Basic multi-point tool positioning algorithm: Selecting cutter contact points. The first cutter contact ... centers at c1 and c2, and intersect the tool axis at p1 and p2.
Computer Aided Geometric Design 17 (2000) 83–100 www.elsevier.com/locate/comaid
Multi-point tool positioning strategy for 5-axis mashining of sculptured surfaces Andrew Warkentin a,∗ , Fathy Ismail b , Sanjeev Bedi b a Department of Mechanical Engineering, Dalhousie University (DalTech) Halifax,
Nova Scotia, Canada B3J 2X4 b 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 of sculptured surfaces. In this technique the desired surface is generated at more than one point on the tool. The concept and viability of MPM was developed by the current authors in previous works. However, the method used to generate the multi-point tool positions was slow and labor intensive. The objective of this paper is to present efficient algorithms to generate multi-point tool positions. A basic multi-point algorithm is presented based on some assumptions about the curvature characteristics of the surface underneath the tool. This basic algorithm is adequate for simple surfaces but will fail for more complex surfaces typical of industrial applications. Accordingly, tool position adjustment algorithms are developed that combine the basic algorithm with non-linear optimization to 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 in the 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 hundreds of 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: S 0 1 6 7 - 8 3 9 6 ( 9 9 ) 0 0 0 4 0 - 0
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Fig. 1.
aimed at reducing this time by resorting to high speed machining and/or switching from 3-axis to 5-axis machining. The object of this paper is to further develop a new 5-axis tool positioning strategy that reduces the time required for finish machining of open concave surfaces. 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 contact between the tool and the workpiece and is called the multi-point machining, or simply MPM. Fig. 1 shows the elements of a conventional tool path used to produce a surface. The ball end mill generates the surface at a single point as it moves along the tool path. The tool path consists of a number of parallel tool passes; the distance between them is referred to as the tool pass interval or cross-feed. Material is left between the tool passes in the form of scallops because the tool geometry is poorly matched to the surface geometry. These scallops must be removed in subsequent grinding and polishing operations. Tool positioning strategies are used to determine how a tool is placed relative to the design surface. The main objective of these strategies is to remove as much material from the workpiece as possible without cutting into the desired surface (gouging) by fitting the tool of the design surface as closely as possible. Improving the geometric match between the tool and the surface results in a smaller scallop height for a given tool pass interval, or the same scallop for a much larger cross-feed; in both cases a large reduction in machining time can be achieved. Most research on 5-axis tool positioning strategies has focused on machining a surface with a flat or toroidal end mill inclined relative to the surface normal as shown in Fig. 2. The angle of inclination is often referred to as the Sturz angle (φ). The approach was followed by several authors and made popular by Vickers et al. (1989). The authors pointed out that the effective cutting shape of a flat end mill is an ellipse when projected onto a plane perpendicular to the feed direction. At the cutter contact point, a circle with an effective
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Fig. 2.
radius of reff is used to approximate this ellipse. By varying the inclination angle, φ, the effective radius of the tool could be varied in the following manner: reff =
R . sin φ
(1)
This means that an inclined flat end mill can be used to machine the surface as effectively as a much larger ball nosed tool. In the above works, the inclination angle. φ, was selected arbitrarily and the results were sometimes inconsistent. Jensen and Anderson (1993) proposed a method for calculating an optimal tool angle based on local surface curvature. Instead of inclining the tool in the feed direction, the tool is inclined in the direction of minimum curvature on the surface. An inclination angle is calculated such that the effective radius of the tool at the cutter location equals 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 the effect of tool path direction on the technique. Rao et al. (1996) also compared the technique to 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 positioning strategies. Surface properties at a single point on a design surface are used to calculate a tool position. In a sense, the surface underneath the tool is represented entirely by this single point. The effectiveness of the tool positioning strategy depends on the accuracy of this assumption. For instance, tool positioning strategies that consider only the position and surface normal are essentially assuming that the surface is a plane in the vicinity of the tool. These tool positioning strategies will require small tool pass intervals and could be 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 information basically assume that the surface beneath the tool is quadratic. The result is a better geometric match between the tool and the surface, which produces tool paths with larger tool pass intervals and less gouging. The question we are attempting to answer is; can further improvements be realized by considering more than one point on the surface? Warkentin et al. (1995) proposed a tool positioning strategy called multi-point machining (MPM) which matches the geometry of the tool to the surface by positioning the tool in 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 match between the tool and the surface could be realized. This idea was explored by Warkentin et al. (1998) using intersection theory to examine the nature of multi-point contact between a tool 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 tool positions can be found and that the cutter contact points are arranged symmetrically around the surface’s direction of minimum curvature, λmin . From a tool positioning perspective, however, we are primarily concerned with how closely the tool geometry is matched to the surface geometry. In other words, how much of the tool is within a specified tolerance of the surface. This concern is illustrated in Fig. 3, which shows the effect of the separation distance, w, between the contact points, cc1 and cc2 , on the deviation between the tool and a surface. These surface deviations are represented by sections of the machined surface normal to the feed direction. The surface deviations vary significantly from those produced by single point machining techniques. Typically single point machining techniques produce deviation profiles which are “U” or “V” shaped. Multi-point tool positions produce “W” shaped deviation profiles because the tool touches the surface at two points. When the separation distance w is small the cutter contact points are close together and the resulting deviation profile is similar to the single point case. As the cutter contact points separation, w, increases the maximum 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 the surface. The methodology used by Warkentin et al. (1998) to generate multi-point tool positions was 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 calculating multi-point tool positions will be presented.
2. Multi-point tool positioning The implementation of the multi-point tool positioning strategy for a toroidal cutter can be divided into two parts. First, a geometric algorithm is used to place the tool on the two points of contact. This basic multi-point algorithm is based on the findings of Warkentin et al. (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 surfaces whose 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 is open. Accessibility of the tool of the surface will not be considered in this paper. To satisfy this criterion, the maximum curvature of all points under the tool must be less than the curvature of a sphere, κ, that just bounds the tool. κ6
1 . R+r
(3)
The parameters R and r define the torus and corner (insert) radius of the cutter, respectively. The second assumption is that the minimum and maximum curvatures, κmin and κmax , and the associated minimum and maximum directions of curvature λmin and λmax , of the region of the surface underneath the tool are constant. Given these assumptions a multi-point tool position can be found using analytic geometry as explained below. The initial tool position is found in three stages. In the first stage, two potential cutter contact points are located on the surface; the first contact point is specified during tool path planning, and the second cutter contact point is located using a curvature approximation of the surface. In the second stage, a tool position is calculated based on the location of the cutter contact points and their normal vectors. The resulting tool position should produce tangential 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 the method the tool is shifted in order to ensure that there is tangential contact at the first cutter contact point. This adjustment will only guarantee that there is tangential contact at a single contact point. The formulations of the three stages of the basic MPM tool positioning algorithm 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 stage as shown in Fig. 4. At this stage, a set of cc1 points on the surface, called the cutter contact path, is specified. The second contact point, cc2 , is located a distance equal to the prescribed separation distance w away from cc1 in the direction of maximum curvature λmax . 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. 5. In this figure a plane containing the normal vector n1 and the direction of maximum curvature λmax at the first cutter contact point cc1 has been intersected with the surface. The resulting intersection curve can be approximated by a circle with a radius equal to 1/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 two cutter contact points is (cc1 − cc2 ). The magnitude of this vector is equal to the separation distance w. This vector can be expressed in terms of n1 and λmax . (4) cc2 − cc1 = (cc2 − cc1 ) · n1 n1 + (cc2 − cc1 ) · λmax λmax , where (cc2 − cc1 ) · n1 and (cc2 − cc1 ) · λmax are the components of (cc2 − cc1 ) on n1 and λmax , respectively. These components may be expressed in terms of w and the angle α. (5) cc2 − cc1 = w sin(α)n1 + cos(α)λmax . The position of cc2 can then be found by rearranging expression (5). cc2 = w sin(α)n1 + cos(α)λmax + cc1 .
(6)
The angle α depends on the maximum radius of curvature and the separation between cutter contact points, according to −1 κmax w . (7) α = sin 2 Note that there will be an error in the location of cc2 if the curvature of the surface changes. In most instances the calculated cc2 will not lie on the surface, as shown in Fig. 5. In this case cc2 is projected onto the surface. 5. Basic multi-point tool positioning algorithm: Calculating the tool position and tool axis 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 positioned such that tangential contact exists between the tool and at least one cutter contact point. Fig. 6(a) shows the tool in tangential contact with cc1 and cc2 . The lines formed by the normal vectors, n1 and n2 , at the cutter contact points, cc1 and cc2 , pass through the insert centers at c1 and c2 , and intersect the tool axis at p1 and p2 . The position and orientation of the tool can be specified by determining the location of two points on the tool axis. Thus, the points p1 and tpos shown in the figure will be found in order to calculate the tool position. The point tpos will specify the location of the tool and the vector taxis = p1 − tpos will specify the orientation of the tool. The point p1 can be found by intersecting the line defined by the points cc1 and c1 with a plane containing the tool axis. One such plane is the plane perpendicular to the line joining c1 and c2 that passes through the midpoint between c1 and c2 . This plane will be referred to as the tool axis plane and is shown in Fig. 6(b). The points c1 and c2 are located a distance r along the normal vectors n1 and n2 from the cutter contact points cc1 and cc2 : c1 = cc1 + rn1 , c2 = cc2 + rn2 .
(8)
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(a)
(b)
Fig. 6. (a) Geometry of tool and contact points. (b) Tool axis plane.
Point a is the midpoint between c1 and c2 a=
c1 + c2 . 2
(9)
A vector normal to the tool axis plane, e3 , can be found by noting that the tool axis plane is normal to the vector joining c1 and c2 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 and p lie in the plane. The line joining cc1 and c1 is now defined. A point p on this line can be found using cc1 and n1 from p = cc1 + ηn1 ,
(12)
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where η is the distance along the line from cc1 . The point p1 can now be obtained by intersecting the tool axis plane with this line by substituting Eq. (12) into (11). The resulting value of η gives the distance between cc1 and p1 : η=
e3 · a − e3 · cc1 . e3 · n1
(13)
Substituting η into Eq. (12) 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 points tpos , p1 and a in the tool axis plane as shown in Fig. 6(b). Note that these three points form a right angle triangle because the plane containing tpos , c1 and c2 is always perpendicular to the tool axis. Since this plane is in an arbitrary orientation, bases vectors at point a must be constructed in order to use planar geometry to locate tpos . A unit vector, e1 , in the direction from a to p1 is given by e1 =
(p1 − a) . |p1 − a|
(14)
A second unit vector, e2 , perpendicular to e1 and e3 may be expressed as e 2 = e1 × e3 . The distance, d, between the center of the tool, tpos , and point a is given by s |c2 − c1 |2 . d = |a − tpos | = R 2 − 2
(15)
(16)
The tool position can now be calculated from tpos = a + d sin(β) · e1 + d cos(β) · e2 , where β = cos−1
d . |p1 − a|
(17)
(18)
Given two points on the tool axis, the tool axis vector, taxis , is calculated by normalizing the vector from tpos to p1 according to taxis =
(p1 − tpos ) . |p1 − tpos |
(19)
Together, the tool axis vector, taxis , and the tool position vector, tpos , define the orientation 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) using cc1 and cc2 will not result in tangential contact between the tool and the workpiece at 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 final stage of the method the tool will be shifted in order to guarantee tangential contact at cc1 . In this process the tool position will be altered but the tool axis direction will remain the same. Basically a point on the torus, pt , that could produce tangential contact at cc1 is located. Then the tool is moved so that the point pt is brought in tangential contact with cc1 . In order for cc1 and pt to be tangential, their normal vectors must be collinear. In Fig. 7, a point on the torus, pt , with a normal vector nt , collinear with the surface normal n1 , 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 point pt in the tool coordinate system is pt = R(taxis × n) − rn1 .
(21)
In order to achieve tangential contact at cc1 , the tool must be translated by the distance between cc1 and pt , which is cc1 − (tpos + pt ).
(22)
Note that tpos was added to pt to convert from the tool coordinate system to the workpiece coordinate system. The tool is now translated by the distance between cc1 and pt t∗pos = tpos + cc1 − (tpos + pt ),
(23)
which reduces to t∗pos = cc1 − pt , where t∗pos is the adjusted tool position.
(24)
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7. Tool position correction algorithm The method of calculating multi-point tool positions discussed above is based on the assumptions that the principle directions of curvature are constant underneath the tool. However, these assumptions cannot always be guaranteed for most of surfaces used in practice. For example, a surface may be defined by a set of points generated by measuring a prototype object with a coordinate measurement machine (CMM). In such a case, normal vectors and curvature information must be approximated numerically. Errors in these approximations will result in errors in the tool positions. Even when curvature information is available, the principle directions of curvature may change in the region underneath the tool. 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 significantly underneath the tool, an algorithm has been developed to adjust a tool position such that multi-point contact is achieved. However, before presenting the algorithm it is important to first gain some insight into the positioning errors generated by the basic multi-point positioning algorithm.
8. Tool positioning error To recapitulate, the basic multi-point positioning algorithm is a three step process: selecting potential cutter contact points cc1 and cc2 , calculating the tool position (tpos , taxis) and shifting tpos such that the tool is tangent to the first cutter contact point cc1 . Ideally, the resulting tool position would be tangent to both cutter contact points. If the selection of potential cutter contact points is incorrect the tool will still be tangent to the surface at cc1 but not at cc2 . In order words the error in the tool position is reflected in the error in the location of cc2 . This error may result in gouging or sub-optimal tool positions. Thus the approach to tool position correction will be to select a potential cc2 point that results in tangential contact at cc1 and cc2 . The error function developed here will be used as a measure of how far a point on the surface is from tangency with the tool and this function is based on the intersection theory 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 space and have collinear normal vectors. Therefore if we calculate the perpendicular distance between points on the surface and the tool that have parallel normal vectors we can determine if the points are tangent or not. Furthermore, the perpendicular distance between these two points is a measure of how far these points are from tangency. The required calculations for this process are illustrated by Fig. 8. First, a point on the tool t2 is identified as a potential tangent point. This point must have the same normal vector as cc2 . Therefore, t2 must lie in a plane containing both the tool axis taxis and the normal vector n2 and cc2 . The normal to this plane is n = n2 × taxis.
(25)
Planar geometry can then be used to locate the position of t2 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 at cc2 is then the distance between cc2 and t2 along the normal vector n2 error = |(cc2 − t2 ) · n2 |.
(27)
In Fig. 9 the contours of the error function have been plotted for a cubic Bézier surface with the control points listed in Table 1. In this figure a local coordinate system consisting of the surface normal n1 , minimum direction of curvature λmin and the maximum direction of curvature λmax was constructed at cc1 . The error was then calculated for a grid of potential cc2 points under the tool. Each error value was subsequently projected onto the tangent plane at cc1 . The error contour lines are plotted every 1 µm in increasing order away from cc1 . For example, the error at point P indicated on the figure is 2 µm. In Fig. 9 the error function has two distinct branches; one branch lies in the direction of maximum curvature and one branch lies in the direction of minimum curvature. A tool position for each branch is illustrated in Fig. 10. In Fig. 10(a), a torus has been placed in tangential contact with two points on the surface such that both points lie in the direction of maximum curvature. In Fig. 10(b) the torus is in tangential contact with two points that lie in the direction of minimum curvature. In both cases tangential contact has been achieved at two points on the surface. However, in Fig. 10(b) the torus is gouging the surface in an unacceptable manner. For tool positioning, we are only interested in non-gouging tool positions, therefore points of contact that lie in the direction of minimum curvature should be avoided.
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Fig. 9.
(a)
(b) Fig. 10.
It is interesting to notice in Fig. 9 that the errors around the direction of minimum curvature λmin are not symmetrical; they are very large on the upper right side compared to the upper left side. The reason for this lack of symmetry is that there are actually two possible solutions for every pair of cutter contact points (cc1 , cc2 ) depending on the tool tilt. If the tool is tilted forward there will be two points of tangential contact on the front of the tool, and if the tool is tilted backward there will be two tangential contact points at the back of the tool. We can control the direction of tilt by switching cc1 and cc2 in the basic
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multi-point algorithm. In the case of Fig. 9, the tool is tilted forward on the right hand side of the figure and backward on the left hand side of the figure. Fortunately, the increased sensitivity to the tilt direction around λmin was not a major concern in the present work since we are only interested in solutions in the vicinity of λmax in order to avoid gouging. In Fig. 9 we notice that the solution branches are not straight lines. They deviate significantly from the directions of curvatures. For this reason, the use of a circle, according to Fig. 5, to approximate the location of cc2 would not produce good tool positions for this surface. For this reason a non-linear search method will be utilized here to locate the second contact 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 a cc2 point that produces zero error. This point could be located by generating a graph similar to Fig. 9 and selecting a point with zero error a distance w from cc1 . This exhaustive search methodology is computationally expensive and might only be used as a last resort. The method presented here will quickly converge to a multi-point tool position provided the solution branches illustrated in Fig. 9 are least piecewise continuous. The proposed method is illustrated in Fig. 11. A coordinate system consisting of the maximum and minimum directions of curvature is constructed at cc1 . Note that the superscript “T” on ccT 2 indicates that these points are located on the tangent plane at cc1 . The approximate location of ccT 2 is located a distance w along the direction of maximum curvature. Since the error function is not zero at this point, the resulting tool position will be erroneous. Instead, the correct location of ccT 2 lies on the λmax branch of the error function at a distance of w from cc1 . This point can be expressed in terms of the curvature directions, λmin and λmax , the separation distance, w and an unknown angle θ in the following manner: (28) ccT 2 = cc1 + w cos(θ )λmax + sin(θ )λmin .
Fig. 11.
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Thus the location of cc2 can be found by searching along the arc defined by Eq. (28), by changing the angle θ until a location that minimizes the error function defined by Eq. (27) is found. For every value of θ , ccT 2 must be projected onto the surface in order to calculate the error function. In this paper, Brent’s method described by Press et al. (1992) is used for the 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 Bézier surface described by the control points listed in Table 1 and shown in Fig. 12. This path was simulated using the “mow the grass” technique described by Jerard et al. (1989). It was also used to machine the actual surface. A tool with r = 3 mm and R = 5 mm was used for the simulations and the cutting test. The tool pass interval was 10.0 mm and the separation between cutter contact points, w, was 8.0 mm. Measured and simulated results are shown in Fig. 13. The thick trace is the measured result and the thin trace is the simulated result. Each scan is parallel to the x-axis. The upper, middle and lower scans are located approximately at y = −40.0 mm, y = 0.0 mm and y = 40.0 mm, respectively. Ideally, the simulated and experimental results would match exactly. However, there are differences that arise mainly from setup error. The setup error occurs because it is difficult to match the mathematical coordinate system used to generate the tool path with the physical 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 the x, y and z components of the coordinate systems on our 5-axis machine. The propagation of this error onto the surface depends on the rotations used to orient the part. The result is a small distortion of the traces. If the experimental and simulated scallops are compared scallop-by-scallop, the agreement between the experimental and simulated results is indeed excellent. On a scallop-by-scallop basis the maximum scallop from the simulation was 76 µm; within 3% of the measured maximum of approximately 78 µm. The simulated and measured scallops both tended to have a parabolic shape, which varied in the same proportions across the entire surface. This variation was due to the changing curvature of the surface. Regions of the 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 5axis 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 smaller scallops than the other techniques. For example, the maximum scallop heights for MPM, PAM and inclined tool were 78, 94 and 177 µm, respectively. Furthermore, if one had machined this surface with a ball nosed end mill with the same diameter (16 mm) and tool pass 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 trace deviates 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 reducing the inclination angle φ. However, reducing the inclination angle would have resulted in gouging of the surface. In fact, the 5◦ inclination angle was optimal for this surface but still produced relatively large scallops. The PAM trace is nearly flat in the region near the cutter contact point because the curvature of the tool is matched to the local curvature of the surface at the cutter contact points. However, away from the cutter contact points the trace veers sharply away from the surface. Furthermore gouging occurs in some locations because the curvature under the tool changes as we move away from the cutter contact point. An anti-gouging algorithm would have to be used in conjunction with this technique for this surface and many others in order to achieve acceptable results. In the case of MPM the curvature of the tool does not match the curvature of the surface optimally. The resulting traces diverge from the surface quickly near the cutter contact points. However, in the case of MPM there is twice as many cutter contact points per tool position. The result is a better match 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 at two points of contact have been developed in this paper. The algorithms were demonstrated numerically for a cubic Bézier, and were verified experimentally using cutting tests on a 5-axis milling machine. The non-linear optimization algorithm helped avoid gouging for the tested surface. For general applications, however, further work will be needed to include dedicated gouge detection and avoidance modules. Also, further testing on multi-patch surfaces, including convex patches, needs to be conducted. Nevertheless, the developed algorithms were shown to produce a significant reduction in scallop heights compared to other 5-axis techniques using the same tool and cross-feed.
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