Computer Aided Geometric Design 17 (2000) 83–100www.elsevier.com/locate/comaid

Multi-point tool positioning strategy for 5-axis mashiningof sculptured surfaces

Andrew Warkentina,∗, Fathy Ismailb, Sanjeev Bediba 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 3GI

Received August 1998; revised April 1999

Abstract

Multi-point machining (MPM) is a tool positioning technique used for finish machining ofsculptured surfaces. In this technique the desired surface is generated at more than one point onthe tool. The concept and viability of MPM was developed by the current authors in previousworks. However, the method used to generate the multi-point tool positions was slow and laborintensive. The objective of this paper is to present efficient algorithms to generate multi-point toolpositions. A basic multi-point algorithm is presented based on some assumptions about the curvaturecharacteristics of the surface underneath the tool. This basic algorithm is adequate for simple surfacesbut will fail for more complex surfaces typical of industrial applications. Accordingly, tool positionadjustment algorithms are developed that combine the basic algorithm with non-linear optimizationto achieve multi-point tool positions on these more complex surfaces. 2000 Elsevier Science B.V.All rights reserved.

Keywords:Five-axis; 5-axis; Machining; Surface; Multi-point; Tool positioning

1. Introduction

There is an increasing demand for products featuring sculptured surfaces particularly inthe mold and die industry. These surfaces are typically produced on numerically controlled(NC) milling machines. The tool path used to produce the surface may require hundredsof hours of machining time to run on expensive equipment. Current research efforts are

∗ Corresponding author.

0167-8396/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0167-8396(99)00040-0

84 A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100

Fig. 1.

aimed at reducing this time by resorting to high speed machining and/or switching from3-axis to 5-axis machining. The object of this paper is to further develop a new 5-axis toolpositioning strategy that reduces the time required for finish machining of open concavesurfaces. This strategy, developed by the current authors, Warkentin et al. (1995, 1998),reduces machining time by generating the surface at more than one point of contactbetween the tool and the workpiece and is called the multi-point machining, or simplyMPM.

Fig. 1 shows the elements of a conventional tool path used to produce a surface. The ballend mill generates the surface at a single point as it moves along the tool path. The toolpath consists of a number of parallel tool passes; the distance between them is referred toas the tool pass interval or cross-feed. Material is left between the tool passes in the formof scallops because the tool geometry is poorly matched to the surface geometry. Thesescallops must be removed in subsequent grinding and polishing operations.

Tool positioning strategies are used to determine how a tool is placed relative to thedesign surface. The main objective of these strategies is to remove as much material fromthe workpiece as possible without cutting into the desired surface (gouging) by fitting thetool of the design surface as closely as possible. Improving the geometric match betweenthe tool and the surface results in a smaller scallop height for a given tool pass interval, orthe same scallop for a much larger cross-feed; in both cases a large reduction in machiningtime can be achieved.

Most research on 5-axis tool positioning strategies has focused on machining a surfacewith a flat or toroidal end mill inclined relative to the surface normal as shown in Fig. 2. Theangle of inclination is often referred to as the Sturz angle (φ). The approach was followedby several authors and made popular by Vickers et al. (1989). The authors pointed outthat the effective cutting shape of a flat end mill is an ellipse when projected onto a planeperpendicular to the feed direction. At the cutter contact point, a circle with an effective

A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100 85

Fig. 2.

radius ofreff is used to approximate this ellipse. By varying the inclination angle,φ, theeffective radius of the tool could be varied in the following manner:

reff = R

sinφ. (1)

This means that an inclined flat end mill can be used to machine the surface as effectivelyas a much larger ball nosed tool.

In the above works, the inclination angle.φ, was selected arbitrarily and the results weresometimes inconsistent. Jensen and Anderson (1993) proposed a method for calculatingan optimal tool angle based on local surface curvature. Instead of inclining the tool in thefeed direction, the tool is inclined in the direction of minimum curvature on the surface. Aninclination angle is calculated such that the effective radius of the tool at the cutter locationequals the minimum radius of curvature, 1/κmax of the surface

reff = 1

κmax= R

sin(φ). (2)

Rao et al. (1996) developed a similar technique that they called the principal axis method(PAM). They used their technique to machine various surface patches and investigated theeffect of tool path direction on the technique. Rao et al. (1996) also compared the techniqueto 3-axis machining with a ball nosed tool of the same dimensions.

All the above tool positioning strategies can be classified as single point tool positioningstrategies. Surface properties at a single point on a design surface are used to calculatea tool position. In a sense, the surface underneath the tool is represented entirely by thissingle point. The effectiveness of the tool positioning strategy depends on the accuracy ofthis assumption. For instance, tool positioning strategies that consider only the positionand surface normal are essentially assuming that the surface is a plane in the vicinity ofthe tool. These tool positioning strategies will require small tool pass intervals and couldbe subject to large amounts of gouging for highly curved surfaces. The tool positioning

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Fig. 3.

strategies that use curvature information in addition to position and normal informationbasically assume that the surface beneath the tool is quadratic. The result is a bettergeometric match between the tool and the surface, which produces tool paths withlarger tool pass intervals and less gouging. The question we are attempting to answeris; can further improvements be realized by considering more than one point on thesurface?

Warkentin et al. (1995) proposed a tool positioning strategy called multi-point machin-ing (MPM) which matches the geometry of the tool to the surface by positioning the toolin a manner that maximizes the number of contact points between the surface and the tool.By using several points on the surface to calculate a tool position, a better geometric matchbetween the tool and the surface could be realized. This idea was explored by Warkentin etal. (1998) using intersection theory to examine the nature of multi-point contact between atool represented by a torus and a 2nd order and a 3rd order Bézier surfaces.

The investigation in (Warkentin et al., 1998) demonstrated that multi-point toolpositions can be found and that the cutter contact points are arranged symmetricallyaround the surface’s direction of minimum curvature,λmin. From a tool positioningperspective, however, we are primarily concerned with how closely the tool geometryis matched to the surface geometry. In other words, how much of the tool is within aspecified tolerance of the surface. This concern is illustrated in Fig. 3, which shows theeffect of the separation distance,w, between the contact points,cc1 and cc2, on thedeviation between the tool and a surface. These surface deviations are represented bysections of the machined surface normal to the feed direction. The surface deviationsvary significantly from those produced by single point machining techniques. Typicallysingle point machining techniques produce deviation profiles which are “U” or “ V”shaped. Multi-point tool positions produce “W” shaped deviation profiles because thetool touches the surface at two points. When the separation distancew is small thecutter contact points are close together and the resulting deviation profile is similarto the single point case. As the cutter contact points separation,w, increases themaximum deviation from the design surface underneath the tool increases and the “W”shape becomes more pronounced. Thus selecting an appropriate separation between

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cutter contact points can increase the proportion of the tool within tolerance of thesurface.

The methodology used by Warkentin et al. (1998) to generate multi-point tool positionswas ideal for examining the nature of contact between the tool and the design surface.However, it was slow and labor intensive. In this paper an efficient method of calculatingmulti-point tool positions will be presented.

2. Multi-point tool positioning

The implementation of the multi-point tool positioning strategy for a toroidal cutter canbe divided into two parts. First, a geometric algorithm is used to place the tool on the twopoints of contact. This basic multi-point algorithm is based on the findings of Warkentin etal. (1998) and is suitable for surfaces whose directions of curvature are relatively constant.An optimization algorithm is used in conjunction with the basic algorithm for surfaceswhose directions of curvature are not constant.

3. Basic multi-point tool positioning algorithm

The initial tool positions will be found based on two assumptions. First, the tool should“fit” inside the curvature of the surface. In other words we are assuming that the surface isopen. Accessibility of the tool of the surface will not be considered in this paper. To satisfythis criterion, the maximum curvature of all points under the tool must be less than thecurvature of a sphere,κ , that just bounds the tool.

κ 6 1

R + r . (3)

The parametersR and r define the torus and corner (insert) radius of the cutter,respectively. The second assumption is that the minimum and maximum curvatures,κmin andκmax, and the associated minimum and maximum directions of curvatureλminand λmax, of the region of the surface underneath the tool are constant. Given theseassumptions a multi-point tool position can be found using analytic geometry as explainedbelow.

The initial tool position is found in three stages. In the first stage, two potential cuttercontact points are located on the surface; the first contact point is specified during tool pathplanning, and the second cutter contact point is located using a curvature approximation ofthe surface. In the second stage, a tool position is calculated based on the location of thecutter contact points and their normal vectors. The resulting tool position should producetangential contact at two points on the surface provided the above assumptions are valid.However, this will not be true for most real surfaces. Therefore, in the third stage of themethod the tool is shifted in order to ensure that there is tangential contact at the firstcutter contact point. This adjustment will only guarantee that there is tangential contact at asingle contact point. The formulations of the three stages of the basic MPM tool positioningalgorithm are explained in details next.

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4. Basic multi-point tool positioning algorithm: Selecting cutter contact points

The first cutter contact point,cc1, is specified during the tool path planning stageas shown in Fig. 4. At this stage, a set ofcc1 points on the surface, called the cuttercontact path, is specified. The second contact point,cc2, is located a distance equal tothe prescribed separation distancew away fromcc1 in the direction of maximum curvatureλmax. 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 inFig. 5. In this figure a plane containing the normal vectorn1 and the direction of maximumcurvatureλmax at the first cutter contact pointcc1 has been intersected with the surface.The resulting intersection curve can be approximated by a circle with a radius equal to1/κmax.

Fig. 4.

Fig. 5.

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If we assume that the second cutter contact point lies on the approximating circle,then its location can be calculated in the following manner. The vector between the twocutter contact points is(cc1− cc2). The magnitude of this vector is equal to the separationdistancew. This vector can be expressed in terms ofn1 andλmax.

cc2− cc1=((cc2− cc1) · n1

)n1+

((cc2− cc1) · λmax

)λmax, (4)

where(cc2− cc1) · n1 and(cc2− cc1) · λmax are the components of(cc2− cc1) onn1 andλmax, respectively. These components may be expressed in terms ofw and the angleα.

cc2− cc1=w(sin(α)n1+ cos(α)λmax

). (5)

The position ofcc2 can then be found by rearranging expression (5).

cc2=w(sin(α)n1+ cos(α)λmax

)+ cc1. (6)

The angleα depends on the maximum radius of curvature and the separation betweencutter contact points, according to

α = sin−1(κmaxw

2

). (7)

Note that there will be an error in the location ofcc2 if the curvature of the surface changes.In most instances the calculatedcc2 will not lie on the surface, as shown in Fig. 5. In thiscasecc2 is projected onto the surface.

5. Basic multi-point tool positioning algorithm: Calculating the tool position andtool axis

Once both potential cutter contact points are located, the tool position can be foundbased entirely on the geometry of the tool and these two cutter contact points. The tool willbe positioned such that tangential contact exists between the tool and at least one cuttercontact point.

Fig. 6(a) shows the tool in tangential contact withcc1 andcc2. The lines formed by thenormal vectors,n1 andn2, at the cutter contact points,cc1 andcc2, pass through the insertcenters atc1 andc2, and intersect the tool axis atp1 andp2. The position and orientationof the tool can be specified by determining the location of two points on the tool axis.Thus, the pointsp1 andtpos shown in the figure will be found in order to calculate the toolposition. The pointtpos will specify the location of the tool and the vectortaxis= p1− tposwill specify the orientation of the tool.

The pointp1 can be found by intersecting the line defined by the pointscc1 andc1 with aplane containing the tool axis. One such plane is the plane perpendicular to the line joiningc1 andc2 that passes through the midpoint betweenc1 andc2. This plane will be referred toas the tool axis plane and is shown in Fig. 6(b). The pointsc1 andc2 are located a distancer along the normal vectorsn1 andn2 from the cutter contact pointscc1 andcc2:

c1= cc1+ rn1,

(8)c2= cc2+ rn2.

90 A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100

(a) (b)

Fig. 6. (a) Geometry of tool and contact points. (b) Tool axis plane.

Pointa is the midpoint betweenc1 andc2

a= c1+ c2

2. (9)

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

e3= (c1− c2)

|c1− c2| . (10)

The equation of the tool axis plane is defined by

e3 · p− e3 · a= 0, (11)

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

andn1 from

p= cc1+ ηn1, (12)

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whereη is the distance along the line fromcc1. The pointp1 can now be obtained byintersecting the tool axis plane with this line by substituting Eq. (12) into (11). Theresulting value ofη gives the distance betweencc1 andp1:

η= e3 · a− e3 · cc1

e3 · n1. (13)

Substitutingη into Eq. (12) will determine the Cartesian coordinate of the intersectionpoint,p1.

With p1 now calculated, the second point,tpos, needs to be determined. This point willbe found by considering the geometry of pointstpos, p1 anda in the tool axis plane asshown in Fig. 6(b). Note that these three points form a right angle triangle because theplane containingtpos, c1 andc2 is always perpendicular to the tool axis. Since this planeis in an arbitrary orientation, bases vectors at pointa must be constructed in order to useplanar geometry to locatetpos. A unit vector,e1, in the direction froma to p1 is given by

e1= (p1− a)|p1− a| . (14)

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

e2= e1× e3. (15)

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

d = |a− tpos| =√R2− |c2− c1|2

2. (16)

The tool position can now be calculated from

tpos= a+ d sin(β) · e1+ d cos(β) · e2, (17)

where

β = cos−1(

d

|p1− a|). (18)

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

taxis= (p1− tpos)

|p1− tpos| . (19)

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

6. Basic multi-point tool positioning algorithm: Ensuring tangency at cc1

For most surfaces, the approach described above to calculate the tool position(tpos, taxis)

usingcc1 andcc2 will not result in tangential contact between the tool and the workpieceat one or both of these points. For instant, if the curvature of the surface changes between

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Fig. 7.

the cutter contact points, tangential contact will not be achieved at both points. In this finalstage of the method the tool will be shifted in order to guarantee tangential contact atcc1.In this process the tool position will be altered but the tool axis direction will remain thesame. Basically a point on the torus,pt , that could produce tangential contact atcc1 islocated. Then the tool is moved so that the pointpt is brought in tangential contact withcc1. In order forcc1 and pt to be tangential, their normal vectors must be collinear. InFig. 7, a point on the torus,pt , with a normal vectornt , collinear with the surface normaln1, is located relative to the tool center.

This point must lie on a plane containing the tool axis,taxis, and the surface normal,n1.The normal to this plane is

n= n1× taxis. (20)

The position of pointpt in the tool coordinate system is

pt = R(taxis× n)− rn1. (21)

In order to achieve tangential contact atcc1, the tool must be translated by the distancebetweencc1 andpt , which is

cc1− (tpos+ pt). (22)

Note thattpos was added topt to convert from the tool coordinate system to the workpiececoordinate system. The tool is now translated by the distance betweencc1 andpt

t∗pos= tpos+ cc1− (tpos+ pt), (23)

which reduces to

t∗pos= cc1− pt, (24)

wheret∗pos is the adjusted tool position.

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7. Tool position correction algorithm

The method of calculating multi-point tool positions discussed above is based on theassumptions that the principle directions of curvature are constant underneath the tool.However, these assumptions cannot always be guaranteed for most of surfaces used inpractice. For example, a surface may be defined by a set of points generated by measuringa prototype object with a coordinate measurement machine (CMM). In such a case, normalvectors and curvature information must be approximated numerically. Errors in theseapproximations will result in errors in the tool positions. Even when curvature informationis available, the principle directions of curvature may change in the region underneath thetool. This may occur when using high order surfaces or at a juncture of two surface patches.Given that curvature information may be poorly approximated or change significantlyunderneath the tool, an algorithm has been developed to adjust a tool position such thatmulti-point contact is achieved. However, before presenting the algorithm it is importantto first gain some insight into the positioning errors generated by the basic multi-pointpositioning algorithm.

8. Tool positioning error

To recapitulate, the basic multi-point positioning algorithm is a three step process:selecting potential cutter contact pointscc1 and cc2, calculating the tool position(tpos, taxis) and shiftingtpos such that the tool is tangent to the first cutter contact pointcc1. Ideally, the resulting tool position would be tangent to both cutter contact points. Ifthe selection of potential cutter contact points is incorrect the tool will still be tangent tothe surface atcc1 but not atcc2. In order words the error in the tool position is reflectedin the error in the location ofcc2. This error may result in gouging or sub-optimal toolpositions. Thus the approach to tool position correction will be to select a potentialcc2point that results in tangential contact atcc1 andcc2.

The error function developed here will be used as a measure of how far a point onthe surface is from tangency with the tool and this function is based on the intersectiontheory described by Krieziz (1990), Markot and Magedson (1989). This theory states that,a tangency point can only occur when two points share the same location in spaceandhavecollinear normal vectors. Therefore if we calculate the perpendicular distance betweenpoints on the surface and the tool that have parallel normal vectors we can determine ifthe points are tangent or not. Furthermore, the perpendicular distance between these twopoints is a measure of how far these points are from tangency. The required calculationsfor this process are illustrated by Fig. 8.

First, a point on the toolt2 is identified as a potential tangent point. This point must havethe same normal vector ascc2. Therefore,t2 must lie in a plane containing both the toolaxis taxis and the normal vectorn2 andcc2. The normal to this plane is

n= n2× taxis. (25)

Planar geometry can then be used to locate the position oft2

t2= tpos+R(nt × n)− rn2. (26)

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Fig. 8.

Table 1

y \ x −60 −30 30 60

60 20 0 0 45

30 25 20 10 55

−30 60 5 5 30

−60 45 5 −10 5

The error atcc2 is then the distance betweencc2 andt2 along the normal vectorn2

error = |(cc2− t2) · n2|. (27)

In Fig. 9 the contours of the error function have been plotted for a cubic Bézier surfacewith the control points listed in Table 1. In this figure a local coordinate system consistingof the surface normaln1, minimum direction of curvatureλmin and the maximum directionof curvatureλmax was constructed atcc1. The error was then calculated for a grid ofpotentialcc2 points under the tool. Each error value was subsequently projected onto thetangent plane atcc1. The error contour lines are plotted every 1µm in increasing orderaway fromcc1. For example, the error at pointP indicated on the figure is 2µm.

In Fig. 9 the error function has two distinct branches; one branch lies in the directionof maximum curvature and one branch lies in the direction of minimum curvature. A toolposition for each branch is illustrated in Fig. 10. In Fig. 10(a), a torus has been placed intangential contact with two points on the surface such that both points lie in the direction ofmaximum curvature. In Fig. 10(b) the torus is in tangential contact with two points that liein the direction of minimum curvature. In both cases tangential contact has been achievedat two points on the surface. However, in Fig. 10(b) the torus is gouging the surface inan unacceptable manner. For tool positioning, we are only interested in non-gouging toolpositions, therefore points of contact that lie in the direction of minimum curvature shouldbe avoided.

A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100 95

Fig. 9.

(a) (b)

Fig. 10.

It is interesting to notice in Fig. 9 that the errors around the direction of minimumcurvatureλmin are not symmetrical; they are very large on the upper right side comparedto the upper left side. The reason for this lack of symmetry is that there are actually twopossible solutions for every pair of cutter contact points(cc1,cc2) depending on the tooltilt. If the tool is tilted forward there will be two points of tangential contact on the front ofthe tool, and if the tool is tilted backward there will be two tangential contact points at theback of the tool. We can control the direction of tilt by switchingcc1 andcc2 in the basic

96 A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100

multi-point algorithm. In the case of Fig. 9, the tool is tilted forward on the right hand sideof the figure and backward on the left hand side of the figure. Fortunately, the increasedsensitivity to the tilt direction aroundλmin was not a major concern in the present worksince we are only interested in solutions in the vicinity ofλmax in order to avoid gouging.

In Fig. 9 we notice that the solution branches are not straight lines. They deviatesignificantly from the directions of curvatures. For this reason, the use of a circle, accordingto Fig. 5, to approximate the location ofcc2 would not produce good tool positions for thissurface. For this reason a non-linear search method will be utilized here to locate the secondcontact point as close as possible to the maximum curvature branch.

9. Locating the second cutter contact point

The location of the second cutter contact point is found by searching the surface for acc2point that produces zero error. This point could be located by generating a graph similar toFig. 9 and selecting a point with zero error a distancew from cc1. This exhaustive searchmethodology is computationally expensive and might only be used as a last resort. Themethod presented here will quickly converge to a multi-point tool position provided thesolution branches illustrated in Fig. 9 are least piecewise continuous.

The proposed method is illustrated in Fig. 11. A coordinate system consisting of themaximum and minimum directions of curvature is constructed atcc1. Note that thesuperscript “T” onccT

2 indicates that these points are located on the tangent plane atcc1.The approximate location ofccT

2 is located a distancew along the direction of maximumcurvature. Since the error function is not zero at this point, the resulting tool position will beerroneous. Instead, the correct location ofccT

2 lies on theλmax branch of the error functionat a distance ofw fromcc1. This point can be expressed in terms of the curvature directions,λmin andλmax, the separation distance,w and an unknown angleθ in the following manner:

ccT2 = cc1+w

(cos(θ)λmax+ sin(θ)λmin

). (28)

Fig. 11.

A. Warkentin et al. / Computer Aided Geometric Design 17 (2000) 83–100 97

Thus the location ofcc2 can be found by searching along the arc defined by Eq. (28), bychanging the angleθ until a location that minimizes the error function defined by Eq. (27)is found. For every value ofθ , ccT

2 must be projected onto the surface in order to calculatethe error function. In this paper, Brent’s method described by Press et al. (1992) is used forthe minimization of the error function along the arc.

10. Result

The multi-point algorithms were used to generate a 5-axis tool path for the cubic Béziersurface described by the control points listed in Table 1 and shown in Fig. 12. This pathwas simulated using the “mow the grass” technique described by Jerard et al. (1989). Itwas also used to machine the actual surface. A tool withr = 3 mm andR = 5 mm wasused for the simulations and the cutting test. The tool pass interval was 10.0 mm and theseparation between cutter contact points,w, was 8.0 mm.

Measured and simulated results are shown in Fig. 13. The thick trace is the measuredresult and the thin trace is the simulated result. Each scan is parallel to thex-axis. Theupper, middle and lower scans are located approximately aty =−40.0 mm,y = 0.0 mmandy = 40.0 mm, respectively.

Ideally, the simulated and experimental results would match exactly. However, there aredifferences that arise mainly from setup error. The setup error occurs because it is difficultto match the mathematical coordinate system used to generate the tool path with thephysical coordinate systems of the 5-axis milling machine and with the 3-axis coordinate

Fig. 12.

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Fig. 13.

measurement machine. This error is approximately±12 µm in each of thex, y and zcomponents of the coordinate systems on our 5-axis machine. The propagation of thiserror onto the surface depends on the rotations used to orient the part. The result is a smalldistortion of the traces.

If the experimental and simulated scallops are compared scallop-by-scallop, theagreement between the experimental and simulated results is indeed excellent. On ascallop-by-scallop basis the maximum scallop from the simulation was 76µm; within 3%of the measured maximum of approximately 78µm. The simulated and measured scallopsboth tended to have a parabolic shape, which varied in the same proportions across theentire surface. This variation was due to the changing curvature of the surface. Regions ofthe surface with high curvature had larger scallops than regions with low curvature.

In Fig. 14 simulations are used to compare the proposed technique with the competing 5-axis techniques, the inclined tool and principal axis methods, in machining the test surface.Note that the same tool (r = 3 mm,R = 5 mm) and the same tool pass interval (10 mm)were used for these simulations. The graph clearly shows that MPM produces much smallerscallops than the other techniques. For example, the maximum scallop heights for MPM,PAM and inclined tool were 78, 94 and 177µm, respectively. Furthermore, if one hadmachined this surface with a ball nosed end mill with the same diameter (16 mm) and toolpass interval (10 mm), the maximum scallop would have been 1445 mm.

Note that each of the traces had the same point of zero surface deviation for each scallop.This point was the cutter location used for tool path planning. The inclined tool tracedeviates quickly from the surface because the effective radius of the tool is poorly matched

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Fig. 14.

to the local curvature of the surface. This match could have been improved by reducingthe inclination angleφ. However, reducing the inclination angle would have resulted ingouging of the surface. In fact, the 5◦ inclination angle was optimal for this surface butstill produced relatively large scallops. The PAM trace is nearly flat in the region near thecutter contact point because the curvature of the tool is matched to the local curvature ofthe surface at the cutter contact points. However, away from the cutter contact points thetrace veers sharply away from the surface. Furthermore gouging occurs in some locationsbecause the curvature under the tool changes as we move away from the cutter contactpoint. An anti-gouging algorithm would have to be used in conjunction with this techniquefor this surface and many others in order to achieve acceptable results. In the case of MPMthe curvature of the tool does not match the curvature of the surface optimally. The resultingtraces diverge from the surface quickly near the cutter contact points. However, in the caseof MPM there is twice as many cutter contact points per tool position. The result is a bettermatch of the tool with the entire surface under the tool resulting in smaller scallops.

11. Concluding remarks

Algorithms to efficiently produce tool paths to machine open concave surface patches attwo points of contact have been developed in this paper. The algorithms were demonstratednumerically for a cubic Bézier, and were verified experimentally using cutting tests on a5-axis milling machine. The non-linear optimization algorithm helped avoid gouging forthe tested surface. For general applications, however, further work will be needed to includededicated gouge detection and avoidance modules. Also, further testing on multi-patchsurfaces, including convex patches, needs to be conducted. Nevertheless, the developedalgorithms were shown to produce a significant reduction in scallop heights compared toother 5-axis techniques using the same tool and cross-feed.

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Acknowledgements

The authors would like to thank the Natural Sciences and Engineering Research Councilof Canada for financial support.

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