.J. Bohez, S.D.R. Senadhera, K. Pole, J.R. Duflou, and T. Tar. A geometric modelling and five-axis machining algorithm for centrifugal impellers. Journal of Manufacturing Systems, 16(6):422–463, 1997. [6] D.M. Tsay and M.J. Her. Accurate 5-axis machining of twisted ruled surfaces. Journal of Manufacturing Science and Engineering, 123:734–738, 2001. [7] J.-M. Redonnet, W. Rubio, and G. Dessein. Side milling of ruled surfaces: Optimum positioning of the milling cutter and calculation of interference. Advanced Manufacturing Technology, 14(7):459–465, 1998. [8] F Monies, J-M Redonnet, W Rubio, and P Lagarrigue. Improved position of a conical mill for machining ruled surfaces: application to turbine blades. Journal of Engineering Manufacture, 214(B):625–634, 2000.
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[9] F Monies, W Rubio, J-M Redonnet, and P Lagarrigue. Comparative study of interference caused by different position settings of a conical milling cutter on a ruled surface. Proceedings of the Institution of Mechanical Engineers-B- JNL of Engineering Manufacture, 215(9):1305, 2001. [10] F Monies, J.N. Felices, W Rubio, J-M Redonnet, and P Lagarrigue. Five-axis nc milling of ruled surface: optimal geometry of a conical tool. International Journal of Production Research, 40(12):2901–2922, 2002. [11] C. Menzel, S. Bedi, and S. Mann. Triple tangent flank milling of ruled surface. Computer-Aided Design, 36, 2004. [12] Chenggang Li. Flank milling of ruled surfaces with conical tools. Master’s thesis, University of Waterloo, 2004.
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Abstract This paper presents and compares methods of error metrics used to measure the error for the flank milling of a machined ruled surface. The aim of this work is to propose a new scheme for error approximation that is easy to implement and gives better error assessment when using optimization techniques to determine tool position. We propose using as the error calculation an approximation to the distance from the desired surface along the direction of the surface normal. Key words: 5-axis machining, error metric, optimization
1
Introduction
Flank milling is used to machine turbine blades, impellers and other engineering objects. Many techniques have been developed over the last ten years for flank milling. The availability of 5-axis Computer Numerically Controlled (CNC) machines has brought flank milling to the forefront and many researchers have focused their attention to developing better methods of tool positioning for flank milling applications. The main goal in these methods is to determine the trajectory of a cylindrical (flat end mill) or a conical tool so that the surface swept by the tool as it moves closely matches the desired surface. In flank milling methods, the tool is first positioned based on surface properties of one or two points on the surface and then the tool position is optimized by lifting and/or twisting the tool to reduce an error metric. So far most researchers have attempted to machine ruled surfaces with flank milling. One reason for using a ruled surface is the simplicity of defining the surface. Another reason is that until recently the shape produced by a tool moving and 1 2 3
[email protected], Graduate student, Mechanical Engineering [email protected], Associate Professor, Computer Science [email protected], Professor, Mechanical Engineering
Preprint submitted to Elsevier Preprint
12 April 2004
rotating at the same time was not known, but observations showed that if the rotation is small the machined surface was close to a ruled surface. Although methods of generating surfaces swept by moving tools are now well known, no work in the literature focuses on machining of doubly curved surfaces. In this work, as well, the focus is limited to machining ruled surface, although much of our error metric analysis should be applicable to more general classes of surfaces. The ultimate goal is for the machined surface to match the desired surface as closely as possible. This suggests using a closest point error metric for the optimization methods. However, there are two difficulties with using a closest point error metric for optimizing the tool path: first, the metric is expensive to compute, as the simulated machined surface has a complex representation. Indeed, a closed form for the machined surface is rarely used, and approximations are computed and used instead. A second problem with using a closest point error metric with an optimization method is that as we optimize the tool path, we change the machined surface. Thus, not only does the distance to the closest point change, but the direction to the closest point also changes, making the optimization task more difficult. In this paper, we look at error metrics used when optimizing tool paths for the flank milling of ruled surfaces. Many error metrics have been developed by different researchers. Use of different error metrics leads to different solutions. This paper studies the error metrics proposed by other researchers and proposes a new error metric that we feel is most appropriate when using optimization methods for tool positioning.
2
Background
A ruled surface is generated between two guiding lines (T (u) and B(u)) by connecting corresponding points on the two lines with a straight line called the generating line of the ruled surface (Figure 1(a)). The machined surface is produced by the 5-axis movement of a cutting tool. It can be generated mathematically by identifying grazing curves at numerous points on the tool path and connecting the connective grazing curves into a wire mesh representing the swept surface (Figure 1(b)). The grazing curve is a function of the tool, cutting surface and the velocity of the tool axis. As a tool sweeps through the stock, the tool cuts the stock at numerous points on its surface. The points that lie towards the front of the tool are machined away as the tool leaves its current position. Only those points for which the moving direction is perpendicular to the normal of the surface of the cutting tool are left behind on the stock as a curve on the machined surface; this is the grazing curve. The grazing points are related to the motion of the tool and are easy to identify. These are the points where the velocity of the tool axis is perpendicular to the surface normal. So, the equation of those points that leave a mark on 2
(a)
(b)
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Fig. 1. (a) The desired ruled surface. (b) The machined surface, made by connecting grazing curves. (c) Ruled Surface and Grazing Surface superimposed.
the stock is P (u, v) = S(u, v) +
dS(u,v) du dS(u,v) du
× T (u, v)
R,
× T (u, v)
(1)
where S(u, v) is any point on tool axis, T (u, v) is the tool axis vector and R is the cutter radius corresponding to S(u, v). Different tool positions will generate different grazing curves, which can be joined together into a surface [1]. Consider the situation in Figure 2. In this figure, T (u) and B(u) are the guiding curves, (3) is a ruled line lying on the ruled surface, and P1 is a point on the ruled line. The surface machined by the moving tool is a collection of grazing curves (one of the grazing curves is superimposed on the tool in Figure 3). The match between the desired shape and the machined surface can be measured by the shortest distance between a point on the desired surface and some point C on the machined surface. The error at point P1 is the distance between the machined surface and the point P1 . The question addressed in this paper is in what direction should we measure this distance? While some times one will want to consider the distance between parametrically equivalent points, and at other times one will want the geometric shortest distance, in this paper we argue that for using optimization methods to find the tool orientation, the distance for the error metric should be along the ruled surface normal direction at P1 . In Figure 2, the surface normal passing through point P1 is (1) and the intersection point between the machined surface and normal (1) is the point C. The length of P1 C is our proposed error of the machined surface at point P1 . What ever form of closest point method is used to compute C, the point C has a complex mathematical relationship with the ruled line and the point P1 . This makes it difficult to calculate the exact error in a closed form manner. Generally, there are two reasons for this difficulty. First, the equation of the machined surface is not known, so the closed form calculation of shortest distance is not possible and an iterative numerical technique is required 3
Fig. 2. (1) surface normal at point P1 . (2) tool axis. (3) ruled line. (4) curve on surface
to calculate it. Second, the machined surface depends on the trajectory of the tool and any change in the trajectory effects the error calculation. Thus, using the error calculation to re-position the tool changes the error in the neighbourhood. For these reasons, focusing on the exact error at large computational cost may be futile and a simple error approximation may be better. Researchers who use error assessment to improve tool positioning have used a variety of different approximations. It is a goal of this paper to put the different error approximations into perspective and propose a new scheme for error approximation that is easy to implement and gives a better error assessment. This new method will be compared with established techniques. We begin with a review of previous techniques. Rehsteiner et al. [2] suggested a technique to position the cutting tool. In their method, the tool axis is first positioned parallel to the rule line and tangent to the middle point of the rule, then the tool is rotated about the surface normal (at middle point of rule) until it touches the boundary curves. The interferences at boundary curves are checked to try to get error-free surfaces. Undercutting will occur with this method. Stute et al. [3] developed another method to position the tool. In their work, the tool axis is positioned at two points. One point is on the surface normal through the top point of the rule line and the other point is on the surface normal through the bottom point of the same rule line. Both of these points are offset by the value of milling cutter radius. The maximum overcut is produced in the middle of the ruled surface. Liu [4] presented a double point offset (DPO) method to generate the cutter location. The idea behind this method is to offset two points at parametric values of 0.25 and 0.75 on the 4
rule line along the surface normal direction. The offset distance is the tool radius. The two offset points are then used to define the tool axis orientation. Bohez et al. [5] developed a different algorithm to locate the tool. They require that the tool to be tangent to a point P on the rule line while the tool axis is parallel to the same rule line. The point P is selected so that the angles between the surface normal at P and the normals on the two guiding curves are equal. The resulting overcut can be further diminished by moving the tool axis away from the surface along the surface normal at point P until the tool is tangent to the two guiding curves. The maximum deviation will be around the middle area of the surface. This error can be further reduced by using multiple machining passes. In Tsay and Her’s method [6], the cutting tool is initially tangent to the boundary curves at the same parametric value u, and then by varying the angle between the tool axis and the rule line as well as the distance between the tool axis and ruling, the tool position is optimized. Statistical analysis is used to minimize the error measured in the plane perpendicular to the rule line. Monies et al. (using a conical tool) and Redonnet et al. (using a cylindrical tool) [7–10] suggest that the tool can be positioned to be tangent to the ruled surface at three points by slightly changing the angle between the tool axis and the rule line, which can further reduce the surface error. Seven transcendental equations are developed in their paper to define each tool position. Bedi et al. [1] developed a strategy to roll a cylindrical cutting tool along two guiding rails. The tool remains tangent to the guiding rails and the contact points have the same parametric value. Each tool position is obtained by solving two transcendental equations simultaneously and the maximum overcut will lie at the middle curve of the surface. The error at each point of the surface can be predicted. Menzel et al. [11] modified this strategy to develop a triple tangent flank milling method. In this method, the tool is initialized by the Bedi et al. strategy, and then the contact points are moved along the rule line and finally the tool is twisted to generate an optimized tool position. The optimized tool position reduces the deviation. This technique can also be extended to a conical tool as well. All of these methods try to reduce the error between the machined surface and the desired ruled surface even though different error measurements are used. These techniques are described in the next section.
3
Previous Methods for Error Measurement
Before presenting our error metrics, we begin with a review of other methods for approximate error measurement. These previous methods broadly fall into two categories: the Radial Methods and the Parametric Methods. In addition, some work has been done using a closest point metric. 5
Fig. 3. Radial Error Calculation. (1) tool axis. (2) ruled line. (3) grazing curve. (4) curve on surface
3.1 The Radial Method
In the radial method, the tool is assumed to be stationary. The error is estimated by calculating the shortest distance from P1 (see Figure 3), which lies on the ruled line of interest, to the surface of the tool. In our discussion, we will consider both cylindrical and conical tools (most of the previous work has focused on cylindrical tools). For a cylindrical tool, we could estimate the error as |P10 − P | = R − |P − P1 |, where P10 is the point on the cylinder closest to P1 , P is the point on the tool axis closest to P1 , and R is the radius of the cylinder. Note that for a cylinder, P , P1 , and P10 are collinear. This is the method used as the error metric in several papers. However, except in some situations, this would be an under estimation of the error, since P10 will in general not lie on the machined surface (i.e., P10 is not on a grazing curve and either has or will be machined away by the moving tool). A more accurate radial error calculation proceeds by taking the grazing point A corresponding to P , finding D, the closest point on the rule line to A, and estimating the error as |DA|. In the case of a cylindrical tool, DA is normal to the cylinder surface. However, for a conical tool, DA is not perpendicular to the tool surface and the error must be calculated by multiplying |DA| with the cosine of the cone angle. Figure 4(a) shows an example of the error along a ruled line calculated using the Radial Method. This method of error measurement was used in Liu’s strategy, Bohez et al.’s strategy and Stute et al.’s strategy. However, in these strategies, the cutting tool is a cylinder and the error is obtained by subtracting the shortest distance (point on ruled line to cylinder tool axis) from the tool radius. In Bedi et al.’s strategy, the error is modified by a constant related to tool parameter because they used a conical cutter. 6
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Fig. 4. (a) Error Calculated Using Radial Method at u = 0.25; (b) Error Calculated Using Parametric Method at u = 0.25; (c) Error Calculated Using Motion Method at u = 0.25
Fig. 5. Parametric Error Calculation. (1) tool axis. (2) ruled line. (3) grazing curve. (4) curve on surface
3.2 The Parametric Method
In the Parametric Method, the error is computed between points on the ruled surface and points on the grazing curves that have the same parameter values. Thus, the error at a point P (ui , vi ) on the ruled surface is given by ε = P (ui , vj ) − G(ui , vj ),
where G(ui , vi ) is the corresponding grazing point. Figure 5 graphically shows the Parametric Method for calculation of error. Point P1 is a point on ruled line and point A is the corresponding grazing point. The length of P1 A is the approximate error ε. Figure 4(b) shows an example of the error along a ruled line calculated using the Parametric Method. 7
100
A
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Fig. 6. Schematic Illustration of Error Measurement Methods: (a) Radial (b) Parametric (c) First Order. Error between point A on the ruled surface and the machined surface is given by the dark line. Parametrically, A corresponds to point C on the tool/machined-surface.
3.3
Closest Point
Monies et al. also suggested a strategy similar to the Parametric Method. However, after using the Parametric Method as a starting point, the authors then search for the smallest error ε by looking over a small surface surrounding the point. This method, exhaustively searching for a proper grazing point to get the smallest error ε, is a computationally challenging job.
4
Methods for Error Measurement
The Radial Method for error measurement has the advantage of being simple to compute. However, the approximation seems poor, since as illustrated in Figure 3 and Figure 6(a), this method is not really calculating the error between the point on the ruled surface and the machined surface. The Parametric Method (Figure 6(b)) appears to answer this problem, and it does give an exact measurement of the error between the parameterized ruled surface and the parameterized machined surface. While the Parametric Method will clearly be useful for many purposes, it is a parametric error measure rather than a geometric error measure. In particular, given the sweeping motion of the tool, the Parametric Method will fail to give an accurate measurement of the shortest distance between ruled surface and the machined surface. The method of Monies et al. [9] addresses the question of computing a geometric error between the two surfaces. In many situations, computing the shortest distance between the ruled surface and the machined surface is the appropriate metric. However, there are two concerns with the closest point method: first, it is computationally expensive, and second, when the distance is used in optimization techniques, the direction of error at each point is not constant. The lack of a constant direction of error means that both the direction vector and the distance along the error direction should be considered in the optimization, making the problem more difficult. As an alternative, we propose using the distance from the point S(u, v) along the normal n(u, v) to the ruled surface. While similar to the Closest Point method of Monies et al., with 8
this method the direction of error is constant. However, computing the distance from the ruled surface to the machined surface along the surface normal has similar computational issues as the Monies et al. method. In particular, since the exact mathematical model of the machined surface is unknown and is often approximated with a piecewise linear surface, we would have to search a group of polygons to find the closest point. As a compromise between the accuracy of the error measurement and speed, we propose the Tangent Plane method, which finds the closest point between S(u, v) along the normal direction n(u, v) to a first order approximation to the machined surface (Figure 6(c)). We also suggest a second method, the Motion Method, which finds the closest point between a rule line and a moving grazing point, in essence making a linear approximation to both the ruled surface and to the machine surface, rather than computing the distance between a fixed point on the ruled surface to a planar approximation to the machined surface.
4.1 The Tangent Plane Method
In the tangent plane method, we construct the tangent plane to the grazing surface at G(ui , vi ) and find the intersection of this plane with the line from P (ui , vi ) in the direction of the normal to the ruled surface, n(ui , vi ). The tangent plane to the grazing surface at a grazing point is easily computed, since we know the direction of motion of the grazing point and we can compute the tangent to the grazing curve. The error plot for this method on our running example is discussed further at the end of Section 4.2.
4.2 The Motion Method
The Motion Method is our second method for approximating the error between a ruled surface and the machined surface. Starting with a point P1 on the ruled line, we compute the corresponding grazing point A. We then compute the shortest distance between the ruled line containing P1 and the line through the grazing point A in the direction motion of this grazing point. For this method, at a point A along the grazing curve, we would like to construct a line M in the direction of motion of the grazing point. If there is no rotation, then the direction of motion is given by dS(u, v)/du, which is readily seen from Equation 1 to be the cross product of the the tool axis with the tool radial line passing through A. However, the true direction of motion is unknown for 5-axis motion, since the 5-axis motion of a point A on the grazing curve differs from dS(u, v)/du by a vector that lies in the tangent plane of the tool at A. If the rotation motion is small compared to the translational motion, then this direction will give an acceptable approximation to the direction of motion of the grazing point. Regardless, 9
Fig. 7. Motion method error calculation. (1) tool axis T. (2) ruled line. (3) grazing curve. (4) curve on surface.
using dS(u, v)/du as the direction of M yields a line that is tangent to the surface swept by the moving tool. The error at this point on the grazing curve is estimated as the shortest distance between the ruled line of the surface and the line M . Figure 7 shows this method graphically. Parametrically, point A on the grazing curve corresponds to point P1 on the ruled surface. From point A, the shortest distance to the tool axis T is calculated as is r = |AB|, where B is the closest point on the tool axis. Then, the line M is constructed to pass through A in the direction N = T × (B − A). The shortest distance between line M and ruled line (2) is our approximation to the error. Note that this approximation computes an error from the line M to a point P2 on the ruled line, where P2 6= P1 ; Note also that Figure 7 is exaggerated – in practice, points P1 and P2 nearly coincide. Figure 4(c) shows error along the ruled line u = 0.25 calculated using the Motion Method. The error for the Tangent Plane method is similar, with the two methods producing results within 1% of one another. In our work, the two methods produced similar enough results that we only give the results for the Motion Method.
5
Discussion and Experimental Results
Several methods for error measurement are described in this paper. In this section, we wish to address several issues, including (1) Are the Tangent Plane Method and the Motion Methods good approximations of the Closest Point Method? (2) How do the various methods compare, both as methods for assessing error in machined surfaces and in being used for optimizing tool positions? To address these questions, we ran a variety of simulations and measured data off one machined part. 10
T0
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−5
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Table 1 Control points for guiding trails [mm]
Fig. 8. Machining Part
In this paper, we give the results for machining a particular ruled surface that is comprised of two B´ezier curves where the control points for guiding rails are given in Table 1. Further, we machined this part and measured the closest point error for the ruled line u = 0.9. We machined this part using both a non-optimization technique and an optimization technique [12], using the motion method to optimize the tool position. Figure 8 shows the test part using the optimization technique. Both the Motion Method and the Tangent Plane Method are approximations of the Closest Point Method. To see how good the approximation is, we compared the Motion Method to the Closet Point Method. Figure 9 shows the error plots of these two methods for one ruled line of our example ruled surface; further, we compare these errors to the error in our machined surface. The graph shows that all three error plots are similar. (Figure 10(b) shows a comparison of the Motion Method to the Radial Method and to the Parametric Method on the same data.) Figure 10 compares the error using different methods to measured data. The tool used was a conical tool and it was positioned with Bedi et al.’s strategy and its extension. The cone parameters are R1 = 19.252mm, R2 = 3.175mm, h1 = 3mm, h0 = 60mm. Measuring experimental error introduces additional difficulties into the error metric question. Errors such as the part fixture error, machine tolerances, etc., must be considered and filtered in the experimental results. These issues aside, a correspondence between points on the machined surface and points on the mathematical surface must be established. Radial error metrics no longer make sense (since the tool is no longer present), and to use a Parametric error metric would require appropriately parameterizing the machined surface which is not 11
0 Calculated error (the closest point method) −0.05 Calculated error (the motion method)
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Fig. 9. Comparison of Motion Method and Closest Point (u = 0.9). u=0.9 u=0.25
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Fig. 10. (a) Errors Comparing (u=0.25); (b) Error Comparing (u=0.9)(c) Error Comparing (u=0.9) (after optimization); (d) Error Comparing (u=0.9) (after optimization)
12
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Fig. 11. Optimization using different metrics.
feasible. Instead, we used a Closest Point error metric, which is typically the form of error metric used for experimental data. (A Tangent Plane error metric could also be used, since the ruled surface is known, but the motivating factors for using the Tangent Plane error metric for tool position optimization are not relevant when measuring experimental error.) Figure 10(a) and Figure 10(b) displays the results before optimization and Figure 10(b) also contains the experimental results. These graphs show that the resulting error from the Motion Method is close to the experimental result. The Radial Method gets the similar results, but the resulting error has some deviation. The Parametric Method can also reflect the error variation, but huge deviations may exist in its results. Figure 10(c) and Figure 10(d) show the same tool position as Figure 10(b) after optimization. In Figure 10(c), we see that the Parametric Method yields a large error estimate compared to the other methods. Figure 10(d) is a view of the error plots at higher resolution. Again, we see that the experimental result is almost the same as the error given by the Motion Method. As a final test, we simulated using the three methods (Radial, Parametric, and Motion) to optimize toolpaths, and compared the resulting errors. In all three cases, although different error metrics were used for the optimization, the final error measurements were computed using the Motion Method. Figure 11 shows the results of these tests. In all three cases, we see that the surface obtained using the Parametric Method has the highest error. We also see that the Radial Method gives results similar to the Motion Method; that is a result of, in our example surface, the Radial and Motion Methods having similar error results for the sample tool positions (i.e., see Figures 10(a)(b)). After optimization, note that the Radial Method fails to give an accurate estimation to the measured error as shown in Figure 10(d).
6
Conclusion
In this paper, three different error measurement methods are studied and compared. Two new strategies, the Tangent Plane Method and the Motion Method, were proposed, both of which 13
closely reflect the surface error. Both new methods essentially make linear approximations in two dimensions; the difference is in whether both dimensions are on the machined surface (as in the Tangent Plane Method) or if only one dimension is on the machined surface with the other being on the ruled surface. Both methods are good approximations to the Closest Point Method, and use of either one should produce similar results. The calculation procedure of both new methods is simple and robust. The Radial Method sometimes gets similar results to the Motion Method, but the calculated error has some deviations especially when the tool is twisting. The Parametric Method can reflect the error changing in general conditions, but has large deviations if the tool is twisted during the optimization. While our work in this paper suggests that the Tangent Plane or Motion Method is most appropriate for optimizing tool paths, note that there are situations in which it is likely not the best method. In particular, in the example in this paper, the desired surface is smooth. If we were to machine a surface with a ridge or corner (a C 1 discontinuity), then use of the Motion Method would potentially smooth the ridge, and thus some other method is likely needed to accurately machine the surface in such regions.
References
[1] S. Bedi, S. Mann, and C. Menzel. Flank milling with flat end cutters. Computer-Aided Design, 35:293–300, 2003. [2] F. Rehsteiner and H.J. Renker. Collision-free five-axis milling of twisted ruled surfaces. CIRP ANNALS, 42(1):457–461, 1993. [3] G. Stute, A. Storr, and W. Sielaff. Nc programming of ruled surface for five axis machining. Ann. CIRP, 28(1):267–271, 1979. [4] X. Liu. Five-axis NC cylindrical milling of sculptured surfaces. Computer-Aided Design, 27(12):887–894, 1995. [5] E.L.J. Bohez, S.D.R. Senadhera, K. Pole, J.R. Duflou, and T. Tar. A geometric modelling and five-axis machining algorithm for centrifugal impellers. Journal of Manufacturing Systems, 16(6):422–463, 1997. [6] D.M. Tsay and M.J. Her. Accurate 5-axis machining of twisted ruled surfaces. Journal of Manufacturing Science and Engineering, 123:734–738, 2001. [7] J.-M. Redonnet, W. Rubio, and G. Dessein. Side milling of ruled surfaces: Optimum positioning of the milling cutter and calculation of interference. Advanced Manufacturing Technology, 14(7):459–465, 1998. [8] F Monies, J-M Redonnet, W Rubio, and P Lagarrigue. Improved position of a conical mill for machining ruled surfaces: application to turbine blades. Journal of Engineering Manufacture, 214(B):625–634, 2000.
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[9] F Monies, W Rubio, J-M Redonnet, and P Lagarrigue. Comparative study of interference caused by different position settings of a conical milling cutter on a ruled surface. Proceedings of the Institution of Mechanical Engineers-B- JNL of Engineering Manufacture, 215(9):1305, 2001. [10] F Monies, J.N. Felices, W Rubio, J-M Redonnet, and P Lagarrigue. Five-axis nc milling of ruled surface: optimal geometry of a conical tool. International Journal of Production Research, 40(12):2901–2922, 2002. [11] C. Menzel, S. Bedi, and S. Mann. Triple tangent flank milling of ruled surface. Computer-Aided Design, 36, 2004. [12] Chenggang Li. Flank milling of ruled surfaces with conical tools. Master’s thesis, University of Waterloo, 2004.
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