comparison between multi-point and other 5-axis tool...

International Journal of Machine Tools & Manufacture 40 (2000) 185–208

Comparison between multi-point and other 5-axis toolpositioning strategies

Andrew Warkentina,*, Fathy Ismailb, Sanjeev Bedib

a Department of Mechanical Engineering, Dalhousie University (DalTech), Halifax, Nova Scotia, Canada B3J 2X4b Automation and Control Group, Department of Mechanical Engineering, University of Waterloo, Waterloo,

Ontario, Canada N2L 3Gl

Received 15 March 1999; accepted 8 June 1999

Abstract

A method of generating sculptured surfaces at multiple points of contact between the tool and the work-piece was developed and proven viable by the current authors in previous work. They denoted this finishmachining method, “Multi Point Machining”, or simply MPM. This paper compares MPM with two other5-axis tool positioning strategies; namely: the inclined tool, and the principal axis method. It is also com-pared with 3-axis ball nose machining. Comparisons are conducted using computer simulations and experi-mental cutting tests. Results obtained show that MPM produced scallop heights that are much smaller thanthose produced by the other tool positioning strategies. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Multi-point; Machining; Tool; Path; Planning; Positioning

1. Introduction

Sculptured surfaces are encountered in many industrial and consumer products. These surfacesare typically produced in moulds and dies that have been machined on multi-axis machiningcenters. A common practice has been to machine these surfaces on 3-axis machines using ballnose cutters as shown in Fig. 1. In this operation the tool generates the surface as it makes manyclosely spaced tool passes across the design surface. This operation removes most but not all ofthe material from the design surface. Surface deviations between the desired surface and themachined surface occur because the tool geometry never matches the surface geometry exactly.

* Corresponding author. Tel.:+1-902-494-3901; fax:+1-902-423-6711.E-mail address:[email protected] (A. Warkentin)

0890-6955/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0890-6955(99)00058-9

186 A. Warkentin et al. / International Journal of Machine Tools & Manufacture 40 (2000) 185–208

Fig. 1. Scallop generation.

The resulting surface is left with a large number of scallops. Grinding and polishing operationsare required to remove these scallops and produce the desired surface finish.

Surface machining is a time consuming and costly operation. Typically molds and dies takeseveral hundred hours to produce. Thus researchers [1,2] have proposed many different tool pathplanning and positioning methods designed to reduce the time and cost associated with surfacemachining. Multi-Point Machining (MPM) is a newly proposed tool positioning strategy designedto reduce this production time and cost. As its name implies, the design surface is generated atmore than one point on the cutting tool. Warkentin [3] presented a thorough description of thealgorithm used to determine multi-point tool positions.

In this paper the results of a study comparing MPM with two 5-axis tool positioning strategies,namely the inclined tool [4] and the Principal Axis Method [5,6] are presented. When using theinclined tool method, the cutting tool is inclined at a fixed angle in the feed direction throughoutthe whole tool path. The inclination angle is selected such that the scallops are minimized withoutgouging the surface. In the “Principal Axis Method” or simply PAM, the tool is inclined in thedirection of minimum curvature instead of the feed direction. Furthermore, the tool inclinationvaries depending on the curvature of the surface. For completeness, MPM will also be comparedwith 3-axis ball nose machining.

A brief description of each strategy will be given. In this description vector algebra will beused to compute the tool position,tpos, and tool axis orientation,taxis, at each cutter location. Thecomparisons will be conducted using both computer simulations and experimental cutting testsfor the surface shown in Fig. 2 and defined by Eq. (1). This surface was selected because it istypical of open form surfaces commonly found in the die industry. Results will show that for the

187A. Warkentin et al. / International Journal of Machine Tools & Manufacture 40 (2000) 185–208

Fig. 2. Test surface.

tested surface, MPM leads to scallop heights that are 200, 25,and 2 times smaller than thoseproduced by ball, inclined tool, and PAM, respectively.

S(u,v)53 −94.4+88.9v+5.6v2

−131.3u+28.1u2

5.9(u2v2+u2v)−3.9v2u+76.2u2+6.7v2−27.3uv−50.8u+25.0v+12.14 (1)

2. Ball nosed tool positioning

Machining using a ball nosed cutter was the first method developed for sculptured surfaces andremains the most popular. Ball nosed tool positioning can be explained using Fig. 3. It shows aball nosed tool in tangential contact with a surface at the cutter contact point,cc. Since the cuttingsurface of the tool is spherical, the tool axistaxis has no effect on scallop geometry. Therefore,this comparison will usetaxis=[0, 0, 1]T. The tool position is offset along the surface normal,n,by a distance equal to the tool radius,r.

tpos5cc1rn (2)

Fig. 4 shows the simulated surface deviations using a ball nose cutter with a radius of 8.0 mmand a 4.0 mm tool pass interval. This figure clearly illustrates the characteristic sharp scallopsgenerated during single point machining. The maximum scallop height was 496µm.


Fig. 3. Positioning a ball nosed tool in 3-axis.

Fig. 4. Surface deviations produced by a ball nosed end mill. Tool path parameters:r=8.0 mm, tool pass interval=4.0mm, feed inx direction.


2.1. Inclined tool

The inclined tool [4] positioning strategy, also known as Strutz milling, is the most commonof the 5-axis tool positioning strategies. It has been implemented in several high-end CAM pack-ages and has been shown to be far superior to the ball nosed technique on many occasions. Forthese reasons it is the most appropriate benchmark for other 5-axis tool positioning strategies.

Fig. 5 shows how the inclined tool method works for a toroidal cutter. The tool axis,taxis, isinclined in the feed direction by an anglef. This inclination angle is often called the Strutz angle.The tool position,tpos is calculated such that the tool is placed in tangential contact with thesurface atcc.

The tool can be positioned by considering the plane containing the tool axis and the feeddirection,f. This tool axis plane is shown in Fig. 6. In order to use planer geometry, a coordinatesystem must be created in this plane at the insert center,c, which is located by:

c5cc1rn (3)

One of the coordinate axes will be the surface normal,n. The other coordinate axis must beperpendicular ton and yet lie in the plane. This vector,e, can be constructed using a triplevector product.

e5n3(n3f) (4)

The tool axis is then calculated by:

taxis5cos(f)n1sin(f)e (5)

Fig. 5. Positioning an inclined toroidal cutter.


Fig. 6. The tool axis plane for an inclined tool.

and the tool position is given by:

tpos5c1Rsin(f)n2R cos(f)e (6)

The tool paths for the incline tool method were generated by calculating the tool position andtool axis using Eqs. (5) and (6) for each cutter contact point. These equations require the valueof the inclination angle,f, be specified prior to tool positioning. A small value off will producegouging and a large value will result in unnecessarily large scallops. A value of 6° was used forthe inclination angle in the simulations. This value was selected by performing a series of simula-tions with progressively larger angles as shown in Table 1. The smallest value that did not resultin gouging was selected.

Fig. 7 shows a section of the surface deviations produced by an inclined tool simulation. Themaximum scallop height was 111.5µm. Even with twice the tool pass interval these values wereless than a quarter of those produced by the ball nosed tool of the same diameter. The scallopsare not very uniform because this method does not include surface curvature information. Thescallop size varies as the curvature changes.

3. Principal axis method

The principal axis method is a modification of the inclined tool method. It was formulated toaccount for surface curvature. In the principal axis method, curvature information is incorporated


Table 1Effect of inclination angle on 5-axis machining with an inclined toroidal cutter.r=3.0 mm, R=5.0 mm, tool passinterval=8.0 mm,f=6°

Incline angle,f (degree) Maximum gouge (µm) Volume of gouged material (mm3)

1 222.6 469.72 144.0 150.83 80.7 31.24 36.4 3.35 6.3 0.16 0.1 0.07 0.1 0.0

Fig. 7. Surface deviations produced by an inclined toroidal cutter.R=5.0 mm,r=3.0 mm, tool pass interval=8.0 mm,f=6°, feed in thex-direction.

into the tool position by modifying the tool axis plane and computing the inclination angle basedon the curvature at the cutter contact point as shown in Fig. 8. Rao et al. [5,6] showed that toolinclination could be optimized by inclining the tool in the direction of minimum curvature,l2,such that the minimum curvature of the tool equals the maximum curvature of the surface,k1.The anglef thus can be computed from:


Fig. 8. Principal axis method.

f5sin−1 1 R1k1

−r2 (7)

The tool axis plane in this method will contain the surface normal,n, as in the inclined toolmethod but it will not contain the feed direction. Instead the plane will include the direction ofminimum curvature,l2, as shown in Fig. 9. Once again, a coordinate system must be created inthis plane at the insert center,c, which is located by:

c5cc1rn (8)

The coordinate axes will be the surface normal,n, and the direction of minimum curvature,l2.

e5l2 (9)

The tool axis is then calculated by:

taxis5cos(f)n1sin(f)e (10)


Fig. 9. The tool axis plane for Principle Axis Method.

and the tool position is given by:

tpos5c1Rsin(f)n2R cos(f)e (11)

All that remains is to calculate the inclination angle,f. The minimum curvature of a toroidalcutter,kt2, is given by [5]:

k2t5sin(f)

R+r sin(f)(12)

Therefore, the inclination angle can be found by substituting the maximum curvature of the sur-face,k1, for k2t and solving forf.

f5sin−1 S k1R1+k1r

D (13)

Fig. 10 shows a section of the surface deviations produced in simulation by the principal axismethod. The maximum scallop height was 23.5µm which is less than a quarter of those producedby the inclined tool for the same tool and tool pass interval. Furthermore, the scallop size is muchmore uniform across the entire surface. Clearly, there is an advantage to incorporating curvatureinformation into tool positioning.


Fig. 10. Surface deviations produced by the principal axis method.R=5.0 mm,r=3.0 mm, tool pass interval=8.0 mm.

4. Multi-point machining

All the previously discussed tool positioning strategies attempt to maximize metal removal byconsidering the local geometry of a point on the surface and a point on the tool. In 1995, Warken-tin et al. [7] proposed a tool positioning strategy called Multi-Point Machining (MPM) whichmatches the geometry of the tool to the surface by positioning the tool in a manner that maximizesthe number of contact points between the surface and the tool. The authors demonstrated thepotential of the idea by using it to machine a spherical surface with virtually no scallops in afraction of the time required by conventional machining techniques.

In 1997 Warkentin et al. [8] extended multi-point machining to general concave surfaces. Byexperimentation it was found that, in general two contact points existed between the cutting tooland a concave surface and that these points are approximately symmetrical about the directionof surface minimum curvature as shown in Fig. 11. The distancew between the contact pointscc1 andcc2 is designated the separation distance.

The basic algorithm to findtpos and taxis is now described. The first cutter contact point,cc1,isspecified during the tool path planning stage as shown in Fig. 12. At this stage a set ofcc1 pointson the surface, called the cutter contact path, is specified. The second contact point,cc2, is locateda distance equal to the prescribed separation distancew away fromcc1 in the direction of surfacemaximum curvature,l1. A tool position is then generated for every pair of cutter contact points.

The second contact point can be found by using the curvature approximation shown in Fig.


Fig. 11. Specification of a MPM tool position.

Fig. 12. Path of cutter contact points.


13. In this figure a plane containing the normal vectorn1 and the direction of maximum curvaturel1 at the first cutter contact pointcc1 has been intersected with the surface. A circle with a radiusequal to 1/k1 can approximate the resulting intersection curve. If we assume that the second cuttercontact point lies on the approximating circle, then its location can be calculated in the followingmanner. The vector between the two cutter contact points is (cc22cc1). The magnitude of thisvector is equal to the separation distancew. This vector can be expressed in terms ofn1 andl1.

cc22cc15((cc22cc1)·n1)n11((cc22cc1)·l1)l1 (14)

where (cc2-cc1)·n1 and (cc2-cc1)·l1 are the components of (cc22cc1) on n1 and l1, respectively.These components may be expressed in terms ofw and the anglea.

cc22cc15w(sin(a)n11cos(a)l1) (15)

The position ofcc2 can then be found by rearranging Eq. (15).

cc25w(sin(a)n11cos(a)l1)1cc1 (16)

The anglea depends on the maximum radius of curvature and the separation between cuttercontact points, according to

a5sin−1Sk1w2 D (17)

Note that there will be an error in the location ofcc2 if the curvature of the surface changes. Inmost instances the calculatedcc2 will not lie on the surface, as shown in Fig. 13. In this casecc2

is projected onto the surface in thez direction.Once both potential cutter contact points are located, the tool position can be found based

entirely on the geometry of the tool and these two cutter contact points. The tool will be positionedsuch that tangential contact exists between the tool and at least one cutter contact point.

Fig. 13. Locating second cutter contact point.


Figure 14(a) shows the tool in tangential contact withcc1 and cc2. The lines formed by thenormal vectors,n1 andn2, at the cutter contact points,cc1 andcc2, pass through the insert centersat c1 and c2, and intersect the tool axis atp1 and p2. Note thatp1 and p2 would be at the samelocation if the surface was symmetric. However, for most surfaces this will not be the case.Therefore, the tool position will be calculated without assuming thatp1 and p2 are at the samelocation.

The position and orientation of the tool can be specified by determining the location of twopoints on the tool axis. Thus, the pointsp1 and tpos will be found in order to calculate the toolposition. The pointtpos will specify the location of the tool and the vectortaxis=p12tpos will specifythe orientation of the tool.

The pointp1 can be found by intersecting the line defined by the pointscc1 andc1 with a planecontaining the tool axis. One such plane is the plane perpendicular to the line joiningc1 andc2

Fig. 14. Geometry of multi-point contact.


that passes through the midpoint betweenc1 and c2. This plane will be referred to as the toolaxis plane and is shown in Fig. 14(b). The pointsc1 and c2 are located a distancer along thenormal vectorsn1 andn2 from the cutter contact pointscc1 andcc2:

c1=cc1+rn1

c2=cc2+rn2

(18)

Point a is the midpoint betweenc1 andc2.

a5c1+c2

2(19)

A vector normal to the tool axis plane,e3, can be found by noting that the tool axis plane isnormal to the vector joiningc1 andc2.

e35(c1−c2)uc1−c2u

(20)

The equation of the tool axis plane is defined by

e3·p2e3·a50 (21)

where the pointsa andp lie in the plane.The line joiningcc1 andc1 is now defined. A pointp on this line can be found usingcc1 and

n1 from:

p5cc11hn1 (22)

whereh is the distance along the line fromcc1. The pointp1 can now be obtained by intersectingthe tool axis plane with this line by substituting Eq. (22) into Eq. (21). The resulting value ofhgives the distance betweencc1 andp1.

h5e3·a−e3·cc1

e3·n1

(23)

Substitutingh into Eq. (22) will determine the Cartesian coordinate of the intersection point,p1.With p1 now calculated, the second point,tpos, needs to be determined. This point will be found

by considering the geometry of pointstpos, p1 anda in the tool axis plane as shown in Fig. 14(b).Note that these three points form a right angle triangle because the plane containingtpos, c1 andc2 is always perpendicular to the tool axis. Since this plane is in an arbitrary orientation, basesvectors at pointa must be constructed in order to use planar geometry to locatetpos. A unit vector,e1, in the direction froma to p1 is given by:

e15(p1−a)up1−au

(24)

A second unit vector,e2, perpendicular toe1 ande3 may be expressed as:

e25e13e3 (25)

The distance,d, between the center of the tool,tpos, and pointa is given by:


d5ua2tposu5!R2−uc2−c1u2

2(26)

The tool position can now be calculated from:

tpos5a1d sin(b)·e11d cos(b)·e2 (27)

where:

b5cos−1S dup1−auD (28)

Given two points on the tool axis, the tool axis vector,taxis, is calculated by normalizing thevector fromtpos to p1 according to:

taxis5(p1−tpos)up1−tposu

(29)

Together, the tool axis vector,taxis, and the tool position vector,tpos, define the orientation andposition of a multi-point tool position.

For most surfaces, the basic approach described above to calculatetpos and taxis usingcc1 andcc2 will not result in tangential contact between the tool and the workpiece at one or both ofthese points. Algorithms for correcting tool positions to ensure contact at the two points aredescribed in detail in [3].

Unlike the other positioning strategies the scallop geometry in MPM is influenced by the toolpass interval,c, and by the separation distance,w. The effects of bothc andw can be combinedinto one parameter, namely the separation ratio (w/c). Different types of scallops will be produceddepending on the separation ratio as shown in Fig. 15. If the separation ratio is equal to one,scallops will form only between the cutter contact points. If the separation ratio is less then one,scallops will form between the cutter contact points and between the tool positions. Finally, if

Fig. 15. Effect of separation ratio in multi-point machining.


the separation ratio is greater than one, the resulting scallops will be produced due to the combi-nation of both scallop formation mechanisms described above.

Surface deviations from the MPM simulations are shown in Fig. 16. The resulting scallopheight was approximately 9.5µm, which is about half the size of its nearest competitor theprincipal axis method. The large round scallops are produced between the cutter contact pointsand the small sharp scallops are produced between tool positions. Two distinct scallop shapes areclearly visible. Note also that the scallop size is fairly even across the surface. The multi-pointtool positioning strategy accounts implicitly for changes in surface curvature because the locationof the two cutter contact points are dependent on that curvature. Thus curvature information isindirectly incorporated into the tool position without actually having to calculate it; a clear advan-tage when machining a poorly defined surface.

5. Comparison of scallop heights

In this section a comparison of the scallop heights produced by MPM and the most popularcompeting tool positioning strategies: ball, inclined tool and the principal axis technique, will bemade. This comparison will be accomplished using simulation and cutting tests. The same toolpath and the same tool diameter, 16.0 mm, will be used in each simulation.

Simulations were conducted for a wide range of tool pass intervals. The results are summarized

Fig. 16. Surface deviations produced by multi-point machining.R=5.0 mm, r=3.0 mm, tool pass interval=8.0 mm,separation ratio=0.8.


in Table 2. In general, the maximum scallop heights for multi-point machining are about 200, 25and 2 times smaller than ball, inclined tool and PAM scallop heights, respectively.

Increased performance comes at the expense of increased computational effort and surfacerequirements. Computational time was assessed by generating tool paths for each method thatwould result in a maximum scallop height of 0.1 mm. It took 121, 58 and 53 seconds on a 166MHz Pentium to perform the tool positioning computations for the multi-point, principal axis andinclined tool methods. Ironically, tool positioning for the ball nose tool required 256 seconds dueto the large number of tool passes required. The principal axis method requires surfaces for whichthe curvature must be calculated accurately while the multi-point, inclined tool and ball nosedtechniques can be implemented with only surface normal information.

Workpieces were machined using the inclined tool, principal axis and multi-point methods inorder to verify the simulated results. Cutting tests were conducted on a Rambaudi milling machinethat had been retrofitted with a tilt–rotary table to provide 5-axis machining capability. Machiningwith a ball nose was not performed because it has already been shown on numerous occasionsto be several times inferior to the inclined tool method in the literature. For comparison purposes,each piece was machined with the same tool and tool pass interval. The 16.0 mm diameter Car-boloy tool MM 16-0.630-R7.6-MD07 with a torus radiusR=5.0 mm and insert radiusr=3.0 mmwas used for the cutting tests. The tool pass interval was 8.0 mm. The spindle speed and feedrate were 1200 RPM and 70.0 mm/min, respectively. The machined surface was measured on aMitotoyo BHN305 Coordinate measuring Machine, CMM. Surface deviations were extracted fromthe resulting surface measurements using the algorithm described in [9].

The photographs of the workpieces are shown in Figs. 17–19. They show that the surfaces arepractically identical in appearance. This is because the tool pass interval was the same for eachworkpiece and each technique produced relatively small scallops. Yet, the surface finish producedby each technique could be distinguished by touch, the inclined tool workpiece felt the roughestand the multi-point workpiece felt the smoothest.

Table 2Comparison of scallop height (µm) for different techniques. Torus dimensions:R=5 mm, r=3 mm, ball dimensions:r=8 mm, separation ratio=0.8 for MPM

Tool pass Interval 3-axis method 5-axis methods(mm)

Ball Inclined tool (6°) Principal axis method Multi-point machining

1 13.5 2.1 0.4 0.42 54.0 7.5 0.6 0.63 127.8 15.3 0.7 0.94 211.8 27.5 1.3 1.25 343.6 49.4 2.7 1.66 498.9 67.5 6.0 3.07 712.6 100.5 12.9 5.38 979.0 132.7 21.9 9.59 1276.4 162.2 37.5 16.210 2061.6 282.1 84.3 27.0


Fig. 17. Test surface machined using inclined tool.R=5.0 mm, r=3.0 mm, tool pass interval=8.0 mm, inclinationangle=6°.

Fig. 18. Test surface machined using principal axis method.R=5.0 mm,r=3.0 mm, tool pass interval=8.0 mm.


Fig. 19. Test surface machined using multi-point method.R=5.0 mm,r=3.0 mm, tool pass interval=8.0 mm, separationratio=0.8.

The surface deviations for the three workpieces are shown in Figs. 20–22, respectively. Thethick lines represent the measured results and the thin lines represent the simulated results. Thesurface deviations are presented as four sections in thez–y plane atx=25.0, 230.0, 260.0 and290.0 mm. Ideally the simulated and experimental results would match exactly. However, thereare two distinct differences between the simulated and experimental results. First, the experimentaldeviations are slightly bow shaped whereas the simulated deviations are straight, and second, theheight of the simulated and experimental scallops differ slightly.

The bow shape, called form error, is largely due to errors in locating the programmed coordinatesystem during workpiece setup. This error was estimated±12 µm in each of thex, y and zcomponents of the vector describing the location of the programmed coordinate system. Thepropagation of this error onto the surface depends on the rotations of the A and C axes of the 5-axis machine which results in the bow shape of the plots.

If the scallops are considered without the form error, the agreement between the experimentaland simulated results is excellent, with the experimental scallops being slightly smaller than thesimulated scallops. These differences may be attributed to errors caused by the simulation tech-nique, measurements and tool deflections. The discrete nature of the metal removal in the simula-tions may cause errors. Surface deviations are calculated every 0.1 mm. As a result features suchas the sharp peaks of single point scallops may be missed and the resulting maximum scallopmay be under-estimated. In addition, metal removal calculations are performed at discrete toolpositions. In reality the tool removes metal as it moves along the tool path. This motion is notaccounted for in the simulations. Therefore, more material is removed during actual machining


Fig. 20. Comparison of experimental and simulated results for inclined tool method.R=5.0 mm,r=3.0 mm, tool passinterval=8.0 mm,f=6°.


Fig. 21. Comparison of experimental and simulated results for principal axis method.R=5.0 mm,r=3.0 mm, tool passinterval=8.0 mm.


Fig. 22. Comparison of experimental and simulated results for multi-point machining.R=5.0 mm, r=3.0 mm, toolpass interval=8.0 mm, separation ratio=0.8.


than in simulated machining. Errors in the shape of the scallop also arise from the CMM measure-ments. As with the simulations, measurements were taken every 0.1 mm. Therefore the sharppeaks of the scallops may be missed. In addition, the coordinate systems used for the measure-ments, machining and the surface definition all vary slightly. The resulting measurements maycontain some offset. Finally, the raw CMM data must be processed to extract surface deviationinformation. The resulting processing may distort the shape of the actual scallops. Tool deflectionsmay be responsible for some of the differences between the experimental and simulated results.Tool deflections during machining were minimized by using a small depth of cut (0.5 mm) anda small feed/tooth (0.03 mm/tooth). The resulting cutting forces were on the order of 10-20 Nwhich produced tool deflections of about 2–4µm.

If the form error is ignored, the shape and size of the simulated surface deviations are closeto the measured surface deviations of each machining method. For example, consider the inclinedtool workpiece shown in Fig. 20. The maximum scallop from the simulation was 133µm; within12% of the measured maximum of approximately 150µm. The simulated and measured scallopsboth tended to have a parabolic shape, which varied in the same proportions across the entiresurface. This variation was due to the changing curvature of the surface. Regions of the surfacewith high curvature had larger scallops than regions with low curvature.

Fig. 21 shows that simulated and measured results for the principal axis method are in goodagreement. Surface deviations appear to be “U” shaped. The scallops are more evenly distributedacross the surface because the principal axis method accounts for variations in curvature. Ignoringthe form error, the maximum measured scallop was approximately 25µm. The maximum simu-lated scallop was 22 mm, which is within 12% of the measured result.

The measured and simulated results for the multi-point workpiece also showed reasonable corre-lation. Both sets of surface deviations show the same distinctive shape of multi-point scallop; alarge rounded scallop followed by a small sharp scallop. The distribution of the scallop size isfairly even because multi-point machining accounts implicitly for curvature when the cutter con-tact points are selected. Information about the principal directions is incorporated into the toolposition because the cutter contact points lie in the direction of surface maximum curvature.Therefore, when the tool is placed on these points the tool axis is approximately lined up withthe direction of minimum curvature. The magnitude of the curvature effects the tool positionbecause the angle between the surface normal vectors at cutter contact points influences the incli-nation angle of the tool.

If the form error is disregarded, the maximum measured scallop height for the multi-pointworkpiece was approximately 10µm. The simulated maximum of 9.5µm was within 5% of themeasured result. Most importantly, both the measured and simulated traces demonstrate the superi-ority of the multi-point method. The multi-point scallop heights were approximately 1.5 and 2.5times smaller than those produced by the inclined tool and principal axis methods.

6. Conclusion

For the tested surface both simulations and experiments showed that multi-point machiningproduces smaller scallops than other 5-axis techniques. It also produces far smaller scallops thanthose produced by ball nose end mills in 3-axis configuration. On the other hand the algorithm


employed in multi-point machining to generate the tool path is the most complicated among thestudied techniques.

It should be realized that the results presented here were for one surface only. Although thatsurface is typical of those found in large dies, investigation of applying the different tool pos-itioning strategies to other surfaces, e.g. those found in turbine blades, is needed. In the presentstudy, detection and elimination of gouging was not included in generating the tool paths. Thisinclusion was not pursued because the tested surface had fairly steady curvatures whose directionsdid not vary significantly from those of thex andy-axes. Such conditions might not exist in othersurfaces and accordingly gouging should be included in further investigations. Also needed areinvestigations into effects of scallop profiles on subsequent finishing operations.

Needless to say that a significant amount of research is still needed to draw general conclusionsregarding the performance and the best range of application of each of the available tool pos-itioning strategies. Yet the results presented here for the novel approach of multi-point machiningare very encouraging; It has a great potential in reducing finish-machining time of sculptured sur-faces.

Acknowledgements

The Natural Sciences and Engineering Research Council of Canada supported this research.

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