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Arc-intersect method for 31212 -axis tool paths on a 5-axis machine
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International Journal of Machine Tools & Manufacture 47 (2007) 182–190 www.elsevier.com/locate/ijmactool
Arc-intersect method for 31212-axis tool paths on a 5-axis machine Paul J. Gray, Fathy Ismail, Sanjeev Bedi Department of Mechanical Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 20 October 2005; received in revised form 25 January 2006; accepted 30 January 2006 Available online 17 April 2006
Abstract An inherent problem with simultaneous 5-axis machining is that it often suffers from dramatic reductions in feed rate when the tool axis is in the vicinity of the singularity point of the machine; during large orientation changes over small distances; during rotary axes reversals and from interpolation of the tool axis vector. 31212-axis machining offers an alternative strategy that can be used to overcome these problems and still maintain some of the salient features of 5-axis machining to improve machining times over 3-axis ballnose machining. In 31212-axis machining, during cutting the machine moves only its three linear axes while the two rotary axes are locked, resulting in a fixed tool orientation. Locking the rotary axes generates fewer fluctuations in the feed rate than simultaneous 5-axis machining and results in a more consistent surface finish with lower variations in cutting force and torque. A new tool positioning strategy called the Arc-Intersect Method (AIM), which can also be applied to simultaneous 5-axis machining, is presented here for 31212axis machining on simultaneous 5-axis machines using toroidal or flat endmills. A cutting test was performed and the part was measured with a CMM to check for accuracy and to measure the cusp heights. Machining times were compared to 5- and 3-axis tool paths and cutting torque measurements were compared between 31212- and 5-axis machining. r 2006 Elsevier Ltd. All rights reserved. Keywords: 31212-axis machining; 5-axis machining; Triangulated surfaces; Cutting torque; Computer graphics
1. Introduction In simultaneous 5-axis surface machining, the tool axis can be dynamically oriented to generate a better match between the cutting geometry of a toroidal or flat endmill to the workpiece. Doing so can increase the machined strip width of each tool pass over 3-axis machining and thus, reduce the total number of tool passes to machine the surface leading to reduced machining times. A well-known approach for 5-axis tool positioning is to match the projected effective radius of the cutting tool to the curvature of the surface at the cutter contact point, ccp [1–3]. However, these methods require iterative gougechecking and correction strategies. Redonnet et al. [4] and Lauwers et al. [5] are examples. To eliminate the need for iterative gouge checking and correction algorithms, some 5-axis tool positioning strategies attempt to match the tool’s cutting geometry to the Corresponding author. Tel.: +1 519 888 4567; fax: +1 519 888 6197.
E-mail address: [email protected] (S. Bedi). 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.01.023
surface around the ccp such as the work by Warkentin et al. [6], and Gray et al. [7] who also used computer graphics to compute tool positions from triangulated surfaces [8]. Generating tool paths from triangulated data is not new; the works in [9–12] present 3-axis machining examples. Li and Jerard [13] developed a strategy for 5-axis Sturz milling using triangular data. Balasubramaniam et al. [14,15] developed methods for 5-axis tool positioning that account for accessibility of the tool using visibility maps of the triangulated data though the tool positions were not optimised for the highest material removal rate. Although 5-axis machining has been shown to reduce machining times over 3-axis [16], it is still plagued with a number of problems, which makes it difficult to gain wider acceptance in industry. In simultaneous 5-axis machining, it is often observed that the feed rate of the tool with respect to the workpiece fluctuates because the rotary axes servo drive systems limit the overall achievable feed rate. This can occur when the tool axis is in the vicinity of the machine’s singularity point, during large orientation changes over short distances along the tool path, and
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when the rotary axes must reverse direction. These fluctuations and reductions in feed rate ultimately lead to longer than expected machining times and result in variations of surface finish over the part. Although 31212-axis machining was originally proposed as a cheaper alternative to simultaneous 5-axis machining of parts via the use of manual indexable rotary-tilt tables on 3-axis machines [17–19], 31212-axis machining can also be used to overcome the problems associated with reduced and fluctuating feed rates commonly encountered in simultaneous 5-axis machining. In 31212-axis machining, the rotary axes are locked while the tool is engaged with the workpiece, thus, the machine only interpolates and moves its linear axes. The result is that cutting forces and surface finish are more consistent than in simultaneous 5-axis machining. To date, there exists no strategy for determining the optimal tool orientation to machine the maximum strip width for 31212-axis machining of complex surfaces using toroidal or flat endmills. Thus, the goal in this paper is to develop a tool positioning strategy that computes an optimal tool orientation for a set of ccps along a pass. Since the work in this paper uses concepts developed in earlier work presented by the present authors for their 5-axis Arc-Intersect Method (AIM) machining strategy, only a brief overview of the method is presented next. 2. Five-axis AIM Similar to the Principal Axis Method [3], the 5-axis AIM, developed by Gray et al. [20] is based on the fact that the widest machined strip width is cut when the tool is tilted along the feed direction (Fig. 1). It can be seen that when the tool tangentially contacts the ccp and the tool axis is tilted along the feed direction, the tool axis will lie in the plane containing the surface normal at the ccp and the feed direction vector. This plane is referred to as the tilting plane indicated in Fig. 1. The concept of AIM is to tilt the tool such that the tool axis remains in the tilting plane and the forward bottom point of the tool remains in tangential contact with the ccp until a second contact point on the surface is touched. This will give the smallest tilt angle and thus, the largest effective radius resulting in the widest machined strip width for the given feed direction at the ccp without gouging the surface. However, instead of rotating
Fig. 1. Five-axis AIM concept and method.
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the tool to find the limiting tilt angle that will touch the second contact point, the surface points beneath the tool are rotated to find the limiting tilt angle. 3. Five-axis arc-intersect algorithm For a given feed direction, a tool position is computed for each ccp along the tool path. With the tool axis restricted to lie in the tilting plane, all tilt angles are measured around the cross vector. The idea is to find the minimum tilt angle of the tool axis about the cross vector at which the tool will contact another point on the surface and maintain its contact with the ccp without gouging the surface as illustrated in Fig. 1. The area beneath the tool is discretized into a set of points referred to as the shadow grid points. Gray et al. [8] presented a method to compute surface points and normals of triangulated surfaces using the computer’s graphics card. The same method is used here for its simplicity. The tolerance for the pixel density to render the scene is described in Appendix A, which is similar to that described by Austin et al. [21] for gouge checking with respect to cloud point data sets. The scene is viewed along the surface normal of the ccp and each rendered pixel is used as a shadow grid point. Fig. 2 shows the viewing volume set-up for rendering the shadow grid points. Each shadow grid point is rotated about the cross vector to generate an arc. If the intersection of the arc to the tool centre plane (the plane containing the cross vector and the normalized feed direction vector) lies within the circular shadow of the tool, then the tool can potentially contact the shadow grid point when it is rotated about the cross vector (Fig. 2). Arcs may intersect either the toroidal geometry of the tool (the edge radius) or the bottomcutting plane of the tool. For the shadow grid points whose arcs intersect the bottom-cutting plane within the radius given by Eq. (1), the intersection to the bottom-cutting plane is computed using Eq. (2). All other arcs will intersect the toroidal geometry of the tool and their intersections are computed using a bisection search. The intersection point of an arc to the tool is used to determine the tilt angle (y)
Fig. 2. Viewing volume for tool position calculation (reproduced from [20]).
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Fig. 3. Arc Intersection calculation.
through which the tool needs to be rotated about the cross vector to contact a particular shadow grid point. The tilt angle, y, is simply the difference between the arc intersection angle, f, and the arc angle, g, between the shadow grid point and the tool centre plane given by Eq. (2) from the geometry of the tool (Fig. 3). The tilt angle, y, is computed using either Eq. (3) or (4) depending on whether the shadow grid point lies above or below the tool centre plane. Bottom_Radius ¼ Rmajor
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2minor ðRminor Centre_ClearanceÞ2 ,
(1) where Bottom_Radius is the radius of the bottom cutting plane of the cutting tool, Rminor the minor radius of the torus, Rmajor the major radius of the torus, Centre_Clearance the perpendicular distance from the bottom point of the tool to the bottom cutting plane. 1 Rminor Centre_Clearance g ¼ sin . (2) Arc_Radius If the shadow grid point is below the tool centre plane then the tilt angle is y ¼ f g.
(3)
If the shadow grid point is above the tool centre plane then the tilt angle is y ¼ ðf þ gÞ,
(4)
where the tool tilt angle, y, is the angle between the surface normal and the tool axis in the tilting plane. A forward tilt of the tool matches concave surface regions and corresponds to a negative tilt angle with this convention. Finally, the largest tilt angle of all the shadow grid points is selected as the tilt angle for the tool position. Having described the 5-axis AIM algorithm, the 31212-axis AIM can now be presented. The next section describes the adaptation of the 5-axis AIM into 31212-axis machining strategy by optimising the tool orientation for many ccps on the surface instead of for only a single ccp.
direction. This can be seen with the help of Fig. 1 by examining the effective radius of the tool when viewing the tool along the tilt direction. All other feed directions will reduce the machined strip width, which can be seen by observing that the projected effective radius of the tool moving perpendicular to the tilting plane is simply the minor radius of the torus. Therefore, the tool is always tilted along the feed direction to cut the widest strip width. The first step in the process is to compute the ccps and their corresponding surface normals for the selected region, from which the average surface normal in this region is computed. Each surface normal is projected onto the projection plane defined by the average surface normal and the feed direction of the tool as shown in Fig. 4. For each ccp, a tool position is computed using the 5-axis AIM with the same viewing volume set-up as in Fig. 2 except the view direction is now along the projected surface normal instead of the actual surface normal. The optimal orientation for the tool axis in the projection plane for each ccp is computed. The tool orientation for the region is then selected as the orientation with the largest tilt angle with respect to the feed direction vector computed from all the ccps in the region. If the tool axis is forced to lie inside the projection plane and the front circular arc of the tool cross-section, (referred to as forward pseudo insert) is forced to touch the ccp, then tangency at the ccp may be violated in places along the tool path where the actual surface normal deviates considerably from the projection plane in the cross feed direction (Fig. 5). These shadow grid points are detected in the AIM bisection search as points that, regardless of the arc intersect angle, will always lie inside the torus. These points are stored but not used in the tool tilt angle calculation. After the tilt for the tool pass is computed, the distance of these stored points to the tilted torus along the final tool axis direction of the region are computed for each tool position. The maximum gouge distance is the value the tool is lifted by along the tool axis. Essentially this means that the tool has been moved to contact an alternate ccp by lifting the tool along the tool axis. This strategy will ensure that the tool will not gouge the surface provided the density
4. The 31212-axis AIM Since the tool orientation is locked during cutting in 31212axis machining, the feed direction of the tool can drastically affect the resulting machined strip width. The widest machined strip width will be generated at the front point of the tool when the tool moves along the tilt
Fig. 4. Projection of surface normals.
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Fig. 6. Machined workpiece.
Fig. 5. Inside shadow grid points.
of shadow grid points is sufficiently high. Appendix A presents the calculations for determining the density of shadow grid points.
Table 1 Triangulation parameters Max. angle between surface normals at triangle vertices Max. aspect ratio Min. edge length Max. edge length Max. dist from triangle edge to surface
0.51 1 0.05 mm 1.5 mm 0.01 mm
5. Machining example An 11 surface patch section of a stamping die for an automobile part was used for a machining example. The workpiece shown in Fig. 6 was slightly modified from the original part to protect the source’s proprietary data for publishing purposes. Triangulation of the surfaces for the AIM was performed with Rhinoceros NURBS modelling software using the parameters listed in Table 1. A comparison between the 31212- and 5-axis AIM for this workpiece is presented in Subsection 5.4. A 2-insert toroidal tool with and outer radius of 12.7 mm and an insert radius of 6 mm was used. The insert radius is the minor radius of the tool’s toroidal surface. The speed was 6700 rpm and the feed rate was set at 2000 mm/min. The part was machined on a Deckel Maho 80P Hi-Dyn tilt/ rotary table 5-axis machine. 5.1. Tool path generation Parallel tool passes along the Y-axis of the workpiece with a constant forward step distance of 1.27 mm were used. Since the CNC machine used in this test is a continuously variable 5-axis machine, the tool orientation was optimised and locked for each tool pass. In this way, each tool pass was considered as a separate region with a fixed surface normal. The cross-feed distance calculations were based on the ‘‘Rolling Ball Method’’, RBM, for 5-axis tool positioning developed by Gray et al. [8] in which the surface normal curvature at each ccp is estimated for a triangulated surface along the cross-feed direction. The estimated curvature was compared to the projected effective radius of the tool at the ccp for each tool position to compute a cross-feed distance. The smallest cross-feed distance computed for the entire tool pass was used as the final cross-feed distance for the pass. The maximum cusp height value was set at 0.1 mm and the maximum allowable cross-feed distance was equal to the outer radius of the tool at 12.7 mm.
5.2. Viewing parameters The ccps were computed using a 201 square pixel window with a width and height of 1 mm. The surface normals were also computed with a 201 square pixel window but with a width and height of 3 mm. Only points within a 3 mm radius of the centre of the view were used for the surface normal calculation. A view depth of 200 mm with a 24-bit depth buffer was used for both windows. The tolerance for the arc intersection point bisection search was set at 0.001 mm and the tool positions were computed using a 315 square pixel window with a width and height of 31.4 mm (equal to the tool outer diameter plus the insert radius) and a view depth of 100 mm. This setting gave a resolution of 0.1 mm/pixel and a depth resolution of 5.96 106 mm per depth plane. The maximum possible penetration of the tool between pixels was 4.14 104 mm according to the method described in Appendix A. 5.3. Machining results Fig. 6 shows the machined part and Fig. 7 shows the coordinate measuring machine, CMM, measurements plotted against the CAD data for various cross-sections. The maximum measured cusp height was 0.128 mm. 5.4. Comparison of methods This subsection presents a comparison of machining times between 3-axis ballnose machining, the 31212- and the 5-axis AIM for the example workpiece. The 5-axis AIM machining results were presented by Gray et al. [20] and are used here for comparison purposes. Because each process was not fully optimised for tool dimensions, feed direction, and rapid traverse parameters between tool passes, and since different part geometries, machines, cutting
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Fig. 7. CMM scans of workpiece.
parameters and cutting tools can affect the performance of each method, a complete comprehensive comparison is not possible. The study can, however, be used to analyse hindrances to machining times for the different processes. The number of passes for a 3-axis ballnose tool path was computed by dividing the width of the part by the crossfeed distance, which was computed using Eq. (5) for a cusp height of 0.1 mm. The total machining time was then computed by multiplying the measured time per pass by the number of passes. Table 2 is a summary of the number of tool passes required for each method. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cross_Feed ¼ 2 R2ball ðRball Cusp_heightÞ2 . (5) The 5-axis tool passes were generated with the method outlined in Section 3 using the cross-feed distance calculation presented in Subsection 5.1 with a maximum allowable cross-feed distance set equal to the radius of the tool. Also, the same strategy of parallel single-direction tool passes along the Y-axis of the workpiece was used as in the 31212-axis AIM. Larger tools will generally be more useful for the 5-axis AIM than for the 31212-axis AIM, since, in 31212axis, the use of larger tools can generate more inside points which would require larger lift distances along the tool axis. Thus, for the 5-axis test, a toroidal tool with an outer radius of 19.05 mm and an insert radius of 6.35 mm was used. The 5-axis AIM tool path was able to make use of the maximum allowable cross-feed distance with the selected tool, indicating that, perhaps, an even larger tool could have been more optimal. In comparison, the 31212-axis AIM did not reach the maximum cross-feed distance anywhere along the part, which suggests that a smaller tool may have been more optimal. The different machining times are presented in Table 3. For all entries in the table, the spindle
Table 2 Number of tool passes for programmed maximum cusp height of 0.1 mm
Number of passes
3-axis ballnose R ¼ 12:7 mm
31212-axis AIM Router ¼ 12:7 mm; Rinsert ¼ 6 mm
5-axis AIM Router ¼ 19:05 mm; Rinsert ¼ 6:35 mm
52
36
16
Table 3 Machining times for different methods and different feed rates Machining times for programmed maximum cusp height of 0.1 mm (min) Feed rate (mm/min)
3-axis ballnose zig–zag path R ¼ 12:7 mm
31212-axis AIM single direction, parallel passes Router ¼ 12:7 mm; Rinsert ¼ 6 mm
5-axis AIM single direction, parallel passes Router ¼ 19:05 mm; Rinsert ¼ 6:35 mm
1000 2000
9.88 5.18
8.12 4.82
4.8 4.12
speed was set to achieve 0.15 mm/tooth chip load for the given feed rates. For the 5-axis test, doubling the programmed feed rate led to a reduction in machining time of only 14%. The reason for the small decrease in machining time is that, in the areas where large changes in the C-axis were necessary, the feed rate dropped dramatically below the programmed feed rate due the C-axis servo drive system limits. Since the machine is only capable of 9.2 rpm in the C-axis and 7.2 rpm in the A-axis, there were two regions along the tool
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pass where the machine could not maintain the programmed feed rate. Thus, doubling the programmed feed rate did not halve the machining time. Another reason for limited gains with the higher programmed feed rate is that a larger percentage of time is spent in rapid traverse between passes with the single direction tool passes over the ballnose zig–zag path, which is not affected by the programmed feed rate. It can be seen that the 31212-axis approach can reduce actual cutting times by reducing the number of tool passes over 3-axis ballnose machining for the same radius tool; however, for this small part, the gains were limited. Considering a feed rate of 2000 mm/min, the average time per tool pass in the 31212-axis test was a little over 8 s, whereas with the ballnose path it was about 6 s. This means that approximately 2 s were lost per pass in rapid traverse and for the orientation change in the 31212-axis approach. When the feed rate was doubled from 1000 to 2000 mm/ min, the 31212-axis AIM reduction in time was 41%, whereas the ballnose path reduction was 48%. This means that by doubling the feed rate, a larger reduction in time was observed for the ballnose tool path, which was expected because the amount of time spent out of cut is smallest for the 3-axis tool path, which is mainly due to its zig–zag path compared to the single direction parallel passes used in the 31212- and 5-axis paths. Conceivably, the 3- and the 31212-axis methods could machine the part faster than the 5-axis method, provided the required spindle speeds could be achieved and tool wear is of no concern. A similar argument can be made that, on a different machine with faster rotary axes, the 5-axis machining could be made faster. To make a fair comparison, the tool path feed directions, tool geometry and tool materials would have to be optimised and compared for different parts and machines. Thus, extrapolation of the present results to other workpieces, materials and machines for comparison purposes is not simple and should be done with caution on a case-by-case basis. 6. Cutting torque comparison To quantify the fluctuations in feed rate during simultaneous 5-axis machining, a second machining test was conducted. Cutting torque measurements were collected for 31212- and 5-axis machining and compared. 6.1. Experimental set-up In this experiment, a Kistler 9123C spindle-mounted rotating force dynamometer was used to measure the cutting forces. Since this experiment is only concerned with demonstrating that the feed rate fluctuates in 5-axis machining, a single measure, in this case cutting torque, is sufficient. Furthermore, measurements were only made for a single tool pass over the surface for the part shown in Fig. 6. The
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tool pass was selected to show the reduction in feed rate during large orientation changes and to avoid the kinematic singularity of the machine. Tool passes were generated at x ¼ 194 mm using the 5-axis AIM and the 31212axis AIM for comparison. A 2-insert toroidal tool with an outer radius of 12.7 mm and an insert radius of 6 mm was used for both cutting tests. The natural frequency of the tool and rotating dynamometer assembly was measured at 360 Hz. The unbalance was found to be 10.8 g mm at 10 000 rpm using the G2.5 class balance quality grade. The test was conducted at a tooth passing frequency of 13 the measured natural frequency, which ensures the measured data to have no more than 10% error according to the dynamometer manufacturer. For the 2-insert cutter, the spindle speed was 3600 rpm. The sampling rate was set at 7200 Hz; giving 60 samples per tooth period, and the feed rate was set at 1080 mm/min, for a target chip load of 0.15 mm/ tooth. The stock material was 6061-T6 aluminum. To prepare the stock material, five semi-finish tool passes were generated using the 5-axis AIM between x ¼ 188 and 204 mm, with a constant cross feed distance of 3 mm. Then a single pass was machined at x ¼ 181:8 mm, with a 1 mm depth of cut. The position of this single tool pass was selected to ensure that the tool’s forward pseudo insert cut the full depth of material at all tool positions in both the 31212- and 5-axis cutting tests. Thus, the maximum chip thickness was cut throughout each tool pass in the torque measuring tests. The axial depth of cut for the test passes was 1 mm. 6.2. Machining results Fig. 8(A) shows the measured torque for both cutting tests. The elapsed cutting times were approximately 9.2 s for the 31212-axis pass and 15.2 s for the 5-axis, respectively, as measured from the torque plots in Fig. 8(A). The 31212axis tool path was approximately 165 mm long (considering the tool will still cut after the tool’s forward point clears the part edge), which, at a feed rate of 1080 mm/min would yield a machining time of 9.17 s. This estimate agrees well with the machining time measured from the torque plots. The middle portion of the 5-axis measurement indicates a reduced chip load in the high curvature areas of the tool path, where large orientation changes occur over short distances. Fig. 8(B) shows the torque at 25% of the machining time for both tests, which approximately corresponds to the point where the surface curvature changes significantly along the tool path. The 31212-axis path remains relatively constant, whereas the force variation associated with the 5-axis path indicates that the tool is essentially cutting with only one tooth. As the radial run-out of the cutting tool was measured at 0.05 mm, it can be deduced that the feed rate for the 5-axis path dropped at that point to around 1/3 or less of the programmed feed rate. In other words, the
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Fig. 8. Measured torque plots: (A) complete plot, (B) 25% cutting time, (C) 50% cutting time, and (D) 75% cutting time.
chip load at that point dropped from the programmed 0.15 mm/ tooth, to a chip load close to the value of the tool runout at 0.05. At the 50% time point shown in Fig. 8(C), the 31212-axis path had moved well beyond the high curvature region, whereas the 5-axis path was still cutting that area. However, the peak torque here for the 5-axis is higher than that at the 25% point, which indicates the feed rate has not dropped as much. Fig. 8(D) shows the torque towards the end of the tool passes, where the tool orientation does not change much in the 5-axis path, resulting in a relatively constant feed rate. The measured machined strip width ranged from 15.9 to 16.8 mm in 5-axis and 8.8 to 11 mm in the 31212-axis cut. Since the machined strip width is smaller in the 31212-axis path, the torque peaks are narrower than the 5-axis path.
7. Discussions From experimental work presented here, it can be seen that simultaneous 5-axis machining suffers greatly when the rotary axes servo drive systems are saturated. This characteristic cannot easily be avoided, except perhaps by selecting the feed direction to minimize orientation changes, by smoothing out the tool orientation changes
along the path, or by changing the orientation of the part set-up on the machine. The 31212-axis tool path resulted in a constant feed rate and more consistent cutting torque, which is useful when machining with slender tools or when machining flexible parts. Controlling the feed rate could be a strong argument in favour of adopting the 31212-axis strategy. Machining times were reduced with the AIM for 31212-axis machining over tool paths for the same sized ballnose tool in 3-axis machining. However, the single direction parallel tool passes used in the 31212-axis method introduced an additional time penalty that did not exist in the zig–zag ballnose 3-axis path. Therefore, the 31212-axis AIM should be used in conjunction with a partitioning scheme like that proposed by Chen et al. [19] to further optimise the process. Selecting the feed direction such that the variation of the surface normal vectors along the direction perpendicular to the projection plane is minimized, can help minimize the lifting distances in the 31212-axis AIM. The size of the cutting tool will also have an effect; larger tools will more likely require higher lifts because the tangency at the ccp will be violated over a larger area. Tool dimensions will also affect the cross-feed distance. Optimisation of the tool dimensions could certainly improve the strategy; however, no simple method exists as of yet.
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Although the 31212-axis AIM can optimise the tool orientation for a given set of ccps, it assumes that the ccps are known beforehand. If adaptive side and forward step algorithms are to be applied when machining a region on the part, the location of the ccps will not be known until the tool orientation is selected. In this case, the 31212-axis AIM as it is presented cannot be applied. A solution to this problem would be to optimise the tool orientation for a sample of points in the selected region. The tool positions could then be computed by projecting the tool onto the surface; an easy task when using the computer’s graphics hardware for tool positioning. It would be beneficial to adapt the 31212-axis AIM for use with indexable rotary axes machines, since they are less expensive and are generally stiffer than continuously variable 5-axis machines. A simple approach would be to compute the orientations for the partitioned regions with the current method and then find the indexing angles that give the closest tool orientation. The tool would then be projected onto the surface at each ccp to give the final tool positions. Although the comparisons conducted here showed that the 31212-axis AIM was slower than 5-axis method, Sections 5.3 and 6 clearly show how 5-axis machining can suffer from saturation of the rotary axes servo drive systems leading to fluctuations and reductions in the feed rate. Saturation of the rotary axes servo drive systems results in longer than expected machining times and variations in surface finish and torque loads on the tool. 31212-axis machining, on the other hand, does not suffer from these effects and is a more promising candidate for high speed machining since the loads remain relatively constant. Finally, although the time gains presented here for 31212axis over 3-axis machining are relatively small for this small part, it has been demonstrated at an industrial plant that the 31212-axis AIM was 1.5–3 times faster than 3-axis high speed machining for a larger automotive panel stamping die. Unfortunately the details of the experiment are confidential. Acknowledgements This research was funded by the Natural Sciences and Engineering Research Council of Canada, by Ontario Innovation Trust, and the Canadian Foundation for Innovation. Appendix A. Pixel spacing and depth buffer resolution Using the concept developed by Austin et al. [21] for computing the maximum penetration of a cutting tool between discrete points representing a surface, the horizontal and vertical pixel resolution for the viewing window can be set according to the required machining accuracy. The viewing volume width for tool positioning can be set to the width of the cutting tool because the tool axis is constrained to lie in a fixed plane in the AIM (Fig. 1). The
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Fig. 9. Pixel spacing (reproduced from [20]).
number of pixels for the width and height of the viewing window can be set by determining the maximum possible penetration of the cutting tool between rendered pixels. This concept was used by Austin et al. [21] for gouge checking of tool positions with respect to discrete surface models. Recognizing that the minimum radius of curvature of a toroidal cutting tool is its minor radius, the maximum penetration of the tool between rendered pixels can be conservatively estimated by determining the penetration distance of a circle with a radius equal to the torus minor radius between the diagonal distance between pixels. This is illustrated in Fig. 9, and expressed below in Eq. (6). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ¼ r r2 d 2 . (6) 4 Since a 24-bit depth buffer was used for the work in this paper, setting the viewing volume depth is not as critical as setting the viewing volume width and height because the depth resolution is generally much higher than the horizontal and vertical pixel resolution of the viewing window. References [1] S. Bedi, S. Gravelle, Y. Chen, Principal curvature alignment technique for machining complex surfaces, Journal of Manufacturing Science and Engineering 119 (1997) 756–765. [2] S. Mullins, C. Jensen, D. Anderson, Scallop elimination based on precise 5-axis tool placement, orientation, and step-over calculations, Advances in Design Automation 2 (1993) 535–544. [3] N. Rao, F. Ismail, S. Bedi, Tool path planning for five-axis machining using the principal axis method, International Journal of Machine Tools and Manufacture 37 (7) (1997) 1025–1040. [4] J.-M. Redonnet, W. Rubio, F. Monies, G. Dessin, Optimising tool positioning for end-mill machining of free-form surfaces on 5-axis machines for both semi-finishing and finishing, International Journal of Advanced Manufacturing Technology 16 (6) (2000) 383–391. [5] B. Lauwers, P. Dejonghe, J. Kruth, Optimal and collision free tool posture in five-axis machining through the tight integration of tool path generation and machine simulation, Computer-Aided Design 35 (5) (2003) 421–432. [6] A. Warkentin, F. Ismail, S. Bedi, Multi-point tool positioning strategy for 5-axis machining of sculptured surfaces, Computer Aided Geometric Design 17 (1) (2000) 83–100. [7] P. Gray, S. Bedi, F. Ismail, Rolling ball method for 5-axis surface machining, Computer-Aided Design 35 (4) (2003) 347–357. [8] P. Gray, F. Ismail, S. Bedi, Graphics-assisted rolling ball method, Computer-Aided Design 36 (7) (2004) 653–663.
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[9] C.-S. Jun, D.-S. Kim, S. Park, A new curve-based approach to polyhedral machining, Computer-Aided Design 34 (5) (2002) 379–389. [10] T. Saito, T. Takahashi, NC machining with G-buffer method, SIGGRAPH Computer Graphics 25 (4) (1991) 207–216. [11] D. Veeramani, Y.-S. Gau, Issues in patch-by-patch machining of compound sculptured surfaces, IIE Transactions 30 (4) (1998) 341–355. [12] J. Hwang, T.-C. Chang, Three-axis machining of compound surfaces using flat and filleted endmills, Computer-Aided Design 30 (8) (1998) 641–647. [13] S. Li, R. Jerard, 5-Axis machining of sculptured surfaces with a flatend cutter, Computer-Aided Design 26 (3) (1994) 165–178. [14] M. Balasubramaniam, P. Laximprasad, S. Sarma, Z. Shaikh, Generating 5-axis NC roughing paths directly from a tessellated representation, Computer-Aided Design 32 (4) (2000) 261–277. [15] M. Balasubramaniam, S. Sarma, K. Marciniak, Collision-free finishing toolpaths from visibility data, Computer-Aided Design 35 (2003) 359–374.
[16] P. Gray, S. Bedi, F. Ismail, N. Rao, G. Morphy, Comparison of 5axis and 3-axis finish machining of hydroforming die inserts, International Journal of Advanced Manufacturing Technology 17 (8) (2001) 562–569. [17] S.-H. Suh, J.-J. Lee, Five-axis part machining with three-axis CNC machine and indexing table, Journal of Manufacturing Science and Engineering 120 (1998) 120–128. [18] S.-H. Suh, J.-J. Lee, S.-K. Kim, Multiaxis machining with additionalaxis NC system: theory and development, International Journal of Advanced Manufacturing Technology 14 (1998) 865–875. [19] Z. Chen, Z. Dong, G. Vickers, Automated surface subdivision and tool path generation for 31212-axis CNC machining of sculptured parts, Computers in Industry 50 (2003) 319–331. [20] P. Gray, S. Bedi, F. Ismail, Arc-intersect method for 5-axis tool positioning, Computer-Aided Design 37 (7) (2005) 663–674. [21] S. Austin, R. Jerard, R. Drysdale, Comparison of discretization algorithms for NURBS surfaces with application to numerically controlled machining, Computer-Aided Design 29 (1) (1997) 71–83.
International Journal of Machine Tools & Manufacture 47 (2007) 182–190 www.elsevier.com/locate/ijmactool
Arc-intersect method for 31212-axis tool paths on a 5-axis machine Paul J. Gray, Fathy Ismail, Sanjeev Bedi Department of Mechanical Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 20 October 2005; received in revised form 25 January 2006; accepted 30 January 2006 Available online 17 April 2006
Abstract An inherent problem with simultaneous 5-axis machining is that it often suffers from dramatic reductions in feed rate when the tool axis is in the vicinity of the singularity point of the machine; during large orientation changes over small distances; during rotary axes reversals and from interpolation of the tool axis vector. 31212-axis machining offers an alternative strategy that can be used to overcome these problems and still maintain some of the salient features of 5-axis machining to improve machining times over 3-axis ballnose machining. In 31212-axis machining, during cutting the machine moves only its three linear axes while the two rotary axes are locked, resulting in a fixed tool orientation. Locking the rotary axes generates fewer fluctuations in the feed rate than simultaneous 5-axis machining and results in a more consistent surface finish with lower variations in cutting force and torque. A new tool positioning strategy called the Arc-Intersect Method (AIM), which can also be applied to simultaneous 5-axis machining, is presented here for 31212axis machining on simultaneous 5-axis machines using toroidal or flat endmills. A cutting test was performed and the part was measured with a CMM to check for accuracy and to measure the cusp heights. Machining times were compared to 5- and 3-axis tool paths and cutting torque measurements were compared between 31212- and 5-axis machining. r 2006 Elsevier Ltd. All rights reserved. Keywords: 31212-axis machining; 5-axis machining; Triangulated surfaces; Cutting torque; Computer graphics
1. Introduction In simultaneous 5-axis surface machining, the tool axis can be dynamically oriented to generate a better match between the cutting geometry of a toroidal or flat endmill to the workpiece. Doing so can increase the machined strip width of each tool pass over 3-axis machining and thus, reduce the total number of tool passes to machine the surface leading to reduced machining times. A well-known approach for 5-axis tool positioning is to match the projected effective radius of the cutting tool to the curvature of the surface at the cutter contact point, ccp [1–3]. However, these methods require iterative gougechecking and correction strategies. Redonnet et al. [4] and Lauwers et al. [5] are examples. To eliminate the need for iterative gouge checking and correction algorithms, some 5-axis tool positioning strategies attempt to match the tool’s cutting geometry to the Corresponding author. Tel.: +1 519 888 4567; fax: +1 519 888 6197.
E-mail address: [email protected] (S. Bedi). 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.01.023
surface around the ccp such as the work by Warkentin et al. [6], and Gray et al. [7] who also used computer graphics to compute tool positions from triangulated surfaces [8]. Generating tool paths from triangulated data is not new; the works in [9–12] present 3-axis machining examples. Li and Jerard [13] developed a strategy for 5-axis Sturz milling using triangular data. Balasubramaniam et al. [14,15] developed methods for 5-axis tool positioning that account for accessibility of the tool using visibility maps of the triangulated data though the tool positions were not optimised for the highest material removal rate. Although 5-axis machining has been shown to reduce machining times over 3-axis [16], it is still plagued with a number of problems, which makes it difficult to gain wider acceptance in industry. In simultaneous 5-axis machining, it is often observed that the feed rate of the tool with respect to the workpiece fluctuates because the rotary axes servo drive systems limit the overall achievable feed rate. This can occur when the tool axis is in the vicinity of the machine’s singularity point, during large orientation changes over short distances along the tool path, and
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when the rotary axes must reverse direction. These fluctuations and reductions in feed rate ultimately lead to longer than expected machining times and result in variations of surface finish over the part. Although 31212-axis machining was originally proposed as a cheaper alternative to simultaneous 5-axis machining of parts via the use of manual indexable rotary-tilt tables on 3-axis machines [17–19], 31212-axis machining can also be used to overcome the problems associated with reduced and fluctuating feed rates commonly encountered in simultaneous 5-axis machining. In 31212-axis machining, the rotary axes are locked while the tool is engaged with the workpiece, thus, the machine only interpolates and moves its linear axes. The result is that cutting forces and surface finish are more consistent than in simultaneous 5-axis machining. To date, there exists no strategy for determining the optimal tool orientation to machine the maximum strip width for 31212-axis machining of complex surfaces using toroidal or flat endmills. Thus, the goal in this paper is to develop a tool positioning strategy that computes an optimal tool orientation for a set of ccps along a pass. Since the work in this paper uses concepts developed in earlier work presented by the present authors for their 5-axis Arc-Intersect Method (AIM) machining strategy, only a brief overview of the method is presented next. 2. Five-axis AIM Similar to the Principal Axis Method [3], the 5-axis AIM, developed by Gray et al. [20] is based on the fact that the widest machined strip width is cut when the tool is tilted along the feed direction (Fig. 1). It can be seen that when the tool tangentially contacts the ccp and the tool axis is tilted along the feed direction, the tool axis will lie in the plane containing the surface normal at the ccp and the feed direction vector. This plane is referred to as the tilting plane indicated in Fig. 1. The concept of AIM is to tilt the tool such that the tool axis remains in the tilting plane and the forward bottom point of the tool remains in tangential contact with the ccp until a second contact point on the surface is touched. This will give the smallest tilt angle and thus, the largest effective radius resulting in the widest machined strip width for the given feed direction at the ccp without gouging the surface. However, instead of rotating
Fig. 1. Five-axis AIM concept and method.
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the tool to find the limiting tilt angle that will touch the second contact point, the surface points beneath the tool are rotated to find the limiting tilt angle. 3. Five-axis arc-intersect algorithm For a given feed direction, a tool position is computed for each ccp along the tool path. With the tool axis restricted to lie in the tilting plane, all tilt angles are measured around the cross vector. The idea is to find the minimum tilt angle of the tool axis about the cross vector at which the tool will contact another point on the surface and maintain its contact with the ccp without gouging the surface as illustrated in Fig. 1. The area beneath the tool is discretized into a set of points referred to as the shadow grid points. Gray et al. [8] presented a method to compute surface points and normals of triangulated surfaces using the computer’s graphics card. The same method is used here for its simplicity. The tolerance for the pixel density to render the scene is described in Appendix A, which is similar to that described by Austin et al. [21] for gouge checking with respect to cloud point data sets. The scene is viewed along the surface normal of the ccp and each rendered pixel is used as a shadow grid point. Fig. 2 shows the viewing volume set-up for rendering the shadow grid points. Each shadow grid point is rotated about the cross vector to generate an arc. If the intersection of the arc to the tool centre plane (the plane containing the cross vector and the normalized feed direction vector) lies within the circular shadow of the tool, then the tool can potentially contact the shadow grid point when it is rotated about the cross vector (Fig. 2). Arcs may intersect either the toroidal geometry of the tool (the edge radius) or the bottomcutting plane of the tool. For the shadow grid points whose arcs intersect the bottom-cutting plane within the radius given by Eq. (1), the intersection to the bottom-cutting plane is computed using Eq. (2). All other arcs will intersect the toroidal geometry of the tool and their intersections are computed using a bisection search. The intersection point of an arc to the tool is used to determine the tilt angle (y)
Fig. 2. Viewing volume for tool position calculation (reproduced from [20]).
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Fig. 3. Arc Intersection calculation.
through which the tool needs to be rotated about the cross vector to contact a particular shadow grid point. The tilt angle, y, is simply the difference between the arc intersection angle, f, and the arc angle, g, between the shadow grid point and the tool centre plane given by Eq. (2) from the geometry of the tool (Fig. 3). The tilt angle, y, is computed using either Eq. (3) or (4) depending on whether the shadow grid point lies above or below the tool centre plane. Bottom_Radius ¼ Rmajor
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2minor ðRminor Centre_ClearanceÞ2 ,
(1) where Bottom_Radius is the radius of the bottom cutting plane of the cutting tool, Rminor the minor radius of the torus, Rmajor the major radius of the torus, Centre_Clearance the perpendicular distance from the bottom point of the tool to the bottom cutting plane. 1 Rminor Centre_Clearance g ¼ sin . (2) Arc_Radius If the shadow grid point is below the tool centre plane then the tilt angle is y ¼ f g.
(3)
If the shadow grid point is above the tool centre plane then the tilt angle is y ¼ ðf þ gÞ,
(4)
where the tool tilt angle, y, is the angle between the surface normal and the tool axis in the tilting plane. A forward tilt of the tool matches concave surface regions and corresponds to a negative tilt angle with this convention. Finally, the largest tilt angle of all the shadow grid points is selected as the tilt angle for the tool position. Having described the 5-axis AIM algorithm, the 31212-axis AIM can now be presented. The next section describes the adaptation of the 5-axis AIM into 31212-axis machining strategy by optimising the tool orientation for many ccps on the surface instead of for only a single ccp.
direction. This can be seen with the help of Fig. 1 by examining the effective radius of the tool when viewing the tool along the tilt direction. All other feed directions will reduce the machined strip width, which can be seen by observing that the projected effective radius of the tool moving perpendicular to the tilting plane is simply the minor radius of the torus. Therefore, the tool is always tilted along the feed direction to cut the widest strip width. The first step in the process is to compute the ccps and their corresponding surface normals for the selected region, from which the average surface normal in this region is computed. Each surface normal is projected onto the projection plane defined by the average surface normal and the feed direction of the tool as shown in Fig. 4. For each ccp, a tool position is computed using the 5-axis AIM with the same viewing volume set-up as in Fig. 2 except the view direction is now along the projected surface normal instead of the actual surface normal. The optimal orientation for the tool axis in the projection plane for each ccp is computed. The tool orientation for the region is then selected as the orientation with the largest tilt angle with respect to the feed direction vector computed from all the ccps in the region. If the tool axis is forced to lie inside the projection plane and the front circular arc of the tool cross-section, (referred to as forward pseudo insert) is forced to touch the ccp, then tangency at the ccp may be violated in places along the tool path where the actual surface normal deviates considerably from the projection plane in the cross feed direction (Fig. 5). These shadow grid points are detected in the AIM bisection search as points that, regardless of the arc intersect angle, will always lie inside the torus. These points are stored but not used in the tool tilt angle calculation. After the tilt for the tool pass is computed, the distance of these stored points to the tilted torus along the final tool axis direction of the region are computed for each tool position. The maximum gouge distance is the value the tool is lifted by along the tool axis. Essentially this means that the tool has been moved to contact an alternate ccp by lifting the tool along the tool axis. This strategy will ensure that the tool will not gouge the surface provided the density
4. The 31212-axis AIM Since the tool orientation is locked during cutting in 31212axis machining, the feed direction of the tool can drastically affect the resulting machined strip width. The widest machined strip width will be generated at the front point of the tool when the tool moves along the tilt
Fig. 4. Projection of surface normals.
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Fig. 6. Machined workpiece.
Fig. 5. Inside shadow grid points.
of shadow grid points is sufficiently high. Appendix A presents the calculations for determining the density of shadow grid points.
Table 1 Triangulation parameters Max. angle between surface normals at triangle vertices Max. aspect ratio Min. edge length Max. edge length Max. dist from triangle edge to surface
0.51 1 0.05 mm 1.5 mm 0.01 mm
5. Machining example An 11 surface patch section of a stamping die for an automobile part was used for a machining example. The workpiece shown in Fig. 6 was slightly modified from the original part to protect the source’s proprietary data for publishing purposes. Triangulation of the surfaces for the AIM was performed with Rhinoceros NURBS modelling software using the parameters listed in Table 1. A comparison between the 31212- and 5-axis AIM for this workpiece is presented in Subsection 5.4. A 2-insert toroidal tool with and outer radius of 12.7 mm and an insert radius of 6 mm was used. The insert radius is the minor radius of the tool’s toroidal surface. The speed was 6700 rpm and the feed rate was set at 2000 mm/min. The part was machined on a Deckel Maho 80P Hi-Dyn tilt/ rotary table 5-axis machine. 5.1. Tool path generation Parallel tool passes along the Y-axis of the workpiece with a constant forward step distance of 1.27 mm were used. Since the CNC machine used in this test is a continuously variable 5-axis machine, the tool orientation was optimised and locked for each tool pass. In this way, each tool pass was considered as a separate region with a fixed surface normal. The cross-feed distance calculations were based on the ‘‘Rolling Ball Method’’, RBM, for 5-axis tool positioning developed by Gray et al. [8] in which the surface normal curvature at each ccp is estimated for a triangulated surface along the cross-feed direction. The estimated curvature was compared to the projected effective radius of the tool at the ccp for each tool position to compute a cross-feed distance. The smallest cross-feed distance computed for the entire tool pass was used as the final cross-feed distance for the pass. The maximum cusp height value was set at 0.1 mm and the maximum allowable cross-feed distance was equal to the outer radius of the tool at 12.7 mm.
5.2. Viewing parameters The ccps were computed using a 201 square pixel window with a width and height of 1 mm. The surface normals were also computed with a 201 square pixel window but with a width and height of 3 mm. Only points within a 3 mm radius of the centre of the view were used for the surface normal calculation. A view depth of 200 mm with a 24-bit depth buffer was used for both windows. The tolerance for the arc intersection point bisection search was set at 0.001 mm and the tool positions were computed using a 315 square pixel window with a width and height of 31.4 mm (equal to the tool outer diameter plus the insert radius) and a view depth of 100 mm. This setting gave a resolution of 0.1 mm/pixel and a depth resolution of 5.96 106 mm per depth plane. The maximum possible penetration of the tool between pixels was 4.14 104 mm according to the method described in Appendix A. 5.3. Machining results Fig. 6 shows the machined part and Fig. 7 shows the coordinate measuring machine, CMM, measurements plotted against the CAD data for various cross-sections. The maximum measured cusp height was 0.128 mm. 5.4. Comparison of methods This subsection presents a comparison of machining times between 3-axis ballnose machining, the 31212- and the 5-axis AIM for the example workpiece. The 5-axis AIM machining results were presented by Gray et al. [20] and are used here for comparison purposes. Because each process was not fully optimised for tool dimensions, feed direction, and rapid traverse parameters between tool passes, and since different part geometries, machines, cutting
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Fig. 7. CMM scans of workpiece.
parameters and cutting tools can affect the performance of each method, a complete comprehensive comparison is not possible. The study can, however, be used to analyse hindrances to machining times for the different processes. The number of passes for a 3-axis ballnose tool path was computed by dividing the width of the part by the crossfeed distance, which was computed using Eq. (5) for a cusp height of 0.1 mm. The total machining time was then computed by multiplying the measured time per pass by the number of passes. Table 2 is a summary of the number of tool passes required for each method. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cross_Feed ¼ 2 R2ball ðRball Cusp_heightÞ2 . (5) The 5-axis tool passes were generated with the method outlined in Section 3 using the cross-feed distance calculation presented in Subsection 5.1 with a maximum allowable cross-feed distance set equal to the radius of the tool. Also, the same strategy of parallel single-direction tool passes along the Y-axis of the workpiece was used as in the 31212-axis AIM. Larger tools will generally be more useful for the 5-axis AIM than for the 31212-axis AIM, since, in 31212axis, the use of larger tools can generate more inside points which would require larger lift distances along the tool axis. Thus, for the 5-axis test, a toroidal tool with an outer radius of 19.05 mm and an insert radius of 6.35 mm was used. The 5-axis AIM tool path was able to make use of the maximum allowable cross-feed distance with the selected tool, indicating that, perhaps, an even larger tool could have been more optimal. In comparison, the 31212-axis AIM did not reach the maximum cross-feed distance anywhere along the part, which suggests that a smaller tool may have been more optimal. The different machining times are presented in Table 3. For all entries in the table, the spindle
Table 2 Number of tool passes for programmed maximum cusp height of 0.1 mm
Number of passes
3-axis ballnose R ¼ 12:7 mm
31212-axis AIM Router ¼ 12:7 mm; Rinsert ¼ 6 mm
5-axis AIM Router ¼ 19:05 mm; Rinsert ¼ 6:35 mm
52
36
16
Table 3 Machining times for different methods and different feed rates Machining times for programmed maximum cusp height of 0.1 mm (min) Feed rate (mm/min)
3-axis ballnose zig–zag path R ¼ 12:7 mm
31212-axis AIM single direction, parallel passes Router ¼ 12:7 mm; Rinsert ¼ 6 mm
5-axis AIM single direction, parallel passes Router ¼ 19:05 mm; Rinsert ¼ 6:35 mm
1000 2000
9.88 5.18
8.12 4.82
4.8 4.12
speed was set to achieve 0.15 mm/tooth chip load for the given feed rates. For the 5-axis test, doubling the programmed feed rate led to a reduction in machining time of only 14%. The reason for the small decrease in machining time is that, in the areas where large changes in the C-axis were necessary, the feed rate dropped dramatically below the programmed feed rate due the C-axis servo drive system limits. Since the machine is only capable of 9.2 rpm in the C-axis and 7.2 rpm in the A-axis, there were two regions along the tool
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pass where the machine could not maintain the programmed feed rate. Thus, doubling the programmed feed rate did not halve the machining time. Another reason for limited gains with the higher programmed feed rate is that a larger percentage of time is spent in rapid traverse between passes with the single direction tool passes over the ballnose zig–zag path, which is not affected by the programmed feed rate. It can be seen that the 31212-axis approach can reduce actual cutting times by reducing the number of tool passes over 3-axis ballnose machining for the same radius tool; however, for this small part, the gains were limited. Considering a feed rate of 2000 mm/min, the average time per tool pass in the 31212-axis test was a little over 8 s, whereas with the ballnose path it was about 6 s. This means that approximately 2 s were lost per pass in rapid traverse and for the orientation change in the 31212-axis approach. When the feed rate was doubled from 1000 to 2000 mm/ min, the 31212-axis AIM reduction in time was 41%, whereas the ballnose path reduction was 48%. This means that by doubling the feed rate, a larger reduction in time was observed for the ballnose tool path, which was expected because the amount of time spent out of cut is smallest for the 3-axis tool path, which is mainly due to its zig–zag path compared to the single direction parallel passes used in the 31212- and 5-axis paths. Conceivably, the 3- and the 31212-axis methods could machine the part faster than the 5-axis method, provided the required spindle speeds could be achieved and tool wear is of no concern. A similar argument can be made that, on a different machine with faster rotary axes, the 5-axis machining could be made faster. To make a fair comparison, the tool path feed directions, tool geometry and tool materials would have to be optimised and compared for different parts and machines. Thus, extrapolation of the present results to other workpieces, materials and machines for comparison purposes is not simple and should be done with caution on a case-by-case basis. 6. Cutting torque comparison To quantify the fluctuations in feed rate during simultaneous 5-axis machining, a second machining test was conducted. Cutting torque measurements were collected for 31212- and 5-axis machining and compared. 6.1. Experimental set-up In this experiment, a Kistler 9123C spindle-mounted rotating force dynamometer was used to measure the cutting forces. Since this experiment is only concerned with demonstrating that the feed rate fluctuates in 5-axis machining, a single measure, in this case cutting torque, is sufficient. Furthermore, measurements were only made for a single tool pass over the surface for the part shown in Fig. 6. The
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tool pass was selected to show the reduction in feed rate during large orientation changes and to avoid the kinematic singularity of the machine. Tool passes were generated at x ¼ 194 mm using the 5-axis AIM and the 31212axis AIM for comparison. A 2-insert toroidal tool with an outer radius of 12.7 mm and an insert radius of 6 mm was used for both cutting tests. The natural frequency of the tool and rotating dynamometer assembly was measured at 360 Hz. The unbalance was found to be 10.8 g mm at 10 000 rpm using the G2.5 class balance quality grade. The test was conducted at a tooth passing frequency of 13 the measured natural frequency, which ensures the measured data to have no more than 10% error according to the dynamometer manufacturer. For the 2-insert cutter, the spindle speed was 3600 rpm. The sampling rate was set at 7200 Hz; giving 60 samples per tooth period, and the feed rate was set at 1080 mm/min, for a target chip load of 0.15 mm/ tooth. The stock material was 6061-T6 aluminum. To prepare the stock material, five semi-finish tool passes were generated using the 5-axis AIM between x ¼ 188 and 204 mm, with a constant cross feed distance of 3 mm. Then a single pass was machined at x ¼ 181:8 mm, with a 1 mm depth of cut. The position of this single tool pass was selected to ensure that the tool’s forward pseudo insert cut the full depth of material at all tool positions in both the 31212- and 5-axis cutting tests. Thus, the maximum chip thickness was cut throughout each tool pass in the torque measuring tests. The axial depth of cut for the test passes was 1 mm. 6.2. Machining results Fig. 8(A) shows the measured torque for both cutting tests. The elapsed cutting times were approximately 9.2 s for the 31212-axis pass and 15.2 s for the 5-axis, respectively, as measured from the torque plots in Fig. 8(A). The 31212axis tool path was approximately 165 mm long (considering the tool will still cut after the tool’s forward point clears the part edge), which, at a feed rate of 1080 mm/min would yield a machining time of 9.17 s. This estimate agrees well with the machining time measured from the torque plots. The middle portion of the 5-axis measurement indicates a reduced chip load in the high curvature areas of the tool path, where large orientation changes occur over short distances. Fig. 8(B) shows the torque at 25% of the machining time for both tests, which approximately corresponds to the point where the surface curvature changes significantly along the tool path. The 31212-axis path remains relatively constant, whereas the force variation associated with the 5-axis path indicates that the tool is essentially cutting with only one tooth. As the radial run-out of the cutting tool was measured at 0.05 mm, it can be deduced that the feed rate for the 5-axis path dropped at that point to around 1/3 or less of the programmed feed rate. In other words, the
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Fig. 8. Measured torque plots: (A) complete plot, (B) 25% cutting time, (C) 50% cutting time, and (D) 75% cutting time.
chip load at that point dropped from the programmed 0.15 mm/ tooth, to a chip load close to the value of the tool runout at 0.05. At the 50% time point shown in Fig. 8(C), the 31212-axis path had moved well beyond the high curvature region, whereas the 5-axis path was still cutting that area. However, the peak torque here for the 5-axis is higher than that at the 25% point, which indicates the feed rate has not dropped as much. Fig. 8(D) shows the torque towards the end of the tool passes, where the tool orientation does not change much in the 5-axis path, resulting in a relatively constant feed rate. The measured machined strip width ranged from 15.9 to 16.8 mm in 5-axis and 8.8 to 11 mm in the 31212-axis cut. Since the machined strip width is smaller in the 31212-axis path, the torque peaks are narrower than the 5-axis path.
7. Discussions From experimental work presented here, it can be seen that simultaneous 5-axis machining suffers greatly when the rotary axes servo drive systems are saturated. This characteristic cannot easily be avoided, except perhaps by selecting the feed direction to minimize orientation changes, by smoothing out the tool orientation changes
along the path, or by changing the orientation of the part set-up on the machine. The 31212-axis tool path resulted in a constant feed rate and more consistent cutting torque, which is useful when machining with slender tools or when machining flexible parts. Controlling the feed rate could be a strong argument in favour of adopting the 31212-axis strategy. Machining times were reduced with the AIM for 31212-axis machining over tool paths for the same sized ballnose tool in 3-axis machining. However, the single direction parallel tool passes used in the 31212-axis method introduced an additional time penalty that did not exist in the zig–zag ballnose 3-axis path. Therefore, the 31212-axis AIM should be used in conjunction with a partitioning scheme like that proposed by Chen et al. [19] to further optimise the process. Selecting the feed direction such that the variation of the surface normal vectors along the direction perpendicular to the projection plane is minimized, can help minimize the lifting distances in the 31212-axis AIM. The size of the cutting tool will also have an effect; larger tools will more likely require higher lifts because the tangency at the ccp will be violated over a larger area. Tool dimensions will also affect the cross-feed distance. Optimisation of the tool dimensions could certainly improve the strategy; however, no simple method exists as of yet.
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Although the 31212-axis AIM can optimise the tool orientation for a given set of ccps, it assumes that the ccps are known beforehand. If adaptive side and forward step algorithms are to be applied when machining a region on the part, the location of the ccps will not be known until the tool orientation is selected. In this case, the 31212-axis AIM as it is presented cannot be applied. A solution to this problem would be to optimise the tool orientation for a sample of points in the selected region. The tool positions could then be computed by projecting the tool onto the surface; an easy task when using the computer’s graphics hardware for tool positioning. It would be beneficial to adapt the 31212-axis AIM for use with indexable rotary axes machines, since they are less expensive and are generally stiffer than continuously variable 5-axis machines. A simple approach would be to compute the orientations for the partitioned regions with the current method and then find the indexing angles that give the closest tool orientation. The tool would then be projected onto the surface at each ccp to give the final tool positions. Although the comparisons conducted here showed that the 31212-axis AIM was slower than 5-axis method, Sections 5.3 and 6 clearly show how 5-axis machining can suffer from saturation of the rotary axes servo drive systems leading to fluctuations and reductions in the feed rate. Saturation of the rotary axes servo drive systems results in longer than expected machining times and variations in surface finish and torque loads on the tool. 31212-axis machining, on the other hand, does not suffer from these effects and is a more promising candidate for high speed machining since the loads remain relatively constant. Finally, although the time gains presented here for 31212axis over 3-axis machining are relatively small for this small part, it has been demonstrated at an industrial plant that the 31212-axis AIM was 1.5–3 times faster than 3-axis high speed machining for a larger automotive panel stamping die. Unfortunately the details of the experiment are confidential. Acknowledgements This research was funded by the Natural Sciences and Engineering Research Council of Canada, by Ontario Innovation Trust, and the Canadian Foundation for Innovation. Appendix A. Pixel spacing and depth buffer resolution Using the concept developed by Austin et al. [21] for computing the maximum penetration of a cutting tool between discrete points representing a surface, the horizontal and vertical pixel resolution for the viewing window can be set according to the required machining accuracy. The viewing volume width for tool positioning can be set to the width of the cutting tool because the tool axis is constrained to lie in a fixed plane in the AIM (Fig. 1). The
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Fig. 9. Pixel spacing (reproduced from [20]).
number of pixels for the width and height of the viewing window can be set by determining the maximum possible penetration of the cutting tool between rendered pixels. This concept was used by Austin et al. [21] for gouge checking of tool positions with respect to discrete surface models. Recognizing that the minimum radius of curvature of a toroidal cutting tool is its minor radius, the maximum penetration of the tool between rendered pixels can be conservatively estimated by determining the penetration distance of a circle with a radius equal to the torus minor radius between the diagonal distance between pixels. This is illustrated in Fig. 9, and expressed below in Eq. (6). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ¼ r r2 d 2 . (6) 4 Since a 24-bit depth buffer was used for the work in this paper, setting the viewing volume depth is not as critical as setting the viewing volume width and height because the depth resolution is generally much higher than the horizontal and vertical pixel resolution of the viewing window. References [1] S. Bedi, S. Gravelle, Y. Chen, Principal curvature alignment technique for machining complex surfaces, Journal of Manufacturing Science and Engineering 119 (1997) 756–765. [2] S. Mullins, C. Jensen, D. Anderson, Scallop elimination based on precise 5-axis tool placement, orientation, and step-over calculations, Advances in Design Automation 2 (1993) 535–544. [3] N. Rao, F. Ismail, S. Bedi, Tool path planning for five-axis machining using the principal axis method, International Journal of Machine Tools and Manufacture 37 (7) (1997) 1025–1040. [4] J.-M. Redonnet, W. Rubio, F. Monies, G. Dessin, Optimising tool positioning for end-mill machining of free-form surfaces on 5-axis machines for both semi-finishing and finishing, International Journal of Advanced Manufacturing Technology 16 (6) (2000) 383–391. [5] B. Lauwers, P. Dejonghe, J. Kruth, Optimal and collision free tool posture in five-axis machining through the tight integration of tool path generation and machine simulation, Computer-Aided Design 35 (5) (2003) 421–432. [6] A. Warkentin, F. Ismail, S. Bedi, Multi-point tool positioning strategy for 5-axis machining of sculptured surfaces, Computer Aided Geometric Design 17 (1) (2000) 83–100. [7] P. Gray, S. Bedi, F. Ismail, Rolling ball method for 5-axis surface machining, Computer-Aided Design 35 (4) (2003) 347–357. [8] P. Gray, F. Ismail, S. Bedi, Graphics-assisted rolling ball method, Computer-Aided Design 36 (7) (2004) 653–663.
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